In this study, we look at some estimation issues for complementary unit Weibull distributions in the context of adaptive progressive type-II hybrid censoring. The point and interval estimations of the model parameters, as well as a number of its reliability indices, are explored. The likelihood frequentist approach is used as a classical strategy to obtain the point and approximate confidence ranges. The median parameter of the distribution is produced in a closed form as a function of the shape parameter, while the shape parameter can be obtained iteratively. The squared error loss function and gamma and beta prior distributions are used for evaluating Bayes estimates. The Markov chain Monte Carlo method is used to solve the difficult posterior distribution expression in order to provide Bayes estimates and the highest posterior density credible ranges. A simulation study is done to evaluate the efficacy of various estimating methodologies making use of different circumstances for sample sizes and progressive censoring strategies. Finally, three real-world datasets from veterinary, industrial, and physical applications are examined to highlight the practical importance of the provided methodologies.

Significant work has been achieved in recent years in enhancing some lifetime distributions and effectively leveraging them to solve some modeling challenges in a variety of sectors, including engineering, medicine, and finance, among many others. When we compare distributions with infinite support to distributions with limited support, we discover significant sparsity, even though there are numerous real-life instances in which observations take values within a constrained range, such as ratios, percentages, and fractions. As a result, Guerra et al.1 suggested a novel two-parameter complementary unit Weibull (CUW) model based on the Weibull lifetime distribution (which is one of the most common models for simulating non-negative random variables). They additionally stated that the CUW distribution provides a good fit when compared to some bounded models in the literature, including the popular beta and Kumaraswamy distributions, using the rate of literacy for Brazilian municipalities in 2010 and Colombian municipalities in 2005.

However, if Y is a random variable with the CUW distribution, indicated by Y ∼CUW(α, γ), where α and γ are the model parameters, then its cumulative distribution function [say F(·)], probability density function [say f(·)], and hazard rate function (HRF) [say h(.)] are all given by
(1)
(2)
and
(3)
where α > 0 is the shape parameter and γ ∈ (0, 1) is the median parameter. From (1), the CUW reliability function (RF), say R(·), is given by R(·) = 1 − F(·).

Taking various options of α and γ, Fig. 1 indicates that the CUW density can be positive (or negative) skewed or approximately symmetric. It also shows that the CUW failure rate allows for monotonically increasing, decreasing, or bathtub shapes, which are important in reliability examinations. Guerra et al.1 used the maximum likelihood method to estimate the distribution parameters using the complete sample scenario.

FIG. 1.

The PDF (left) and HRF (right shapes of the CUW distribution.

FIG. 1.

The PDF (left) and HRF (right shapes of the CUW distribution.

Close modal

The use of a whole sample in reliability and life testing tests is impractical because many modern goods are more reliable and require a long time to fail. As a result, several censoring strategies are presented in the literature in order to complete the experiment before the entire sample fails. Type-I, type-II, and progressive type-II censoring (PT2C) are some of the most often employed techniques. The PT2C enables the researcher to exclude some working units from the study in order to conduct additional studies or conserve testing time. For more detail, see the works of Ng et al.,2 Dey et al.,3 and Balakrishnan and Cramer.4 Kundu and Joarder5 suggested a progressive type-I hybrid censoring plan in which the test ends at random when the minimum of a predetermined time or the time of a certain amount of failures occurs. The primary drawback of this technique is that it may result in a relatively small number of failures, rendering statistical inference conclusions obtained using this scheme unsuitable or inefficient.

To address this limitation, Ng et al.6 offered an adaptive progressive type-II hybrid censoring (APT2-HC) technique to improve statistical analysis accuracy. In this system, the number of failures m is set in before starting the experiment, and the testing period can go beyond a scheduled duration T. Furthermore, we have the progressive censoring plan R1, R2, …, Rm, but the quantities for some Ri could be modified as the outcome of the test. This scheme can be explained as follows: Consider n units are subjected to a life test, with m < n representing the required number of failures. Once the first failure occurs, Y1:m:n units are eliminated from the test at random. Similarly, when the second failure Y2:m:n occurs, R2 units are randomly removed from the test, and so on. If the mth failure happens before time T (i.e., Ym:m:m < T), the test is terminated and the typical PT2C occurs. However, if Yd:m:n < T < Yd+1:m:n, where d + 1 < m and Yd:m:n are the dth failure times that take place prior time T, we will not remove any living element from the test by placing Rd+1, Rd+2, , Rm−1 = 0, as well as R*=nmi=1dRi. This configuration makes sure that the experiment ends when we reach the desired number of failures m and that the overall test duration will not deviate too much from the ideal length of time T. This approach was utilized by several writers to estimate many lifetime models, including Nassar and Abo-Kasem,7 Nassar et al.,8 Sobhi and Soliman,9 Chen and Gui,10 Alotaibi et al.,11 Dutta et al.,12 and Elshahhat et al.13 Let y1:m:n < … < yd:m:n < T < yd+1:m:n < … < ym:m:n be an APT2-HC sample from a continuous population with a probability density function f(y) and a cumulative distribution function F(y) and a progressive censoring design R1, , Rd, 0, …, 0, R*. Therefore, the likelihood function of the data that was observed can be represented as follows:
(4)

We are motivated to investigate the estimation challenges of the CUW distribution when the data are APT2-HC because of its usefulness in modeling real-world data when compared with some classical models. Even though Guerra et al.1 looked at the estimation of the parameters of this distribution, they mentioned nothing regarding the Bayesian estimation for the parameters, the estimation of reliability measures, or the performance of the estimates in the presence of any censoring situation. Starting with these facts, we can list this study’s objectives as follows:

  • Examining the maximum likelihood estimation of the CUW distribution, including deriving the point and approximate confidence intervals (ACIs) for the unknown parameters and any parameter of them.

  • Evaluating the Bayesian estimation of the various parameters utilizing the squared error loss function.

  • Assessing the effectiveness of the two estimation approaches to see which strategy produces more accurate estimations.

  • Deciding on the optimal progressive censoring strategy based on some metrics.

  • Reviewing the CUW distribution’s efficiency in suiting several real-world datasets and comparing it to some common competitor models.

  • Demonstrating the practical implications of the proposed estimation methods.

  • It is to be mentioned here that because of the complicated form of the posterior distribution, the Bayes estimators cannot be computed explicitly. As a result, we recommend employing the Markov Chain Monte Carlo (MCMC) approach to obtain both the required point estimates and the highest posterior density (HPD) credible intervals.

The remaining part of this work is arranged as follows. Section II presents the maximum likelihood estimates (MLEs) and ACIs of the parameters, RF, and HRF. Section III deduces the Bayesian estimates using the MCMC procedure. Section IV provides simulation results to highlight the performance of the various estimates based on the APT2-HC data. Section V then goes over how to determine optimal censoring strategies by applying different optimality criteria. Section VI provides three practical scenarios to demonstrate the utility of the CUW distribution. This paper concludes with Sec. VII.

In this section, we will delve into the intricacies of the maximum likelihood estimation method of the CUW distribution. Besides the point estimates of the unknown parameters and the reliability measures, our discourse will extend to the construction of interval estimates of these parameters. The focus will be under the framework of APT2-HC. Based on an APT2-HC sample taken from the CUW population, one can write the likelihood function using (1), (2), and (4) as
(5)
where zi = −log(1 − yi:m:n), z̲=(z1,,zm), and
The log-likelihood function of (5) can be formulated as
(6)
The maximum likelihood estimates (MLEs) of α and γ are the values that maximize the log-likelihood function. Maximizing (6) with regard to the unknown parameters yields these values. Alternatively, one can solve the following normal equations simultaneously to get the MLEs:
(7)
and
(8)
where v(z̲;α,γ)=ẃ(z̲;α)log[log(1γ)] and
From Eq. (8), one can easily write the MLE of γ as a function of α as
(9)
By substituting γ̂(α) in (6), the profile log-likelihood function of α takes the form
(10)
From (10), we can get the MLE of α as α = p(α), where
(11)
To determine the solution of α = p(α) in (11), we recommend implementing the Newton–Raphson iterative method. Once we have the MLE of α, denoted by α̂, we can estimate the MLE γ̂ of γ as γ̂(α̂) from (9). Using the MLEs’ invariance property, we can estimate any function of the parameters without re-estimating the entire model. Using this feature, we can obtain the MLEs of the RF and HRF at mission time t of the CUW distribution as follows:
and
It is critical to obtain interval estimates of unknown parameters in order to express the uncertainty inherent in point estimates. Unlike the MLE, which provides a single-point estimate for the unknown parameter, interval estimation provides a range within which the true parameter value is expected to fall. Because of this significance, the ACIs of the unknown parameters, RF and HRF of the CUW distribution, are investigated in the remainder of this section. The MLEs’ asymptotic normality is used for creating the ACIs. To acquire such intervals, we first need to estimate the variance-covariance matrix by calculating the inverse of the observed Fisher information matrix as follows:
(12)
where
and
where
Then, the ACIs of α and γ can be obtained, respectively, as
where zτ/2 is the percentile 100(1 − τ/2)% of the standard normal distribution.
To get such intervals for the RF and HRF, we need to estimate the variances of the MLEs R̂(t) and ĥ(t). One common approach is the delta method, which can be used to approximate the required estimated variances. To apply this procedure, we first should differentiate the RF and HRF with respect to the various parameters as follows:
and
Then, the estimated variances of R̂(t) and ĥ(t) can be approximated as follows:
(13)
and
(14)
where all the quantities in (13) and (14) are evaluated at the MLEs α̂ and γ̂. Consequently, the ACIs of R(t) and h(t), are expressed, respectively, as
Small sample sizes have a significant impact on the accuracy of traditional likelihood-based estimates and may result in incorrect inferential inferences. As a result, the Bayesian approach is used as an alternative to classical estimation, which could include additional prior knowledge in this case. In this section, we look at Bayesian inference for the unknown parameters of the CUW distribution using the squared error loss function. The primary feature of the beta distribution is the flexibility of its density shape. Thus, since the median parameter γ takes values between 0 and 1, a natural choice for the prior distribution of γ is the beta distribution. Furthermore, we pick the gamma prior distribution for α since it responds to parameter support and is flexible, which may not result in many complex computational and inferential challenges. Subsequently, we suppose that α and γ are independent random variables with the gamma prior distribution for α and beta prior distribution for γ. Based on these assumptions, we can write the joint prior as follows:
(15)
where aj, bj > 0, j = 1, 2. Without loss of generality, one can easily consider other prior distributions, such as uniform and Hartigan, among others. The posterior distribution of α and γ, denoted by Q(α,γ|z̲), can be acquired via combining (LFG1) and (15),
(16)
where A is defined as
To obtain the Bayes estimator of any unknown parameter, we need a suitable loss function. Here, we consider the squared error loss function in which the Bayes estimator is the posterior mean. Let ξ(α, γ) be an unknown parameter to be estimated. Therefore, we can get its Bayes estimator to obtain the expected value of the posterior distribution in (17) as follows:
(17)
At first glance, it is clear that the Bayes estimator in (17) cannot be stated in a closed form. As a result, some numerical computation approaches are required to provide the needed estimates. One method is to use the MCMC technique, which generates random samples from the posterior distribution and then computes the Bayes estimates and HPD credible ranges. To implement the MCMC method, we must first compute the so-called full conditional distribution of the unknown parameters α and γ. These distributions are simply obtained from (17) for α and γ, respectively, as
(18)
and
(19)

It is evident from Q1(α|γ,z̲) and Q2(γ|α,z̲) given by (18) and (19), respectively, that the full conditional distributions of α and γ are unknown. As a result, no standard form can be used to create random numbers from them. To create the requisite samples, the Metropolis–Hastings (M–H) techniques might be used. To apply this strategy, we require a candidate distribution to produce samples from, which is known as proposal distribution. In this work, we employ the normal distribution as a proposal distribution, with mean and variance equal to those derived via the maximum likelihood method, where the mean is the MLE of the unknown parameter. The steps utilized to generate samples using the M–H process are as follows:

  1. Make ς = 1 and set the initial thoughts to (α(0),γ(0))=(α̂,γ̂).

  2. Following the M–H phases, extract α(ς), γ(ς) from (18) and (19), respectively, with a normal suggesting distribution.

  3. Set ς = ς + 1.

  4. Carry out step 2; M walks until achieving a sequence of α(ς), γ(ς), ς = 1, dots, M.

  5. Compute R(ς) = R(ς) (t) and h(ς) = h(ς) (t).

To ensure the convergence of the generated series, after omitting O walk as a burn-in period, the Bayes estimate for any parameter, say σ, can be determined as follows:
On the other hand, the following steps can be implemented to acquire the HPD credible interval of the unknown parameter σ.
  1. Sort the series of σ(ς), ς = O + 1, , M, to become σ(O+1) < σ(O+2) < … < σ(M).

  2. Calculate the HPD as

where ς* = O + 1, , M is selected to achieve,
where [l] is the largest integer that is either less than or equal to l.

This section demonstrates simulations to validate the accuracy of the obtained point and interval estimations of the CUW parameters, including α, γ, R(t), and h(t) for t > 0 based on various scenarios.

To assess the performance of the proposed estimators for α, γ, R(t), and h(t) acquired in Secs. II and III, extensive Monte Carlo simulation studies are implemented. To establish this goal, based on different choices of T, n, m, and Ri, i = 1, , m, we produce 1000 APT2-HC samples from two different CUW populations, called set-I:CUW(0.8.0.4) and set-II:CUW(1.5.0.7). Taking t = 0.1, the plausible values of (R(t), h(t)) at set-I and set-II are (0.8219, 1.6539) and (0.9822, 0.2838), respectively. All estimates of α, γ, R(t), or h(t) are examined when T(=0.3, 0.7), n(=40, 80), and m is taken as a failure proportion (FP) mn×100%=50 and 75%, and various schemes are as follows:

Clearly, the proposed schemes 1, 2, and 3 behave in a similar way to the left, middle, and right censoring plans, respectively. After assigning the values of T, n, m, and Ri, i = 1, 2, , m, to get an APT2-HC sample, do the following procedure:

  • Step 1: Produce a traditional PT2C sample as follows:

    1. Create Si, i = 1, 2, , m, independent observations.

    2. Set Qi=Sii+j=mi+1mRj1,i=1,2,,m.

    3. Let ui = 1 − QmQm−1…Qmi+1 for i = 1, 2, , m.

    4. Set Yi = F−1(uiα, γ), i = 1, 2, , m, as a PT2C sample from CUW(α, γ).

  • Step 2: Find d at T and discard the staying items Yi for i = d + 2, , Ym.

  • Step 3: Find Yd+2, , Ym with size ndj=1dRj1 from fy;α,γRyd+1;α,γ1.

As soon as the 1000 samples are collected, via the package ‘maxLik’ by Henningsen and Toomet,14 the MLE and 95% ACI estimates of α, γ, R(t), or h(t) are evaluated. To demonstrate the effects of the considered priors, we have used different informative priors of α and γ when those hyperparameters are chosen in such a way that the prior mean became the mean value of each model parameter. Hence, for each set of α and γ, two sets of the hyperparameters (a1, a2, b1, b2) are utilized, namely, prior 1: (4, 4, 5, 6) for set-I and (7.5, 7, 5, 3) for set-II; prior 2: (8, 6, 10, 9) for set-I and (15, 10.5, 10, 4.5) for set-II.

According to the MH technique, via the package ‘coda’ by Plummer et al.,15 for each unknown quantity, 2000 (out of 10 000) MCMC iterations are used to carry out the MCMC and 95% HPD interval estimates. In Bayes’ MCMC calculations, to obtain a representative sample of the posterior distribution, assessment of convergence includes verifying that the sequence or series is the primary issue. Thus, four convergence diagnostics are used for this purpose, which are (i) autocorrelation, (ii) trace, (iii) Brooks–Gelman–Rubin (BGR), and (iv) trace (with fifth thinness).

In Bayes MCMC calculations, to get a representative sample from the objective posterior distribution, the convergence assessment involves checking that the sequence, or chain, is the main issue. Four convergence tools are used for this purpose, namely, (i) auto-correlation, (ii) trace, (iii) Brooks–Gelman–Rubin (BGR), and (iv) trace (with fifth thinning) diagnostics. Without loss of generality, from set-I, these diagnostics are created and shown in Figs. 24 when (T, n, m) = (0.8, 40, 20) and scheme [1] (for instance).

FIG. 2.

Auto-correlation (top) and trace (bottom) diagrams of α, γ, R(t), and h(t).

FIG. 2.

Auto-correlation (top) and trace (bottom) diagrams of α, γ, R(t), and h(t).

Close modal
FIG. 3.

The BGR diagram in Monte Carlo simulation.

FIG. 3.

The BGR diagram in Monte Carlo simulation.

Close modal
FIG. 4.

Trace/density (along its Gaussian kernel) plots of α and γ in Monte Carlo simulation.

FIG. 4.

Trace/density (along its Gaussian kernel) plots of α and γ in Monte Carlo simulation.

Close modal

Figure 2 shows that (i) the similarity between data points in the simulated chain and the convergence of the results show strong mixing and accurate results for the posterior distribution. Figure 3 exhibits that there is no substantial difference between the variance-within and variance-between the generated Markovian chains. Figure 4 shows that the sampled Markovian chains of α or γ are suitably composite and node. Thus, the calculated results of α, γ, R(t), or h(t) are reasonable and efficient.

Specifically, the average estimates (Av.Es) of α (for short) are given by
where α̌(j) is the evaluated estimate of α at the jth simulated sample.

The comparison between the proposed point estimators of α (for short) is made based on the following two accuracy metrics:

  • Root Mean Squared-Error (RMSE):
  • Mean Absolute Bias (MAB):

Additionally, the comparison between the proposed interval estimators of α (for short) is made based on the following two accuracy metrics:

  • Average Confidence Length (ACL):
  • Coverage probability (CP):

where 1(·) is the indicator function and L() and U() denote the lower and upper bounds, respectively, of (1 − ϵ)%.

From Tables IXVI, in terms of the lowest levels of RMSEs, MABs, and ACLs as well as the largest level of CPs, we report the following observations:

  • All offered estimates of α, γ, R(t), or h(t) behave adequately.

  • As n(or mn×100%) increases, all estimates behave well. A similar note is also reached when nm decreases.

  • Comparing the two prior sets 1 and 2 on the Bayes evaluations, it is observed that estimates developed based on the latter are better than those developed based on the former. This is an anticipated result because the variance of prior 2 is less than that of prior 1.

  • Comparing schemes 1–3, we have noted the following:

    • Estimates of α or R(t) become satisfactorily based on scheme [3] (when the staying nm live subjects are removed in the mth stage) than others.

    • Estimates of γ or h(t) become satisfactorily based on scheme [1] (when the staying nm live subjects are removed in the first stage) than others.

  • As T increases, for set-I and set-II, we have noted the following:

    • The estimation results of α, γ, R(t), or h(t) derived from the likelihood or Bayes approach become better.

    • The RMSEs, MABs, and ACLs of α, γ, R(t), or h(t) narrowed down. The opposite comment is observed when the comparison is made in terms of their CPs.

  • As α and γ grow, for set-I and set-II, it is clear that we have the following:

    • The RMSE and MAB results of α and γ derived from the likelihood or Bayes approach increased, while those of the reliability indices R(t) and h(t) decreased.

    • The ACL results of α, γ, R(t), or h(t) derived from the asymptotic or credible approach decreased, whereas their CPs grew.

  • As a recommendation, the M–H algorithm is advised to estimate the unknown parameters and the reliability characteristics of the CUW distribution when the sample is collected from the proposed censored strategy.

TABLE I.

The Av.Es (first column), RMSEs (second column), and MABs (third column) of α from set-I.

n(FP%)SchemeMLEMCMC-P1MCMC-P2
T = 0.3
40 (50%) [1] 0.8368 0.4126 0.4076 0.8931 0.3813 0.3868 0.7874 0.3625 0.3611 
[2] 0.8217 0.3715 0.3625 0.8414 0.3408 0.3530 0.7900 0.3308 0.3266 
[3] 0.7541 0.3295 0.3285 0.8538 0.3094 0.2761 0.6754 0.2912 0.2691 
40 (75%) [1] 0.8348 0.2866 0.2762 0.6920 0.2644 0.2249 0.7574 0.2361 0.1432 
[2] 0.7143 0.2382 0.2492 0.6819 0.2163 0.1652 0.7156 0.1921 0.1304 
[3] 0.7552 0.2176 0.1843 0.7437 0.1897 0.1526 0.7447 0.1520 0.1258 
80 (50%) [1] 0.8179 0.1953 0.1305 0.8463 0.1757 0.1233 0.7493 0.1225 0.1151 
[2] 0.8165 0.1862 0.1250 0.8254 0.1573 0.1133 0.7586 0.1049 0.0942 
[3] 0.7844 0.1763 0.1044 0.3924 0.1364 0.0897 0.7389 0.0943 0.0838 
80 (75%) [1] 0.8169 0.1695 0.0973 0.8609 0.1191 0.0844 0.8304 0.0761 0.0731 
[2] 0.7120 0.1495 0.0822 0.8299 0.1056 0.0736 0.8376 0.0630 0.0645 
[3] 0.7511 0.1284 0.0755 0.7515 0.0937 0.0632 0.8109 0.0582 0.0522 
T = 0.7 
40 (50%) [1] 0.8301 0.1940 0.1753 0.8967 0.1878 0.1544 0.7879 0.1503 0.1384 
[2] 0.8275 0.1779 0.1518 0.8634 0.1676 0.1423 0.7833 0.1368 0.1259 
[3] 0.8316 0.1724 0.1381 0.9173 0.1463 0.1287 0.7753 0.1238 0.1188 
40 (75%) [1] 0.8323 0.1663 0.1274 0.7349 0.1373 0.1124 0.7386 0.1104 0.1063 
[2] 0.8327 0.1660 0.1236 0.7361 0.1293 0.1087 0.7376 0.1025 0.1016 
[3] 0.7537 0.1510 0.1135 0.7143 0.1253 0.1053 0.6741 0.0879 0.0953 
80 (50%) [1] 0.8145 0.1496 0.1095 0.8270 0.1241 0.1019 0.7473 0.0869 0.0896 
[2] 0.8116 0.1219 0.1032 0.7986 0.1221 0.0950 0.7511 0.0823 0.0854 
[3] 0.7972 0.1119 0.0986 0.7376 0.1077 0.0855 0.7147 0.0772 0.0778 
80 (75%) [1] 0.8126 0.1054 0.0894 0.8881 0.0904 0.0800 0.8476 0.0681 0.0641 
[2] 0.8104 0.0995 0.0790 0.8544 0.0854 0.0736 0.8440 0.0500 0.0572 
[3] 0.7041 0.0965 0.0727 0.6261 0.0764 0.0695 0.6617 0.0447 0.0470 
n(FP%)SchemeMLEMCMC-P1MCMC-P2
T = 0.3
40 (50%) [1] 0.8368 0.4126 0.4076 0.8931 0.3813 0.3868 0.7874 0.3625 0.3611 
[2] 0.8217 0.3715 0.3625 0.8414 0.3408 0.3530 0.7900 0.3308 0.3266 
[3] 0.7541 0.3295 0.3285 0.8538 0.3094 0.2761 0.6754 0.2912 0.2691 
40 (75%) [1] 0.8348 0.2866 0.2762 0.6920 0.2644 0.2249 0.7574 0.2361 0.1432 
[2] 0.7143 0.2382 0.2492 0.6819 0.2163 0.1652 0.7156 0.1921 0.1304 
[3] 0.7552 0.2176 0.1843 0.7437 0.1897 0.1526 0.7447 0.1520 0.1258 
80 (50%) [1] 0.8179 0.1953 0.1305 0.8463 0.1757 0.1233 0.7493 0.1225 0.1151 
[2] 0.8165 0.1862 0.1250 0.8254 0.1573 0.1133 0.7586 0.1049 0.0942 
[3] 0.7844 0.1763 0.1044 0.3924 0.1364 0.0897 0.7389 0.0943 0.0838 
80 (75%) [1] 0.8169 0.1695 0.0973 0.8609 0.1191 0.0844 0.8304 0.0761 0.0731 
[2] 0.7120 0.1495 0.0822 0.8299 0.1056 0.0736 0.8376 0.0630 0.0645 
[3] 0.7511 0.1284 0.0755 0.7515 0.0937 0.0632 0.8109 0.0582 0.0522 
T = 0.7 
40 (50%) [1] 0.8301 0.1940 0.1753 0.8967 0.1878 0.1544 0.7879 0.1503 0.1384 
[2] 0.8275 0.1779 0.1518 0.8634 0.1676 0.1423 0.7833 0.1368 0.1259 
[3] 0.8316 0.1724 0.1381 0.9173 0.1463 0.1287 0.7753 0.1238 0.1188 
40 (75%) [1] 0.8323 0.1663 0.1274 0.7349 0.1373 0.1124 0.7386 0.1104 0.1063 
[2] 0.8327 0.1660 0.1236 0.7361 0.1293 0.1087 0.7376 0.1025 0.1016 
[3] 0.7537 0.1510 0.1135 0.7143 0.1253 0.1053 0.6741 0.0879 0.0953 
80 (50%) [1] 0.8145 0.1496 0.1095 0.8270 0.1241 0.1019 0.7473 0.0869 0.0896 
[2] 0.8116 0.1219 0.1032 0.7986 0.1221 0.0950 0.7511 0.0823 0.0854 
[3] 0.7972 0.1119 0.0986 0.7376 0.1077 0.0855 0.7147 0.0772 0.0778 
80 (75%) [1] 0.8126 0.1054 0.0894 0.8881 0.0904 0.0800 0.8476 0.0681 0.0641 
[2] 0.8104 0.0995 0.0790 0.8544 0.0854 0.0736 0.8440 0.0500 0.0572 
[3] 0.7041 0.0965 0.0727 0.6261 0.0764 0.0695 0.6617 0.0447 0.0470 
TABLE II.

The Av.Es (first column), RMSEs (second column), and MABs (third column) of α from set-II.

n (FP%)SchemeMLEMCMC-P1MCMC-P2
T = 0.3
40 (50%) [1] 1.2822 0.5385 0.5925 1.2907 0.3488 0.3308 1.1807 0.3030 0.3085 
[2] 0.9716 0.5189 0.4996 1.1596 0.3077 0.2899 1.1193 0.2609 0.2452 
[3] 0.9701 0.5122 0.4851 1.0349 0.2162 0.1995 1.1768 0.1685 0.1629 
40 (75%) [1] 1.2744 0.5048 0.4729 1.1607 0.2020 0.1791 1.1283 0.1642 0.1348 
[2] 0.9856 0.4754 0.4154 0.9691 0.1919 0.1651 1.0427 0.1199 0.1181 
[3] 1.0852 0.4251 0.3851 0.9691 0.1857 0.1534 1.0427 0.0986 0.0977 
80 (50%) [1] 1.2406 0.3738 0.3504 1.2636 0.1728 0.1342 1.1527 0.0816 0.0787 
[2] 0.9731 0.3692 0.3266 1.0009 0.1418 0.1255 1.0599 0.0788 0.0672 
[3] 0.9608 0.2362 0.1828 1.0089 0.1129 0.1193 1.1239 0.0569 0.0531 
80 (75%) [1] 1.2327 0.2060 0.1581 1.3324 0.0911 0.1081 1.2529 0.0538 0.0473 
[2] 0.9985 0.1560 0.1210 1.0501 0.0747 0.0974 1.1234 0.0390 0.0311 
[3] 1.1785 0.1425 0.1117 1.0501 0.0743 0.0841 1.1234 0.0354 0.0288 
T = 0.7 
40 (50%) [1] 1.2524 0.4449 0.4338 1.3158 0.2605 0.2490 1.1864 0.1711 0.1657 
[2] 1.2570 0.3936 0.3667 1.3268 0.2426 0.2309 1.1886 0.1667 0.1573 
[3] 1.0762 0.3822 0.3098 1.1602 0.1904 0.1962 1.0421 0.1439 0.1401 
40 (75%) [1] 1.2471 0.3748 0.3030 1.1091 0.1826 0.1580 1.1226 0.1101 0.1281 
[2] 1.2164 0.2618 0.1965 1.1094 0.1381 0.1270 1.1194 0.0881 0.1077 
[3] 0.8402 0.2420 0.1862 0.9548 0.1265 0.1159 0.8692 0.0848 0.0977 
80 (50%) [1] 1.2233 0.2141 0.1664 1.2663 0.1121 0.1084 1.1564 0.0716 0.0766 
[2] 1.2242 0.2040 0.1582 1.2486 0.1081 0.0980 1.1536 0.0532 0.0565 
[3] 0.9906 0.1657 0.1327 0.8915 0.1019 0.0908 0.9102 0.0506 0.0437 
80 (75%) [1] 1.2230 0.1592 0.1246 1.3067 0.0890 0.0825 1.2377 0.0423 0.0379 
[2] 1.2147 0.1468 0.1147 1.3621 0.0670 0.0683 1.2677 0.0309 0.0274 
[3] 0.7663 0.1320 0.1036 1.0793 0.0484 0.0501 1.0372 0.0298 0.0227 
n (FP%)SchemeMLEMCMC-P1MCMC-P2
T = 0.3
40 (50%) [1] 1.2822 0.5385 0.5925 1.2907 0.3488 0.3308 1.1807 0.3030 0.3085 
[2] 0.9716 0.5189 0.4996 1.1596 0.3077 0.2899 1.1193 0.2609 0.2452 
[3] 0.9701 0.5122 0.4851 1.0349 0.2162 0.1995 1.1768 0.1685 0.1629 
40 (75%) [1] 1.2744 0.5048 0.4729 1.1607 0.2020 0.1791 1.1283 0.1642 0.1348 
[2] 0.9856 0.4754 0.4154 0.9691 0.1919 0.1651 1.0427 0.1199 0.1181 
[3] 1.0852 0.4251 0.3851 0.9691 0.1857 0.1534 1.0427 0.0986 0.0977 
80 (50%) [1] 1.2406 0.3738 0.3504 1.2636 0.1728 0.1342 1.1527 0.0816 0.0787 
[2] 0.9731 0.3692 0.3266 1.0009 0.1418 0.1255 1.0599 0.0788 0.0672 
[3] 0.9608 0.2362 0.1828 1.0089 0.1129 0.1193 1.1239 0.0569 0.0531 
80 (75%) [1] 1.2327 0.2060 0.1581 1.3324 0.0911 0.1081 1.2529 0.0538 0.0473 
[2] 0.9985 0.1560 0.1210 1.0501 0.0747 0.0974 1.1234 0.0390 0.0311 
[3] 1.1785 0.1425 0.1117 1.0501 0.0743 0.0841 1.1234 0.0354 0.0288 
T = 0.7 
40 (50%) [1] 1.2524 0.4449 0.4338 1.3158 0.2605 0.2490 1.1864 0.1711 0.1657 
[2] 1.2570 0.3936 0.3667 1.3268 0.2426 0.2309 1.1886 0.1667 0.1573 
[3] 1.0762 0.3822 0.3098 1.1602 0.1904 0.1962 1.0421 0.1439 0.1401 
40 (75%) [1] 1.2471 0.3748 0.3030 1.1091 0.1826 0.1580 1.1226 0.1101 0.1281 
[2] 1.2164 0.2618 0.1965 1.1094 0.1381 0.1270 1.1194 0.0881 0.1077 
[3] 0.8402 0.2420 0.1862 0.9548 0.1265 0.1159 0.8692 0.0848 0.0977 
80 (50%) [1] 1.2233 0.2141 0.1664 1.2663 0.1121 0.1084 1.1564 0.0716 0.0766 
[2] 1.2242 0.2040 0.1582 1.2486 0.1081 0.0980 1.1536 0.0532 0.0565 
[3] 0.9906 0.1657 0.1327 0.8915 0.1019 0.0908 0.9102 0.0506 0.0437 
80 (75%) [1] 1.2230 0.1592 0.1246 1.3067 0.0890 0.0825 1.2377 0.0423 0.0379 
[2] 1.2147 0.1468 0.1147 1.3621 0.0670 0.0683 1.2677 0.0309 0.0274 
[3] 0.7663 0.1320 0.1036 1.0793 0.0484 0.0501 1.0372 0.0298 0.0227 
TABLE III.

The Av.Es (first column), RMSEs (second column), and MABs (third column) of γ from set-I.

n (FP%)SchemeMLEMCMC-P1MCMC-P2
T = 0.3
40 (50%) [1] 0.4860 0.2839 0.1363 0.4814 0.2391 0.1085 0.5193 0.1479 0.0779 
[2] 0.4176 0.3413 0.1495 0.4130 0.3045 0.1288 0.4387 0.1978 0.0860 
[3] 0.4085 0.3890 0.2074 0.4384 0.3247 0.1301 0.4229 0.2654 0.0978 
40 (75%) [1] 0.5064 0.1434 0.0876 0.5154 0.1022 0.0661 0.4449 0.0785 0.0608 
[2] 0.4797 0.1550 0.1078 0.4018 0.1097 0.0766 0.5085 0.0793 0.0705 
[3] 0.4121 0.2494 0.1184 0.3978 0.2185 0.0819 0.4596 0.0869 0.0722 
80 (50%) [1] 0.4618 0.0856 0.0600 0.8411 0.0584 0.0471 0.4589 0.0430 0.0377 
[2] 0.4109 0.0987 0.0637 0.4206 0.0696 0.0525 0.4300 0.0643 0.0502 
[3] 0.4164 0.1118 0.0732 0.4525 0.0947 0.0645 0.4322 0.0705 0.0563 
80 (75%) [1] 0.4632 0.0649 0.0525 0.4666 0.0477 0.0383 0.4605 0.0330 0.0274 
[2] 0.4710 0.0674 0.0525 0.4367 0.0511 0.0408 0.4392 0.0366 0.0289 
[3] 0.4260 0.0756 0.0582 0.4882 0.0545 0.0432 0.4116 0.0408 0.0319 
T = 0.7 
40 (50%) [1] 0.5167 0.1500 0.1326 0.5317 0.1110 0.0853 0.4721 0.0851 0.0722 
[2] 0.4025 0.1860 0.1339 0.4456 0.1475 0.1288 0.3978 0.0987 0.0779 
[3] 0.4014 0.2022 0.1774 0.4245 0.1579 0.1301 0.3855 0.1097 0.0860 
40 (75%) [1] 0.4674 0.0930 0.0758 0.4368 0.0876 0.0661 0.3661 0.0743 0.0608 
[2] 0.4025 0.1094 0.0983 0.3562 0.0988 0.0683 0.3372 0.0767 0.0683 
[3] 0.4043 0.1298 0.1137 0.3573 0.1095 0.0819 0.3391 0.0799 0.0705 
80 (50%) [1] 0.5256 0.0750 0.0540 0.6074 0.0590 0.0424 0.4664 0.0422 0.0339 
[2] 0.4030 0.0802 0.0574 0.4279 0.0683 0.0473 0.3320 0.0582 0.0452 
[3] 0.4027 0.0902 0.0659 0.4334 0.0744 0.0580 0.3281 0.0592 0.0507 
80 (75%) [1] 0.4807 0.0601 0.0472 0.5283 0.0405 0.0345 0.4857 0.0303 0.0246 
[2] 0.4039 0.0675 0.0493 0.4385 0.0466 0.0367 0.3888 0.0346 0.0260 
[3] 0.4038 0.0737 0.0524 0.4431 0.0490 0.0389 0.3840 0.0370 0.0287 
n (FP%)SchemeMLEMCMC-P1MCMC-P2
T = 0.3
40 (50%) [1] 0.4860 0.2839 0.1363 0.4814 0.2391 0.1085 0.5193 0.1479 0.0779 
[2] 0.4176 0.3413 0.1495 0.4130 0.3045 0.1288 0.4387 0.1978 0.0860 
[3] 0.4085 0.3890 0.2074 0.4384 0.3247 0.1301 0.4229 0.2654 0.0978 
40 (75%) [1] 0.5064 0.1434 0.0876 0.5154 0.1022 0.0661 0.4449 0.0785 0.0608 
[2] 0.4797 0.1550 0.1078 0.4018 0.1097 0.0766 0.5085 0.0793 0.0705 
[3] 0.4121 0.2494 0.1184 0.3978 0.2185 0.0819 0.4596 0.0869 0.0722 
80 (50%) [1] 0.4618 0.0856 0.0600 0.8411 0.0584 0.0471 0.4589 0.0430 0.0377 
[2] 0.4109 0.0987 0.0637 0.4206 0.0696 0.0525 0.4300 0.0643 0.0502 
[3] 0.4164 0.1118 0.0732 0.4525 0.0947 0.0645 0.4322 0.0705 0.0563 
80 (75%) [1] 0.4632 0.0649 0.0525 0.4666 0.0477 0.0383 0.4605 0.0330 0.0274 
[2] 0.4710 0.0674 0.0525 0.4367 0.0511 0.0408 0.4392 0.0366 0.0289 
[3] 0.4260 0.0756 0.0582 0.4882 0.0545 0.0432 0.4116 0.0408 0.0319 
T = 0.7 
40 (50%) [1] 0.5167 0.1500 0.1326 0.5317 0.1110 0.0853 0.4721 0.0851 0.0722 
[2] 0.4025 0.1860 0.1339 0.4456 0.1475 0.1288 0.3978 0.0987 0.0779 
[3] 0.4014 0.2022 0.1774 0.4245 0.1579 0.1301 0.3855 0.1097 0.0860 
40 (75%) [1] 0.4674 0.0930 0.0758 0.4368 0.0876 0.0661 0.3661 0.0743 0.0608 
[2] 0.4025 0.1094 0.0983 0.3562 0.0988 0.0683 0.3372 0.0767 0.0683 
[3] 0.4043 0.1298 0.1137 0.3573 0.1095 0.0819 0.3391 0.0799 0.0705 
80 (50%) [1] 0.5256 0.0750 0.0540 0.6074 0.0590 0.0424 0.4664 0.0422 0.0339 
[2] 0.4030 0.0802 0.0574 0.4279 0.0683 0.0473 0.3320 0.0582 0.0452 
[3] 0.4027 0.0902 0.0659 0.4334 0.0744 0.0580 0.3281 0.0592 0.0507 
80 (75%) [1] 0.4807 0.0601 0.0472 0.5283 0.0405 0.0345 0.4857 0.0303 0.0246 
[2] 0.4039 0.0675 0.0493 0.4385 0.0466 0.0367 0.3888 0.0346 0.0260 
[3] 0.4038 0.0737 0.0524 0.4431 0.0490 0.0389 0.3840 0.0370 0.0287 
TABLE IV.

The Av.Es (first column), RMSEs (second column), and MABs (third column) of γ from set-II.

n (FP%)SchemeMLEMCMC-P1MCMC-P2
T = 0.3
40 (50%) [1] 0.9776 0.2726 0.2693 0.9012 0.2211 0.2186 0.8848 0.1882 0.1848 
[2] 0.9674 0.2784 0.2776 0.9439 0.2473 0.2439 0.9085 0.2165 0.2085 
[3] 0.7156 0.2863 0.2860 0.6524 0.2709 0.2697 0.6749 0.2536 0.2351 
40 (75%) [1] 0.8579 0.1970 0.1877 0.8840 0.1917 0.1676 0.8104 0.1433 0.1410 
[2] 0.8552 0.2397 0.1977 0.8840 0.2025 0.1677 0.8104 0.1572 0.1571 
[3] 0.7130 0.2500 0.2273 0.7125 0.2118 0.2012 0.6760 0.1772 0.1714 
80 (50%) [1] 0.9860 0.1263 0.1477 0.9697 0.1174 0.1160 0.9511 0.0913 0.1073 
[2] 0.9183 0.1614 0.1569 0.9186 0.1492 0.1358 0.8069 0.1122 0.1269 
[3] 0.7052 0.1862 0.1784 0.7137 0.1542 0.1579 0.6271 0.1213 0.1363 
80 (75%) [1] 0.8677 0.0833 0.1126 0.8977 0.0746 0.0840 0.8714 0.0630 0.0725 
[2] 0.8675 0.0938 0.1274 0.8977 0.0850 0.0955 0.8714 0.0743 0.0835 
[3] 0.7044 0.1071 0.1375 0.7470 0.0966 0.1056 0.7021 0.0865 0.0967 
T = 0.7 
40 (50%) [1] 0.8032 0.1552 0.1848 0.9039 0.1839 0.1635 0.8971 0.1395 0.1193 
[2] 0.6997 0.2178 0.2039 0.6756 0.2021 0.1971 0.6889 0.1486 0.1275 
[3] 0.7001 0.2464 0.2363 0.6644 0.2288 0.2153 0.6822 0.1611 0.1352 
40 (75%) [1] 0.8149 0.1135 0.1259 0.8272 0.1072 0.1166 0.7803 0.0751 0.0840 
[2] 0.7079 0.1239 0.1463 0.6910 0.1123 0.1286 0.6719 0.0908 0.0942 
[3] 0.7030 0.1416 0.1672 0.6841 0.1562 0.1388 0.6698 0.1249 0.1049 
80 (50%) [1] 0.8248 0.0760 0.0929 0.9634 0.0594 0.0875 0.9615 0.0470 0.0593 
[2] 0.7013 0.0866 0.1054 0.7156 0.0700 0.0905 0.6344 0.0508 0.0663 
[3] 0.7020 0.1030 0.1158 0.6921 0.0841 0.1041 0.6126 0.0672 0.0740 
80 (75%) [1] 0.8352 0.0498 0.0690 0.8476 0.0360 0.0590 0.8349 0.0323 0.0296 
[2] 0.7030 0.0545 0.0762 0.6977 0.0389 0.0645 0.6626 0.0359 0.0361 
[3] 0.7014 0.0647 0.0872 0.7552 0.0447 0.0737 0.7080 0.0372 0.0486 
n (FP%)SchemeMLEMCMC-P1MCMC-P2
T = 0.3
40 (50%) [1] 0.9776 0.2726 0.2693 0.9012 0.2211 0.2186 0.8848 0.1882 0.1848 
[2] 0.9674 0.2784 0.2776 0.9439 0.2473 0.2439 0.9085 0.2165 0.2085 
[3] 0.7156 0.2863 0.2860 0.6524 0.2709 0.2697 0.6749 0.2536 0.2351 
40 (75%) [1] 0.8579 0.1970 0.1877 0.8840 0.1917 0.1676 0.8104 0.1433 0.1410 
[2] 0.8552 0.2397 0.1977 0.8840 0.2025 0.1677 0.8104 0.1572 0.1571 
[3] 0.7130 0.2500 0.2273 0.7125 0.2118 0.2012 0.6760 0.1772 0.1714 
80 (50%) [1] 0.9860 0.1263 0.1477 0.9697 0.1174 0.1160 0.9511 0.0913 0.1073 
[2] 0.9183 0.1614 0.1569 0.9186 0.1492 0.1358 0.8069 0.1122 0.1269 
[3] 0.7052 0.1862 0.1784 0.7137 0.1542 0.1579 0.6271 0.1213 0.1363 
80 (75%) [1] 0.8677 0.0833 0.1126 0.8977 0.0746 0.0840 0.8714 0.0630 0.0725 
[2] 0.8675 0.0938 0.1274 0.8977 0.0850 0.0955 0.8714 0.0743 0.0835 
[3] 0.7044 0.1071 0.1375 0.7470 0.0966 0.1056 0.7021 0.0865 0.0967 
T = 0.7 
40 (50%) [1] 0.8032 0.1552 0.1848 0.9039 0.1839 0.1635 0.8971 0.1395 0.1193 
[2] 0.6997 0.2178 0.2039 0.6756 0.2021 0.1971 0.6889 0.1486 0.1275 
[3] 0.7001 0.2464 0.2363 0.6644 0.2288 0.2153 0.6822 0.1611 0.1352 
40 (75%) [1] 0.8149 0.1135 0.1259 0.8272 0.1072 0.1166 0.7803 0.0751 0.0840 
[2] 0.7079 0.1239 0.1463 0.6910 0.1123 0.1286 0.6719 0.0908 0.0942 
[3] 0.7030 0.1416 0.1672 0.6841 0.1562 0.1388 0.6698 0.1249 0.1049 
80 (50%) [1] 0.8248 0.0760 0.0929 0.9634 0.0594 0.0875 0.9615 0.0470 0.0593 
[2] 0.7013 0.0866 0.1054 0.7156 0.0700 0.0905 0.6344 0.0508 0.0663 
[3] 0.7020 0.1030 0.1158 0.6921 0.0841 0.1041 0.6126 0.0672 0.0740 
80 (75%) [1] 0.8352 0.0498 0.0690 0.8476 0.0360 0.0590 0.8349 0.0323 0.0296 
[2] 0.7030 0.0545 0.0762 0.6977 0.0389 0.0645 0.6626 0.0359 0.0361 
[3] 0.7014 0.0647 0.0872 0.7552 0.0447 0.0737 0.7080 0.0372 0.0486 
TABLE V.

The Av.Es (first column), RMSEs (second column), and MABs (third column) of R(t) from set-I.

n (FP%)SchemeMLEMCMC-P1MCMC-P2
T = 0.3
40 (50%) [1] 0.8280 0.1087 0.0993 0.8529 0.0984 0.0889 0.8119 0.0753 0.0480 
[2] 0.8259 0.0775 0.0662 0.8537 0.0597 0.0695 0.8121 0.0564 0.0455 
[3] 0.8292 0.0726 0.0618 0.8260 0.0575 0.0583 0.8230 0.0548 0.0449 
40 (75%) [1] 0.8304 0.0555 0.0570 0.7829 0.0526 0.0536 0.7793 0.0514 0.0418 
[2] 0.8178 0.0547 0.0515 0.7828 0.0511 0.0466 0.7798 0.0489 0.0408 
[3] 0.8087 0.0520 0.0486 0.7843 0.0471 0.0402 0.7227 0.0364 0.0294 
80 (50%) [1] 0.8259 0.0511 0.0454 0.8485 0.0459 0.0370 0.7750 0.0349 0.0277 
[2] 0.8248 0.0495 0.0399 0.8329 0.0448 0.0359 0.7762 0.0333 0.0265 
[3] 0.8254 0.0416 0.0369 0.7970 0.0406 0.0323 0.7980 0.0284 0.0228 
80 (75%) [1] 0.8261 0.0405 0.0350 0.8645 0.0370 0.0303 0.8334 0.0247 0.0188 
[2] 0.8151 0.0386 0.0330 0.8406 0.0365 0.0294 0.8273 0.0170 0.0162 
[3] 0.8013 0.0376 0.0313 0.8103 0.0310 0.0284 0.7770 0.0137 0.0130 
T = 0.7 
40 (50%) [1] 0.8211 0.0795 0.0671 0.8485 0.0653 0.0532 0.8110 0.0602 0.0427 
[2] 0.8211 0.0738 0.0667 0.8487 0.0589 0.0498 0.8155 0.0491 0.0393 
[3] 0.8600 0.0721 0.0621 0.8874 0.0506 0.0494 0.8406 0.0454 0.0366 
40 (75%) [1] 0.8245 0.0553 0.0579 0.7769 0.0504 0.0480 0.7742 0.0388 0.0338 
[2] 0.8243 0.0540 0.0457 0.7779 0.0494 0.0429 0.7728 0.0385 0.0306 
[3] 0.8261 0.0522 0.0455 0.8067 0.0483 0.0424 0.7693 0.0307 0.0269 
80 (50%) [1] 0.8221 0.0498 0.0421 0.8370 0.0445 0.0338 0.7722 0.0259 0.0195 
[2] 0.8221 0.0489 0.0409 0.8290 0.0382 0.0333 0.7754 0.0223 0.0190 
[3] 0.8597 0.0476 0.0354 0.8672 0.0329 0.0289 0.8224 0.0205 0.0178 
80 (75%) [1] 0.8231 0.0415 0.0334 0.8556 0.0320 0.0231 0.8259 0.0191 0.0160 
[2] 0.8229 0.0387 0.0296 0.8470 0.0267 0.0205 0.8272 0.0160 0.0138 
[3] 0.8190 0.0384 0.0293 0.8133 0.0213 0.0169 0.8131 0.0126 0.0115 
n (FP%)SchemeMLEMCMC-P1MCMC-P2
T = 0.3
40 (50%) [1] 0.8280 0.1087 0.0993 0.8529 0.0984 0.0889 0.8119 0.0753 0.0480 
[2] 0.8259 0.0775 0.0662 0.8537 0.0597 0.0695 0.8121 0.0564 0.0455 
[3] 0.8292 0.0726 0.0618 0.8260 0.0575 0.0583 0.8230 0.0548 0.0449 
40 (75%) [1] 0.8304 0.0555 0.0570 0.7829 0.0526 0.0536 0.7793 0.0514 0.0418 
[2] 0.8178 0.0547 0.0515 0.7828 0.0511 0.0466 0.7798 0.0489 0.0408 
[3] 0.8087 0.0520 0.0486 0.7843 0.0471 0.0402 0.7227 0.0364 0.0294 
80 (50%) [1] 0.8259 0.0511 0.0454 0.8485 0.0459 0.0370 0.7750 0.0349 0.0277 
[2] 0.8248 0.0495 0.0399 0.8329 0.0448 0.0359 0.7762 0.0333 0.0265 
[3] 0.8254 0.0416 0.0369 0.7970 0.0406 0.0323 0.7980 0.0284 0.0228 
80 (75%) [1] 0.8261 0.0405 0.0350 0.8645 0.0370 0.0303 0.8334 0.0247 0.0188 
[2] 0.8151 0.0386 0.0330 0.8406 0.0365 0.0294 0.8273 0.0170 0.0162 
[3] 0.8013 0.0376 0.0313 0.8103 0.0310 0.0284 0.7770 0.0137 0.0130 
T = 0.7 
40 (50%) [1] 0.8211 0.0795 0.0671 0.8485 0.0653 0.0532 0.8110 0.0602 0.0427 
[2] 0.8211 0.0738 0.0667 0.8487 0.0589 0.0498 0.8155 0.0491 0.0393 
[3] 0.8600 0.0721 0.0621 0.8874 0.0506 0.0494 0.8406 0.0454 0.0366 
40 (75%) [1] 0.8245 0.0553 0.0579 0.7769 0.0504 0.0480 0.7742 0.0388 0.0338 
[2] 0.8243 0.0540 0.0457 0.7779 0.0494 0.0429 0.7728 0.0385 0.0306 
[3] 0.8261 0.0522 0.0455 0.8067 0.0483 0.0424 0.7693 0.0307 0.0269 
80 (50%) [1] 0.8221 0.0498 0.0421 0.8370 0.0445 0.0338 0.7722 0.0259 0.0195 
[2] 0.8221 0.0489 0.0409 0.8290 0.0382 0.0333 0.7754 0.0223 0.0190 
[3] 0.8597 0.0476 0.0354 0.8672 0.0329 0.0289 0.8224 0.0205 0.0178 
80 (75%) [1] 0.8231 0.0415 0.0334 0.8556 0.0320 0.0231 0.8259 0.0191 0.0160 
[2] 0.8229 0.0387 0.0296 0.8470 0.0267 0.0205 0.8272 0.0160 0.0138 
[3] 0.8190 0.0384 0.0293 0.8133 0.0213 0.0169 0.8131 0.0126 0.0115 
TABLE VI.

The Av.Es (first column), RMSEs (second column), and MABs (third column) of R(t) from set-II.

n (FP%)SchemeMLEMCMC-P1MCMC-P2
T = 0.3
40 (50%) [1] 0.9668 0.0457 0.0414 0.9645 0.0353 0.0340 0.9579 0.0201 0.0207 
[2] 0.9443 0.0429 0.0353 0.9852 0.0196 0.0204 0.9795 0.0183 0.0169 
[3] 0.9438 0.0325 0.0290 0.9723 0.0183 0.0176 0.9803 0.0160 0.0161 
40 (75%) [1] 0.9668 0.0313 0.0261 0.9604 0.0173 0.0168 0.9529 0.0126 0.0149 
[2] 0.9408 0.0301 0.0249 0.9633 0.0149 0.0160 0.9614 0.0118 0.0131 
[3] 0.9404 0.0292 0.0236 0.9633 0.0135 0.0153 0.9614 0.0106 0.0123 
80 (50%) [1] 0.9649 0.0277 0.0223 0.9693 0.0129 0.0137 0.9485 0.0099 0.0107 
[2] 0.9376 0.0268 0.0217 0.9710 0.0110 0.0114 0.9630 0.0079 0.0093 
[3] 0.9301 0.0197 0.0158 0.9800 0.0097 0.0102 0.9842 0.0083 0.0071 
80 (75%) [1] 0.9647 0.0182 0.0144 0.9771 0.0092 0.0091 0.9678 0.0071 0.0057 
[2] 0.9323 0.0142 0.0115 0.9703 0.0082 0.0081 0.9757 0.0051 0.0045 
[3] 0.9322 0.0129 0.0110 0.9703 0.0072 0.0064 0.9757 0.0039 0.0027 
T = 0.7 
40 (50%) [1] 0.9615 0.0364 0.0335 0.9679 0.0249 0.0259 0.9595 0.0183 0.0177 
[2] 0.9621 0.0350 0.0313 0.9699 0.0176 0.0217 0.9606 0.0167 0.0165 
[3] 0.9548 0.0277 0.0278 0.9811 0.0170 0.0168 0.9707 0.0145 0.0144 
40 (75%) [1] 0.9628 0.0274 0.0254 0.9512 0.0159 0.0158 0.9513 0.0143 0.0133 
[2] 0.9610 0.0235 0.0234 0.9522 0.0136 0.0150 0.9512 0.0115 0.0124 
[3] 0.9298 0.0221 0.0213 0.9531 0.0125 0.0139 0.9305 0.0104 0.0112 
80 (50%) [1] 0.9625 0.0211 0.0179 0.9671 0.0115 0.0128 0.9466 0.0087 0.0094 
[2] 0.9628 0.0200 0.0164 0.9685 0.0102 0.0116 0.9498 0.0085 0.0072 
[3] 0.9503 0.0166 0.0153 0.9662 0.0089 0.0113 0.9690 0.0057 0.0061 
80 (75%) [1] 0.9630 0.0159 0.0130 0.9761 0.0082 0.0077 0.9673 0.0053 0.0055 
[2] 0.9626 0.0147 0.0113 0.9745 0.0068 0.0074 0.9645 0.0043 0.0042 
[3] 0.9221 0.0141 0.0102 0.9672 0.0058 0.0064 0.9634 0.0032 0.0030 
n (FP%)SchemeMLEMCMC-P1MCMC-P2
T = 0.3
40 (50%) [1] 0.9668 0.0457 0.0414 0.9645 0.0353 0.0340 0.9579 0.0201 0.0207 
[2] 0.9443 0.0429 0.0353 0.9852 0.0196 0.0204 0.9795 0.0183 0.0169 
[3] 0.9438 0.0325 0.0290 0.9723 0.0183 0.0176 0.9803 0.0160 0.0161 
40 (75%) [1] 0.9668 0.0313 0.0261 0.9604 0.0173 0.0168 0.9529 0.0126 0.0149 
[2] 0.9408 0.0301 0.0249 0.9633 0.0149 0.0160 0.9614 0.0118 0.0131 
[3] 0.9404 0.0292 0.0236 0.9633 0.0135 0.0153 0.9614 0.0106 0.0123 
80 (50%) [1] 0.9649 0.0277 0.0223 0.9693 0.0129 0.0137 0.9485 0.0099 0.0107 
[2] 0.9376 0.0268 0.0217 0.9710 0.0110 0.0114 0.9630 0.0079 0.0093 
[3] 0.9301 0.0197 0.0158 0.9800 0.0097 0.0102 0.9842 0.0083 0.0071 
80 (75%) [1] 0.9647 0.0182 0.0144 0.9771 0.0092 0.0091 0.9678 0.0071 0.0057 
[2] 0.9323 0.0142 0.0115 0.9703 0.0082 0.0081 0.9757 0.0051 0.0045 
[3] 0.9322 0.0129 0.0110 0.9703 0.0072 0.0064 0.9757 0.0039 0.0027 
T = 0.7 
40 (50%) [1] 0.9615 0.0364 0.0335 0.9679 0.0249 0.0259 0.9595 0.0183 0.0177 
[2] 0.9621 0.0350 0.0313 0.9699 0.0176 0.0217 0.9606 0.0167 0.0165 
[3] 0.9548 0.0277 0.0278 0.9811 0.0170 0.0168 0.9707 0.0145 0.0144 
40 (75%) [1] 0.9628 0.0274 0.0254 0.9512 0.0159 0.0158 0.9513 0.0143 0.0133 
[2] 0.9610 0.0235 0.0234 0.9522 0.0136 0.0150 0.9512 0.0115 0.0124 
[3] 0.9298 0.0221 0.0213 0.9531 0.0125 0.0139 0.9305 0.0104 0.0112 
80 (50%) [1] 0.9625 0.0211 0.0179 0.9671 0.0115 0.0128 0.9466 0.0087 0.0094 
[2] 0.9628 0.0200 0.0164 0.9685 0.0102 0.0116 0.9498 0.0085 0.0072 
[3] 0.9503 0.0166 0.0153 0.9662 0.0089 0.0113 0.9690 0.0057 0.0061 
80 (75%) [1] 0.9630 0.0159 0.0130 0.9761 0.0082 0.0077 0.9673 0.0053 0.0055 
[2] 0.9626 0.0147 0.0113 0.9745 0.0068 0.0074 0.9645 0.0043 0.0042 
[3] 0.9221 0.0141 0.0102 0.9672 0.0058 0.0064 0.9634 0.0032 0.0030 
TABLE VII.

The Av.Es (first column), RMSEs (second column), and MABs (third column) of h(t) from set-I.

n (FP%)SchemeMLEMCMC-P1MCMC-P2
T = 0.3
40 (50%) [1] 1.0483 0.5576 0.5301 1.0668 0.4826 0.4739 1.2055 0.4131 0.3758 
[2] 1.6053 0.7227 0.6639 1.5443 0.6309 0.5871 1.7318 0.4653 0.4485 
[3] 1.5906 0.8122 0.7827 1.4657 0.7416 0.7349 1.7278 0.6490 0.6448 
40 (75%) [1] 1.2055 0.4267 0.3599 1.3194 0.3412 0.3249 1.4986 0.3168 0.2640 
[2] 1.4989 0.4342 0.4126 1.7115 0.3772 0.3616 1.8711 0.3294 0.3056 
[3] 1.5778 0.4831 0.4488 1.7271 0.4468 0.3832 1.8758 0.3973 0.3618 
80 (50%) [1] 0.8859 0.3732 0.3040 0.9190 0.2971 0.2391 1.0091 0.2335 0.1749 
[2] 1.6283 0.3906 0.3134 1.5641 0.3141 0.2460 2.0290 0.2710 0.2136 
[3] 1.6139 0.4014 0.3239 1.4269 0.3341 0.2782 2.0150 0.3010 0.2360 
80 (75%) [1] 1.1801 0.2874 0.2321 1.1244 0.2375 0.1881 1.3483 0.1462 0.1193 
[2] 1.5234 0.3098 0.2541 1.5007 0.2585 0.2016 1.6761 0.1750 0.1272 
[3] 1.6157 0.3677 0.2946 1.2957 0.2592 0.2151 1.5968 0.2055 0.1548 
T = 0.7 
40 (50%) [1] 1.2722 0.5099 0.4218 1.1321 0.4225 0.3577 1.4120 0.4015 0.3104 
[2] 1.6587 0.5815 0.5241 1.4641 0.4565 0.4175 1.6814 0.4389 0.3628 
[3] 1.6428 0.5959 0.5655 1.5205 0.4945 0.4331 1.7377 0.4504 0.3855 
40 (75%) [1] 1.4484 0.3774 0.3123 1.5733 0.3234 0.2758 1.8564 0.3047 0.2411 
[2] 1.6301 0.3856 0.3320 1.9027 0.3635 0.2877 2.0027 0.3212 0.2725 
[3] 1.6217 0.4695 0.3880 1.9009 0.3982 0.3475 1.9918 0.3865 0.2891 
80 (50%) [1] 1.2474 0.3542 0.2791 1.0886 0.2889 0.2337 1.4650 0.2185 0.1629 
[2] 1.6474 0.3601 0.2914 1.5516 0.2941 0.2438 2.0160 0.2399 0.1882 
[3] 1.6400 0.3755 0.3035 1.5121 0.3189 0.2677 2.0387 0.2870 0.2198 
80 (75%) [1] 1.4351 0.2757 0.2221 1.3433 0.2045 0.1596 1.4366 0.1310 0.1058 
[2] 1.6347 0.2931 0.2457 1.4744 0.2158 0.1863 1.6894 0.1544 0.1195 
[3] 1.6329 0.3347 0.2685 1.4319 0.2280 0.2082 1.7103 0.1765 0.1466 
n (FP%)SchemeMLEMCMC-P1MCMC-P2
T = 0.3
40 (50%) [1] 1.0483 0.5576 0.5301 1.0668 0.4826 0.4739 1.2055 0.4131 0.3758 
[2] 1.6053 0.7227 0.6639 1.5443 0.6309 0.5871 1.7318 0.4653 0.4485 
[3] 1.5906 0.8122 0.7827 1.4657 0.7416 0.7349 1.7278 0.6490 0.6448 
40 (75%) [1] 1.2055 0.4267 0.3599 1.3194 0.3412 0.3249 1.4986 0.3168 0.2640 
[2] 1.4989 0.4342 0.4126 1.7115 0.3772 0.3616 1.8711 0.3294 0.3056 
[3] 1.5778 0.4831 0.4488 1.7271 0.4468 0.3832 1.8758 0.3973 0.3618 
80 (50%) [1] 0.8859 0.3732 0.3040 0.9190 0.2971 0.2391 1.0091 0.2335 0.1749 
[2] 1.6283 0.3906 0.3134 1.5641 0.3141 0.2460 2.0290 0.2710 0.2136 
[3] 1.6139 0.4014 0.3239 1.4269 0.3341 0.2782 2.0150 0.3010 0.2360 
80 (75%) [1] 1.1801 0.2874 0.2321 1.1244 0.2375 0.1881 1.3483 0.1462 0.1193 
[2] 1.5234 0.3098 0.2541 1.5007 0.2585 0.2016 1.6761 0.1750 0.1272 
[3] 1.6157 0.3677 0.2946 1.2957 0.2592 0.2151 1.5968 0.2055 0.1548 
T = 0.7 
40 (50%) [1] 1.2722 0.5099 0.4218 1.1321 0.4225 0.3577 1.4120 0.4015 0.3104 
[2] 1.6587 0.5815 0.5241 1.4641 0.4565 0.4175 1.6814 0.4389 0.3628 
[3] 1.6428 0.5959 0.5655 1.5205 0.4945 0.4331 1.7377 0.4504 0.3855 
40 (75%) [1] 1.4484 0.3774 0.3123 1.5733 0.3234 0.2758 1.8564 0.3047 0.2411 
[2] 1.6301 0.3856 0.3320 1.9027 0.3635 0.2877 2.0027 0.3212 0.2725 
[3] 1.6217 0.4695 0.3880 1.9009 0.3982 0.3475 1.9918 0.3865 0.2891 
80 (50%) [1] 1.2474 0.3542 0.2791 1.0886 0.2889 0.2337 1.4650 0.2185 0.1629 
[2] 1.6474 0.3601 0.2914 1.5516 0.2941 0.2438 2.0160 0.2399 0.1882 
[3] 1.6400 0.3755 0.3035 1.5121 0.3189 0.2677 2.0387 0.2870 0.2198 
80 (75%) [1] 1.4351 0.2757 0.2221 1.3433 0.2045 0.1596 1.4366 0.1310 0.1058 
[2] 1.6347 0.2931 0.2457 1.4744 0.2158 0.1863 1.6894 0.1544 0.1195 
[3] 1.6329 0.3347 0.2685 1.4319 0.2280 0.2082 1.7103 0.1765 0.1466 
TABLE VIII.

The Av.Es (first column), RMSEs (second column), and MABs (third column) of h(t) from set-II.

n (FP%)SchemeMLEMCMC-P1MCMC-P2
T = 0.3
40 (50%) [1] 0.4008 0.2427 0.2252 0.3060 0.2292 0.2183 0.2463 0.1934 0.1576 
[2] 0.4095 0.2640 0.2594 0.1810 0.2368 0.2266 0.2450 0.1988 0.1667 
[3] 0.4187 0.2973 0.2909 0.4907 0.2887 0.2488 0.5339 0.2093 0.1696 
40 (75%) [1] 0.5244 0.1800 0.1710 0.3785 0.1767 0.1654 0.4306 0.1456 0.1206 
[2] 0.5254 0.1896 0.1801 0.3785 0.1850 0.1766 0.4306 0.1638 0.1382 
[3] 0.4224 0.2043 0.1980 0.4922 0.1985 0.1832 0.5730 0.1855 0.1502 
80 (50%) [1] 0.4526 0.1387 0.1184 0.2122 0.1197 0.0998 0.1867 0.1096 0.0791 
[2] 0.4749 0.1452 0.1368 0.3061 0.1299 0.1081 0.4218 0.1196 0.0842 
[3] 0.4481 0.1535 0.1512 0.4120 0.1440 0.1251 0.6425 0.1253 0.1128 
80 (75%) [1] 0.5662 0.1194 0.0855 0.3160 0.0849 0.0537 0.2915 0.0698 0.0406 
[2] 0.5662 0.1291 0.1024 0.3160 0.0905 0.0716 0.2915 0.0831 0.0617 
[3] 0.4520 0.1344 0.1182 0.3233 0.1094 0.0855 0.4317 0.0988 0.0691 
T = 0.7 
40 (50%) [1] 0.4365 0.2173 0.1981 0.2315 0.1961 0.1561 0.1629 0.1544 0.1059 
[2] 0.4631 0.2245 0.2132 0.4264 0.2080 0.1576 0.1774 0.1736 0.1159 
[3] 0.4678 0.2452 0.2240 0.4523 0.2375 0.1848 0.2011 0.1883 0.1288 
40 (75%) [1] 0.6055 0.1733 0.1687 0.4775 0.1541 0.1298 0.1298 0.1250 0.0847 
[2] 0.4725 0.1833 0.1554 0.5696 0.1765 0.1382 0.1502 0.1361 0.0926 
[3] 0.4602 0.1986 0.1593 0.5819 0.1912 0.1412 0.1540 0.1433 0.0986 
80 (50%) [1] 0.4561 0.1282 0.1276 0.3054 0.1063 0.1103 0.0831 0.0962 0.0711 
[2] 0.4667 0.1482 0.1350 0.4190 0.1280 0.1150 0.0985 0.1175 0.0730 
[3] 0.4686 0.1656 0.1437 0.4425 0.1396 0.1232 0.1192 0.1228 0.0765 
80 (75%) [1] 0.6372 0.0978 0.0819 0.3637 0.0672 0.0701 0.0701 0.0525 0.0403 
[2] 0.4690 0.1062 0.0894 0.3688 0.0787 0.0831 0.0730 0.0718 0.0459 
[3] 0.4664 0.1142 0.0984 0.3293 0.0861 0.0855 0.0765 0.0924 0.0632 
n (FP%)SchemeMLEMCMC-P1MCMC-P2
T = 0.3
40 (50%) [1] 0.4008 0.2427 0.2252 0.3060 0.2292 0.2183 0.2463 0.1934 0.1576 
[2] 0.4095 0.2640 0.2594 0.1810 0.2368 0.2266 0.2450 0.1988 0.1667 
[3] 0.4187 0.2973 0.2909 0.4907 0.2887 0.2488 0.5339 0.2093 0.1696 
40 (75%) [1] 0.5244 0.1800 0.1710 0.3785 0.1767 0.1654 0.4306 0.1456 0.1206 
[2] 0.5254 0.1896 0.1801 0.3785 0.1850 0.1766 0.4306 0.1638 0.1382 
[3] 0.4224 0.2043 0.1980 0.4922 0.1985 0.1832 0.5730 0.1855 0.1502 
80 (50%) [1] 0.4526 0.1387 0.1184 0.2122 0.1197 0.0998 0.1867 0.1096 0.0791 
[2] 0.4749 0.1452 0.1368 0.3061 0.1299 0.1081 0.4218 0.1196 0.0842 
[3] 0.4481 0.1535 0.1512 0.4120 0.1440 0.1251 0.6425 0.1253 0.1128 
80 (75%) [1] 0.5662 0.1194 0.0855 0.3160 0.0849 0.0537 0.2915 0.0698 0.0406 
[2] 0.5662 0.1291 0.1024 0.3160 0.0905 0.0716 0.2915 0.0831 0.0617 
[3] 0.4520 0.1344 0.1182 0.3233 0.1094 0.0855 0.4317 0.0988 0.0691 
T = 0.7 
40 (50%) [1] 0.4365 0.2173 0.1981 0.2315 0.1961 0.1561 0.1629 0.1544 0.1059 
[2] 0.4631 0.2245 0.2132 0.4264 0.2080 0.1576 0.1774 0.1736 0.1159 
[3] 0.4678 0.2452 0.2240 0.4523 0.2375 0.1848 0.2011 0.1883 0.1288 
40 (75%) [1] 0.6055 0.1733 0.1687 0.4775 0.1541 0.1298 0.1298 0.1250 0.0847 
[2] 0.4725 0.1833 0.1554 0.5696 0.1765 0.1382 0.1502 0.1361 0.0926 
[3] 0.4602 0.1986 0.1593 0.5819 0.1912 0.1412 0.1540 0.1433 0.0986 
80 (50%) [1] 0.4561 0.1282 0.1276 0.3054 0.1063 0.1103 0.0831 0.0962 0.0711 
[2] 0.4667 0.1482 0.1350 0.4190 0.1280 0.1150 0.0985 0.1175 0.0730 
[3] 0.4686 0.1656 0.1437 0.4425 0.1396 0.1232 0.1192 0.1228 0.0765 
80 (75%) [1] 0.6372 0.0978 0.0819 0.3637 0.0672 0.0701 0.0701 0.0525 0.0403 
[2] 0.4690 0.1062 0.0894 0.3688 0.0787 0.0831 0.0730 0.0718 0.0459 
[3] 0.4664 0.1142 0.0984 0.3293 0.0861 0.0855 0.0765 0.0924 0.0632 
TABLE IX.

The ACLs (first column) and CPs (second column) of 95% interval estimates of α from set-I.

n (FP%)SchemeACIHPD-P1HPD-P2
T = 0.3
40 (50%) [1] 0.593 0.928 0.486 0.942 0.337 0.951 
[2] 0.587 0.930 0.445 0.944 0.315 0.953 
[3] 0.547 0.933 0.439 0.947 0.288 0.956 
40 (75%) [1] 0.520 0.935 0.429 0.949 0.264 0.958 
[2] 0.486 0.938 0.416 0.952 0.225 0.961 
[3] 0.438 0.941 0.408 0.955 0.209 0.965 
80 (50%) [1] 0.404 0.943 0.386 0.957 0.195 0.967 
[2] 0.388 0.944 0.351 0.958 0.187 0.968 
[3] 0.365 0.946 0.326 0.960 0.172 0.970 
80 (75%) [1] 0.343 0.948 0.319 0.962 0.146 0.972 
[2] 0.318 0.950 0.297 0.964 0.136 0.974 
[3] 0.302 0.951 0.278 0.965 0.121 0.975 
T = 0.7 
40 (50%) [1] 0.525 0.937 0.460 0.951 0.098 0.961 
[2] 0.501 0.939 0.433 0.953 0.103 0.963 
[3] 0.448 0.942 0.412 0.956 0.187 0.966 
40 (75%) [1] 0.415 0.944 0.398 0.959 0.112 0.968 
[2] 0.411 0.947 0.379 0.962 0.114 0.971 
[3] 0.386 0.950 0.339 0.965 0.202 0.974 
80 (50%) [1] 0.365 0.952 0.329 0.967 0.089 0.976 
[2] 0.350 0.953 0.303 0.968 0.092 0.977 
[3] 0.318 0.955 0.296 0.970 0.212 0.979 
80 (75%) [1] 0.317 0.957 0.279 0.972 0.074 0.981 
[2] 0.296 0.960 0.248 0.974 0.073 0.983 
[3] 0.283 0.961 0.237 0.975 0.194 0.985 
n (FP%)SchemeACIHPD-P1HPD-P2
T = 0.3
40 (50%) [1] 0.593 0.928 0.486 0.942 0.337 0.951 
[2] 0.587 0.930 0.445 0.944 0.315 0.953 
[3] 0.547 0.933 0.439 0.947 0.288 0.956 
40 (75%) [1] 0.520 0.935 0.429 0.949 0.264 0.958 
[2] 0.486 0.938 0.416 0.952 0.225 0.961 
[3] 0.438 0.941 0.408 0.955 0.209 0.965 
80 (50%) [1] 0.404 0.943 0.386 0.957 0.195 0.967 
[2] 0.388 0.944 0.351 0.958 0.187 0.968 
[3] 0.365 0.946 0.326 0.960 0.172 0.970 
80 (75%) [1] 0.343 0.948 0.319 0.962 0.146 0.972 
[2] 0.318 0.950 0.297 0.964 0.136 0.974 
[3] 0.302 0.951 0.278 0.965 0.121 0.975 
T = 0.7 
40 (50%) [1] 0.525 0.937 0.460 0.951 0.098 0.961 
[2] 0.501 0.939 0.433 0.953 0.103 0.963 
[3] 0.448 0.942 0.412 0.956 0.187 0.966 
40 (75%) [1] 0.415 0.944 0.398 0.959 0.112 0.968 
[2] 0.411 0.947 0.379 0.962 0.114 0.971 
[3] 0.386 0.950 0.339 0.965 0.202 0.974 
80 (50%) [1] 0.365 0.952 0.329 0.967 0.089 0.976 
[2] 0.350 0.953 0.303 0.968 0.092 0.977 
[3] 0.318 0.955 0.296 0.970 0.212 0.979 
80 (75%) [1] 0.317 0.957 0.279 0.972 0.074 0.981 
[2] 0.296 0.960 0.248 0.974 0.073 0.983 
[3] 0.283 0.961 0.237 0.975 0.194 0.985 
TABLE X.

The ACLs (first column) and CPs (second column) of 95% interval estimates of α from set-II.

n (FP%)SchemeACIHPD-P1HPD-P2
T = 0.3
40 (50%) [1] 0.536 0.933 0.379 0.947 0.318 0.956 
[2] 0.518 0.935 0.367 0.949 0.299 0.958 
[3] 0.498 0.936 0.334 0.950 0.271 0.959 
40 (75%) [1] 0.492 0.939 0.324 0.953 0.255 0.962 
[2] 0.470 0.941 0.306 0.955 0.224 0.965 
[3] 0.414 0.944 0.273 0.958 0.199 0.968 
80 (50%) [1] 0.394 0.946 0.251 0.960 0.179 0.970 
[2] 0.385 0.947 0.235 0.961 0.165 0.971 
[3] 0.367 0.949 0.218 0.963 0.144 0.973 
80 (75%) [1] 0.337 0.950 0.201 0.964 0.127 0.974 
[2] 0.313 0.952 0.181 0.966 0.124 0.976 
[3] 0.298 0.955 0.170 0.969 0.100 0.979 
T = 0.7 
40 (50%) [1] 0.525 0.940 0.370 0.954 0.212 0.963 
[2] 0.512 0.942 0.341 0.956 0.188 0.965 
[3] 0.486 0.943 0.329 0.957 0.158 0.967 
40 (75%) [1] 0.468 0.946 0.319 0.960 0.142 0.970 
[2] 0.449 0.949 0.293 0.962 0.130 0.972 
[3] 0.398 0.951 0.256 0.965 0.122 0.975 
80 (50%) [1] 0.388 0.953 0.236 0.967 0.103 0.977 
[2] 0.372 0.954 0.219 0.968 0.098 0.978 
[3] 0.345 0.956 0.202 0.970 0.092 0.980 
80 (75%) [1] 0.299 0.957 0.188 0.971 0.089 0.981 
[2] 0.269 0.959 0.180 0.974 0.079 0.983 
[3] 0.242 0.962 0.163 0.976 0.073 0.986 
n (FP%)SchemeACIHPD-P1HPD-P2
T = 0.3
40 (50%) [1] 0.536 0.933 0.379 0.947 0.318 0.956 
[2] 0.518 0.935 0.367 0.949 0.299 0.958 
[3] 0.498 0.936 0.334 0.950 0.271 0.959 
40 (75%) [1] 0.492 0.939 0.324 0.953 0.255 0.962 
[2] 0.470 0.941 0.306 0.955 0.224 0.965 
[3] 0.414 0.944 0.273 0.958 0.199 0.968 
80 (50%) [1] 0.394 0.946 0.251 0.960 0.179 0.970 
[2] 0.385 0.947 0.235 0.961 0.165 0.971 
[3] 0.367 0.949 0.218 0.963 0.144 0.973 
80 (75%) [1] 0.337 0.950 0.201 0.964 0.127 0.974 
[2] 0.313 0.952 0.181 0.966 0.124 0.976 
[3] 0.298 0.955 0.170 0.969 0.100 0.979 
T = 0.7 
40 (50%) [1] 0.525 0.940 0.370 0.954 0.212 0.963 
[2] 0.512 0.942 0.341 0.956 0.188 0.965 
[3] 0.486 0.943 0.329 0.957 0.158 0.967 
40 (75%) [1] 0.468 0.946 0.319 0.960 0.142 0.970 
[2] 0.449 0.949 0.293 0.962 0.130 0.972 
[3] 0.398 0.951 0.256 0.965 0.122 0.975 
80 (50%) [1] 0.388 0.953 0.236 0.967 0.103 0.977 
[2] 0.372 0.954 0.219 0.968 0.098 0.978 
[3] 0.345 0.956 0.202 0.970 0.092 0.980 
80 (75%) [1] 0.299 0.957 0.188 0.971 0.089 0.981 
[2] 0.269 0.959 0.180 0.974 0.079 0.983 
[3] 0.242 0.962 0.163 0.976 0.073 0.986 
TABLE XI.

The ACLs (first column) and CPs (second column) of 95% interval estimates of γ from set-I.

n (FP%)SchemeACIHPD-P1HPD-P2
T = 0.3
40 (50%) [1] 0.421 0.941 0.374 0.955 0.211 0.965 
[2] 0.467 0.938 0.419 0.952 0.235 0.962 
[3] 0.496 0.936 0.435 0.950 0.246 0.960 
40 (75%) [1] 0.366 0.946 0.284 0.960 0.139 0.969 
[2] 0.386 0.944 0.339 0.958 0.140 0.968 
[3] 0.407 0.942 0.356 0.956 0.206 0.966 
80 (50%) [1] 0.285 0.952 0.255 0.966 0.124 0.976 
[2] 0.292 0.951 0.271 0.965 0.131 0.975 
[3] 0.337 0.949 0.282 0.963 0.133 0.973 
80 (75%) [1] 0.239 0.957 0.211 0.971 0.105 0.981 
[2] 0.259 0.955 0.226 0.969 0.107 0.979 
[3] 0.270 0.954 0.246 0.968 0.113 0.978 
T = 0.7 
40 (50%) [1] 0.366 0.947 0.309 0.962 0.182 0.971 
[2] 0.389 0.944 0.318 0.960 0.197 0.969 
[3] 0.408 0.940 0.332 0.954 0.220 0.964 
40 (75%) [1] 0.310 0.953 0.271 0.968 0.160 0.977 
[2] 0.334 0.951 0.284 0.966 0.161 0.975 
[3] 0.357 0.950 0.295 0.964 0.173 0.974 
80 (50%) [1] 0.278 0.961 0.241 0.975 0.118 0.985 
[2] 0.285 0.958 0.255 0.973 0.128 0.983 
[3] 0.291 0.955 0.261 0.970 0.134 0.980 
80 (75%) [1] 0.224 0.965 0.210 0.979 0.105 0.989 
[2] 0.240 0.964 0.222 0.978 0.109 0.988 
[3] 0.260 0.962 0.231 0.976 0.111 0.986 
n (FP%)SchemeACIHPD-P1HPD-P2
T = 0.3
40 (50%) [1] 0.421 0.941 0.374 0.955 0.211 0.965 
[2] 0.467 0.938 0.419 0.952 0.235 0.962 
[3] 0.496 0.936 0.435 0.950 0.246 0.960 
40 (75%) [1] 0.366 0.946 0.284 0.960 0.139 0.969 
[2] 0.386 0.944 0.339 0.958 0.140 0.968 
[3] 0.407 0.942 0.356 0.956 0.206 0.966 
80 (50%) [1] 0.285 0.952 0.255 0.966 0.124 0.976 
[2] 0.292 0.951 0.271 0.965 0.131 0.975 
[3] 0.337 0.949 0.282 0.963 0.133 0.973 
80 (75%) [1] 0.239 0.957 0.211 0.971 0.105 0.981 
[2] 0.259 0.955 0.226 0.969 0.107 0.979 
[3] 0.270 0.954 0.246 0.968 0.113 0.978 
T = 0.7 
40 (50%) [1] 0.366 0.947 0.309 0.962 0.182 0.971 
[2] 0.389 0.944 0.318 0.960 0.197 0.969 
[3] 0.408 0.940 0.332 0.954 0.220 0.964 
40 (75%) [1] 0.310 0.953 0.271 0.968 0.160 0.977 
[2] 0.334 0.951 0.284 0.966 0.161 0.975 
[3] 0.357 0.950 0.295 0.964 0.173 0.974 
80 (50%) [1] 0.278 0.961 0.241 0.975 0.118 0.985 
[2] 0.285 0.958 0.255 0.973 0.128 0.983 
[3] 0.291 0.955 0.261 0.970 0.134 0.980 
80 (75%) [1] 0.224 0.965 0.210 0.979 0.105 0.989 
[2] 0.240 0.964 0.222 0.978 0.109 0.988 
[3] 0.260 0.962 0.231 0.976 0.111 0.986 
TABLE XII.

The ACLs (first column) and CPs (second column) of 95% interval estimates of γ from set-II.

n (FP%)SchemeACIHPD-P1HPD-P2
T = 0.3
40 (50%) [1] 0.329 0.946 0.215 0.960 0.175 0.970 
[2] 0.342 0.944 0.223 0.958 0.184 0.968 
[3] 0.376 0.942 0.238 0.956 0.218 0.966 
40 (75%) [1] 0.275 0.952 0.177 0.966 0.131 0.976 
[2] 0.286 0.949 0.198 0.963 0.141 0.973 
[3] 0.313 0.947 0.204 0.961 0.151 0.971 
80 (50%) [1] 0.233 0.958 0.156 0.972 0.116 0.982 
[2] 0.243 0.955 0.159 0.969 0.117 0.979 
[3] 0.265 0.953 0.166 0.967 0.120 0.977 
80 (75%) [1] 0.170 0.953 0.129 0.967 0.095 0.977 
[2] 0.198 0.961 0.136 0.975 0.102 0.985 
[3] 0.227 0.959 0.147 0.973 0.106 0.983 
T = 0.7 
40 (50%) [1] 0.301 0.955 0.207 0.970 0.148 0.979 
[2] 0.326 0.953 0.218 0.968 0.175 0.977 
[3] 0.355 0.951 0.235 0.966 0.214 0.975 
40 (75%) [1] 0.242 0.962 0.170 0.976 0.123 0.986 
[2] 0.277 0.958 0.185 0.973 0.126 0.983 
[3] 0.299 0.956 0.195 0.971 0.137 0.981 
80 (50%) [1] 0.140 0.968 0.124 0.982 0.108 0.992 
[2] 0.224 0.965 0.148 0.979 0.115 0.989 
[3] 0.234 0.963 0.159 0.977 0.120 0.987 
80 (75%) [1] 0.118 0.963 0.105 0.977 0.090 0.996 
[2] 0.122 0.971 0.109 0.985 0.097 0.995 
[3] 0.138 0.969 0.111 0.983 0.101 0.993 
n (FP%)SchemeACIHPD-P1HPD-P2
T = 0.3
40 (50%) [1] 0.329 0.946 0.215 0.960 0.175 0.970 
[2] 0.342 0.944 0.223 0.958 0.184 0.968 
[3] 0.376 0.942 0.238 0.956 0.218 0.966 
40 (75%) [1] 0.275 0.952 0.177 0.966 0.131 0.976 
[2] 0.286 0.949 0.198 0.963 0.141 0.973 
[3] 0.313 0.947 0.204 0.961 0.151 0.971 
80 (50%) [1] 0.233 0.958 0.156 0.972 0.116 0.982 
[2] 0.243 0.955 0.159 0.969 0.117 0.979 
[3] 0.265 0.953 0.166 0.967 0.120 0.977 
80 (75%) [1] 0.170 0.953 0.129 0.967 0.095 0.977 
[2] 0.198 0.961 0.136 0.975 0.102 0.985 
[3] 0.227 0.959 0.147 0.973 0.106 0.983 
T = 0.7 
40 (50%) [1] 0.301 0.955 0.207 0.970 0.148 0.979 
[2] 0.326 0.953 0.218 0.968 0.175 0.977 
[3] 0.355 0.951 0.235 0.966 0.214 0.975 
40 (75%) [1] 0.242 0.962 0.170 0.976 0.123 0.986 
[2] 0.277 0.958 0.185 0.973 0.126 0.983 
[3] 0.299 0.956 0.195 0.971 0.137 0.981 
80 (50%) [1] 0.140 0.968 0.124 0.982 0.108 0.992 
[2] 0.224 0.965 0.148 0.979 0.115 0.989 
[3] 0.234 0.963 0.159 0.977 0.120 0.987 
80 (75%) [1] 0.118 0.963 0.105 0.977 0.090 0.996 
[2] 0.122 0.971 0.109 0.985 0.097 0.995 
[3] 0.138 0.969 0.111 0.983 0.101 0.993 
TABLE XIII.

The ACLs (first column) and CPs (second column) of 95% interval estimates of R(t) from set-I.

n (FP%)SchemeACIHPD-P1HPD-P2
T = 0.3
40 (50%) [1] 0.250 0.943 0.192 0.948 0.140 0.952 
[2] 0.242 0.944 0.178 0.949 0.121 0.953 
[3] 0.224 0.946 0.168 0.951 0.116 0.955 
40 (75%) [1] 0.189 0.949 0.166 0.953 0.085 0.959 
[2] 0.164 0.951 0.158 0.956 0.084 0.960 
[3] 0.159 0.952 0.145 0.957 0.078 0.962 
80 (50%) [1] 0.150 0.953 0.133 0.958 0.068 0.963 
[2] 0.145 0.955 0.133 0.960 0.068 0.965 
[3] 0.142 0.956 0.130 0.961 0.064 0.966 
80 (75%) [1] 0.132 0.958 0.117 0.963 0.058 0.968 
[2] 0.127 0.959 0.113 0.964 0.054 0.969 
[3] 0.124 0.961 0.109 0.965 0.051 0.970 
T = 0.7 
40 (50%) [1] 0.240 0.946 0.186 0.951 0.109 0.956 
[2] 0.221 0.947 0.169 0.953 0.099 0.957 
[3] 0.197 0.949 0.156 0.955 0.097 0.959 
40 (75%) [1] 0.171 0.952 0.144 0.958 0.086 0.962 
[2] 0.157 0.954 0.139 0.960 0.080 0.964 
[3] 0.153 0.955 0.136 0.961 0.074 0.965 
80 (50%) [1] 0.148 0.956 0.129 0.962 0.070 0.966 
[2] 0.142 0.958 0.122 0.964 0.065 0.968 
[3] 0.137 0.959 0.111 0.965 0.059 0.969 
80 (75%) [1] 0.129 0.961 0.107 0.967 0.054 0.971 
[2] 0.121 0.962 0.103 0.968 0.049 0.972 
[3] 0.114 0.963 0.093 0.969 0.046 0.974 
n (FP%)SchemeACIHPD-P1HPD-P2
T = 0.3
40 (50%) [1] 0.250 0.943 0.192 0.948 0.140 0.952 
[2] 0.242 0.944 0.178 0.949 0.121 0.953 
[3] 0.224 0.946 0.168 0.951 0.116 0.955 
40 (75%) [1] 0.189 0.949 0.166 0.953 0.085 0.959 
[2] 0.164 0.951 0.158 0.956 0.084 0.960 
[3] 0.159 0.952 0.145 0.957 0.078 0.962 
80 (50%) [1] 0.150 0.953 0.133 0.958 0.068 0.963 
[2] 0.145 0.955 0.133 0.960 0.068 0.965 
[3] 0.142 0.956 0.130 0.961 0.064 0.966 
80 (75%) [1] 0.132 0.958 0.117 0.963 0.058 0.968 
[2] 0.127 0.959 0.113 0.964 0.054 0.969 
[3] 0.124 0.961 0.109 0.965 0.051 0.970 
T = 0.7 
40 (50%) [1] 0.240 0.946 0.186 0.951 0.109 0.956 
[2] 0.221 0.947 0.169 0.953 0.099 0.957 
[3] 0.197 0.949 0.156 0.955 0.097 0.959 
40 (75%) [1] 0.171 0.952 0.144 0.958 0.086 0.962 
[2] 0.157 0.954 0.139 0.960 0.080 0.964 
[3] 0.153 0.955 0.136 0.961 0.074 0.965 
80 (50%) [1] 0.148 0.956 0.129 0.962 0.070 0.966 
[2] 0.142 0.958 0.122 0.964 0.065 0.968 
[3] 0.137 0.959 0.111 0.965 0.059 0.969 
80 (75%) [1] 0.129 0.961 0.107 0.967 0.054 0.971 
[2] 0.121 0.962 0.103 0.968 0.049 0.972 
[3] 0.114 0.963 0.093 0.969 0.046 0.974 
TABLE XIV.

The ACLs (first column) and CPs (second column) of 95% interval estimates of R(t) from set-II.

n (FP%)SchemeACIHPD-P1HPD-P2
T = 0.3
40 (50%) [1] 0.107 0.954 0.072 0.959 0.050 0.961 
[2] 0.100 0.955 0.059 0.960 0.049 0.962 
[3] 0.098 0.956 0.049 0.961 0.039 0.963 
40 (75%) [1] 0.090 0.958 0.038 0.963 0.032 0.965 
[2] 0.085 0.960 0.035 0.965 0.029 0.967 
[3] 0.081 0.961 0.033 0.966 0.024 0.968 
80 (50%) [1] 0.072 0.963 0.029 0.968 0.021 0.970 
[2] 0.069 0.964 0.027 0.969 0.020 0.971 
[3] 0.064 0.965 0.025 0.970 0.017 0.972 
80 (75%) [1] 0.053 0.967 0.023 0.972 0.015 0.974 
[2] 0.044 0.969 0.019 0.974 0.014 0.976 
[3] 0.039 0.971 0.019 0.976 0.013 0.978 
T = 0.7 
40 (50%) [1] 0.089 0.957 0.063 0.962 0.029 0.964 
[2] 0.069 0.960 0.054 0.965 0.024 0.967 
[3] 0.062 0.961 0.049 0.966 0.021 0.968 
40 (75%) [1] 0.059 0.963 0.038 0.968 0.019 0.970 
[2] 0.049 0.965 0.033 0.970 0.017 0.972 
[3] 0.048 0.966 0.030 0.971 0.016 0.973 
80 (50%) [1] 0.046 0.968 0.027 0.973 0.015 0.975 
[2] 0.045 0.969 0.026 0.974 0.014 0.976 
[3] 0.041 0.970 0.023 0.975 0.013 0.977 
80 (75%) [1] 0.037 0.972 0.021 0.977 0.012 0.979 
[2] 0.033 0.974 0.018 0.979 0.011 0.981 
[3] 0.030 0.976 0.017 0.981 0.010 0.983 
n (FP%)SchemeACIHPD-P1HPD-P2
T = 0.3
40 (50%) [1] 0.107 0.954 0.072 0.959 0.050 0.961 
[2] 0.100 0.955 0.059 0.960 0.049 0.962 
[3] 0.098 0.956 0.049 0.961 0.039 0.963 
40 (75%) [1] 0.090 0.958 0.038 0.963 0.032 0.965 
[2] 0.085 0.960 0.035 0.965 0.029 0.967 
[3] 0.081 0.961 0.033 0.966 0.024 0.968 
80 (50%) [1] 0.072 0.963 0.029 0.968 0.021 0.970 
[2] 0.069 0.964 0.027 0.969 0.020 0.971 
[3] 0.064 0.965 0.025 0.970 0.017 0.972 
80 (75%) [1] 0.053 0.967 0.023 0.972 0.015 0.974 
[2] 0.044 0.969 0.019 0.974 0.014 0.976 
[3] 0.039 0.971 0.019 0.976 0.013 0.978 
T = 0.7 
40 (50%) [1] 0.089 0.957 0.063 0.962 0.029 0.964 
[2] 0.069 0.960 0.054 0.965 0.024 0.967 
[3] 0.062 0.961 0.049 0.966 0.021 0.968 
40 (75%) [1] 0.059 0.963 0.038 0.968 0.019 0.970 
[2] 0.049 0.965 0.033 0.970 0.017 0.972 
[3] 0.048 0.966 0.030 0.971 0.016 0.973 
80 (50%) [1] 0.046 0.968 0.027 0.973 0.015 0.975 
[2] 0.045 0.969 0.026 0.974 0.014 0.976 
[3] 0.041 0.970 0.023 0.975 0.013 0.977 
80 (75%) [1] 0.037 0.972 0.021 0.977 0.012 0.979 
[2] 0.033 0.974 0.018 0.979 0.011 0.981 
[3] 0.030 0.976 0.017 0.981 0.010 0.983 
TABLE XV.

The ACLs (first column) and CPs (second column) of 95% interval estimates of h(t) from set-I.

n (FP%)SchemeACIHPD-P1HPD-P2
T = 0.3
40 (50%) [1] 1.432 0.884 1.407 0.889 0.904 0.897 
[2] 1.586 0.881 1.558 0.885 0.914 0.893 
[3] 1.689 0.876 1.677 0.880 0.942 0.888 
40 (75%) [1] 1.181 0.906 1.172 0.910 0.611 0.919 
[2] 1.329 0.890 1.292 0.894 0.642 0.902 
[3] 1.361 0.888 1.320 0.892 0.767 0.900 
80 (50%) [1] 0.987 0.913 0.971 0.917 0.554 0.926 
[2] 1.075 0.912 0.996 0.916 0.573 0.925 
[3] 1.110 0.909 1.058 0.913 0.581 0.922 
80 (75%) [1] 0.778 0.922 0.732 0.927 0.450 0.936 
[2] 0.904 0.918 0.842 0.924 0.518 0.933 
[3] 0.965 0.916 0.936 0.920 0.547 0.929 
T = 0.7 
40 (50%) [1] 1.342 0.887 1.303 0.890 0.634 0.903 
[2] 1.497 0.884 1.459 0.887 0.685 0.900 
[3] 1.604 0.879 1.551 0.882 0.695 0.895 
40 (75%) [1] 1.057 0.908 1.011 0.911 0.582 0.924 
[2] 1.149 0.894 1.137 0.897 0.600 0.910 
[3] 1.283 0.891 1.202 0.894 0.618 0.907 
80 (50%) [1] 0.964 0.915 0.941 0.917 0.499 0.930 
[2] 1.001 0.912 0.995 0.915 0.523 0.928 
[3] 1.010 0.911 1.007 0.914 0.528 0.927 
80 (75%) [1] 0.580 0.924 0.370 0.930 0.267 0.942 
[2] 0.726 0.921 0.679 0.924 0.389 0.938 
[3] 0.896 0.918 0.863 0.921 0.432 0.935 
n (FP%)SchemeACIHPD-P1HPD-P2
T = 0.3
40 (50%) [1] 1.432 0.884 1.407 0.889 0.904 0.897 
[2] 1.586 0.881 1.558 0.885 0.914 0.893 
[3] 1.689 0.876 1.677 0.880 0.942 0.888 
40 (75%) [1] 1.181 0.906 1.172 0.910 0.611 0.919 
[2] 1.329 0.890 1.292 0.894 0.642 0.902 
[3] 1.361 0.888 1.320 0.892 0.767 0.900 
80 (50%) [1] 0.987 0.913 0.971 0.917 0.554 0.926 
[2] 1.075 0.912 0.996 0.916 0.573 0.925 
[3] 1.110 0.909 1.058 0.913 0.581 0.922 
80 (75%) [1] 0.778 0.922 0.732 0.927 0.450 0.936 
[2] 0.904 0.918 0.842 0.924 0.518 0.933 
[3] 0.965 0.916 0.936 0.920 0.547 0.929 
T = 0.7 
40 (50%) [1] 1.342 0.887 1.303 0.890 0.634 0.903 
[2] 1.497 0.884 1.459 0.887 0.685 0.900 
[3] 1.604 0.879 1.551 0.882 0.695 0.895 
40 (75%) [1] 1.057 0.908 1.011 0.911 0.582 0.924 
[2] 1.149 0.894 1.137 0.897 0.600 0.910 
[3] 1.283 0.891 1.202 0.894 0.618 0.907 
80 (50%) [1] 0.964 0.915 0.941 0.917 0.499 0.930 
[2] 1.001 0.912 0.995 0.915 0.523 0.928 
[3] 1.010 0.911 1.007 0.914 0.528 0.927 
80 (75%) [1] 0.580 0.924 0.370 0.930 0.267 0.942 
[2] 0.726 0.921 0.679 0.924 0.389 0.938 
[3] 0.896 0.918 0.863 0.921 0.432 0.935 
TABLE XVI.

The ACLs (first column) and CPs (second column) of 95% interval estimates of h(t) from set-II.

n (FP%)SchemeACIHPD-P1HPD-P2
T = 0.3
40 (50%) [1] 0.579 0.925 0.429 0.930 0.296 0.936 
[2] 0.598 0.924 0.450 0.929 0.323 0.935 
[3] 0.637 0.922 0.471 0.927 0.360 0.933 
40 (75%] [1] 0.471 0.941 0.335 0.946 0.257 0.952 
[2] 0.503 0.930 0.372 0.935 0.276 0.941 
[3] 0.539 0.928 0.394 0.933 0.286 0.939 
80 (50%) [1] 0.386 0.948 0.293 0.953 0.225 0.959 
[2] 0.413 0.946 0.308 0.951 0.233 0.957 
[3] 0.438 0.944 0.314 0.949 0.253 0.955 
80 (75%) [1] 0.329 0.954 0.235 0.959 0.170 0.965 
[2] 0.346 0.953 0.250 0.958 0.186 0.964 
[3] 0.366 0.951 0.276 0.956 0.219 0.962 
T = 0.7 
40 (50%) [1] 0.467 0.932 0.388 0.937 0.238 0.941 
[2] 0.487 0.931 0.416 0.936 0.277 0.939 
[3] 0.519 0.929 0.426 0.934 0.294 0.937 
40 (75%) [1] 0.361 0.949 0.330 0.953 0.177 0.956 
[2] 0.436 0.937 0.351 0.942 0.186 0.945 
[3] 0.460 0.935 0.364 0.940 0.213 0.943 
80 (50%) [1] 0.323 0.956 0.283 0.960 0.151 0.963 
[2] 0.337 0.954 0.297 0.958 0.159 0.961 
[3] 0.344 0.952 0.313 0.956 0.169 0.959 
80 (75%) [1] 0.242 0.962 0.187 0.966 0.129 0.969 
[2] 0.265 0.961 0.249 0.965 0.130 0.968 
[3] 0.299 0.959 0.268 0.963 0.137 0.967 
n (FP%)SchemeACIHPD-P1HPD-P2
T = 0.3
40 (50%) [1] 0.579 0.925 0.429 0.930 0.296 0.936 
[2] 0.598 0.924 0.450 0.929 0.323 0.935 
[3] 0.637 0.922 0.471 0.927 0.360 0.933 
40 (75%] [1] 0.471 0.941 0.335 0.946 0.257 0.952 
[2] 0.503 0.930 0.372 0.935 0.276 0.941 
[3] 0.539 0.928 0.394 0.933 0.286 0.939 
80 (50%) [1] 0.386 0.948 0.293 0.953 0.225 0.959 
[2] 0.413 0.946 0.308 0.951 0.233 0.957 
[3] 0.438 0.944 0.314 0.949 0.253 0.955 
80 (75%) [1] 0.329 0.954 0.235 0.959 0.170 0.965 
[2] 0.346 0.953 0.250 0.958 0.186 0.964 
[3] 0.366 0.951 0.276 0.956 0.219 0.962 
T = 0.7 
40 (50%) [1] 0.467 0.932 0.388 0.937 0.238 0.941 
[2] 0.487 0.931 0.416 0.936 0.277 0.939 
[3] 0.519 0.929 0.426 0.934 0.294 0.937 
40 (75%) [1] 0.361 0.949 0.330 0.953 0.177 0.956 
[2] 0.436 0.937 0.351 0.942 0.186 0.945 
[3] 0.460 0.935 0.364 0.940 0.213 0.943 
80 (50%) [1] 0.323 0.956 0.283 0.960 0.151 0.963 
[2] 0.337 0.954 0.297 0.958 0.159 0.961 
[3] 0.344 0.952 0.313 0.956 0.169 0.959 
80 (75%) [1] 0.242 0.962 0.187 0.966 0.129 0.969 
[2] 0.265 0.961 0.249 0.965 0.130 0.968 
[3] 0.299 0.959 0.268 0.963 0.137 0.967 

The optimal progressive censoring is utilized frequently in reliability testing, survival analysis, and quality control. Its importance is in boosting the efficiency and cost-effectiveness of experimentation and saving the experimental time. The important issue is how to decide on a certain censorship technique. Should we pick a plan just for its convenience of use or on statistical criteria? An investigator must select the optimal censoring strategy from a variety of alternatives, which is best in terms of providing the most information about the unknown parameters. Identifying an optimal censoring strategy for different contexts has recently been investigated by many researchers, for example, Wang and Yu,16 Pradhan and Kundu,17 and Elbatal et al.18 To decide on the optimal progressive censoring plan for the CUW distribution, we consider the following different criteria:

  1. The first criterion, denoted by O1, maximizes the trace of the observed Fisher information matrix, i.e., O1=Max(I1+I2), where I1=2(α,γ|z̲)α2(α,γ)=(α̂,γ̂) and I2=2(α,γ|z̲)γ2(α,γ)=(α̂,γ̂).

  2. The second criterion, denoted by O2, minimizes the trace of the estimated variance-covariance matrix as
  3. The third criterion, denoted by O3, minimizes the determinant of the estimated variance-covariance matrix as
  4. The fourth criterion, denoted by O3, minimizes the estimated variance of the MLE of the logarithm of qth quantile, where 0 < q < 1, as
    where
    and
    with ϱ = −log(1 − 1)/log(2).

To highlight how the proposed estimators can be used, in practice, for illustrative purposes, we shall present in this section the analysis of three real datasets from the veterinary medicine, industrial, and physical sectors. In Table XVII, we list the following datasets:

  • Veterinary medicine data (say data [1]): This dataset represents the total milk production at the first birth of one hundred and seven cows from the SINDI race. These cows are property of the Carnaúba farm, Agropecuária Manoel Dantas Ltda, Taperoá, Paraíba, Brazil. Recently, Cordeiro and dos Santos19 have reported and analyzed it.

  • Industrial data (say data [2]): This dataset consists of 50 records on burr [in the unit of millimeters (mm)], the hole diameter is 12 mm, and the sheet thickness is 3.15 mm. This dataset was given by Dasgupta20 and discussed by Gündüz and Korkmaz.21 

  • Physical data (say data [3]): This dataset provides 20 values of February water capacity from 1991 to 2010 at Shasta reservoir, California, USA; see the work of Nadar et al.22 

TABLE XVII.

Observation values of three real datasets.

DataObservations
[1] 0.4365 0.4260 0.5140 0.6907 0.7471 0.2605 0.6196 0.8781 0.4990 0.6058 
0.6891 0.5770 0.5394 0.1479 0.2356 0.6012 0.1525 0.5483 0.6927 0.7261 
0.3323 0.0671 0.2361 0.4800 0.5707 0.7131 0.5853 0.6768 0.5350 0.4151 
0.6789 0.4576 0.3259 0.2303 0.7687 0.4371 0.3383 0.6114 0.3480 0.4564 
0.7804 0.3406 0.4823 0.5912 0.5744 0.5481 0.1131 0.7290 0.0168 0.5529 
0.4530 0.3891 0.4752 0.3134 0.3175 0.1167 0.6750 0.5113 0.5447 0.4143 
0.5627 0.5150 0.0776 0.3945 0.4553 0.4470 0.5285 0.5232 0.6465 0.0650 
0.8492 0.8147 0.3627 0.3906 0.4438 0.4612 0.3188 0.2160 0.6707 0.6220 
0.5629 0.4675 0.6844 0.3413 0.4332 0.0854 0.3821 0.4694 0.3635 0.4111 
0.5349 0.3751 0.1546 0.4517 0.2681 0.4049 0.5553 0.5878 0.4741 0.3598 
0.7629 0.5941 0.6174 0.6860 0.0609 0.6488 0.2747    
[2] 0.04 0.02 0.06 0.12 0.14 0.08 0.22 0.12 0.08 0.26 
0.24 0.04 0.14 0.16 0.08 0.26 0.32 0.28 0.14 0.16 
0.24 0.22 0.12 0.18 0.24 0.32 0.16 0.14 0.08 0.16 
0.24 0.16 0.32 0.18 0.24 0.22 0.16 0.12 0.24 0.06 
0.02 0.18 0.22 0.14 0.06 0.04 0.14 0.26 0.18 0.16 
[3] 0.338 936 0.768 007 0.431 915 0.843 485 0.759 932 0.787 408 0.724 626 0.849 868 0.757 583 0.695 970 
0.811 556 0.842 316 0.785 339 0.828 689 0.783 660 0.580 194 0.815 627 0.430 681 0.847 413 0.742 563 
DataObservations
[1] 0.4365 0.4260 0.5140 0.6907 0.7471 0.2605 0.6196 0.8781 0.4990 0.6058 
0.6891 0.5770 0.5394 0.1479 0.2356 0.6012 0.1525 0.5483 0.6927 0.7261 
0.3323 0.0671 0.2361 0.4800 0.5707 0.7131 0.5853 0.6768 0.5350 0.4151 
0.6789 0.4576 0.3259 0.2303 0.7687 0.4371 0.3383 0.6114 0.3480 0.4564 
0.7804 0.3406 0.4823 0.5912 0.5744 0.5481 0.1131 0.7290 0.0168 0.5529 
0.4530 0.3891 0.4752 0.3134 0.3175 0.1167 0.6750 0.5113 0.5447 0.4143 
0.5627 0.5150 0.0776 0.3945 0.4553 0.4470 0.5285 0.5232 0.6465 0.0650 
0.8492 0.8147 0.3627 0.3906 0.4438 0.4612 0.3188 0.2160 0.6707 0.6220 
0.5629 0.4675 0.6844 0.3413 0.4332 0.0854 0.3821 0.4694 0.3635 0.4111 
0.5349 0.3751 0.1546 0.4517 0.2681 0.4049 0.5553 0.5878 0.4741 0.3598 
0.7629 0.5941 0.6174 0.6860 0.0609 0.6488 0.2747    
[2] 0.04 0.02 0.06 0.12 0.14 0.08 0.22 0.12 0.08 0.26 
0.24 0.04 0.14 0.16 0.08 0.26 0.32 0.28 0.14 0.16 
0.24 0.22 0.12 0.18 0.24 0.32 0.16 0.14 0.08 0.16 
0.24 0.16 0.32 0.18 0.24 0.22 0.16 0.12 0.24 0.06 
0.02 0.18 0.22 0.14 0.06 0.04 0.14 0.26 0.18 0.16 
[3] 0.338 936 0.768 007 0.431 915 0.843 485 0.759 932 0.787 408 0.724 626 0.849 868 0.757 583 0.695 970 
0.811 556 0.842 316 0.785 339 0.828 689 0.783 660 0.580 194 0.815 627 0.430 681 0.847 413 0.742 563 

Before going to evaluate the offered estimates of α, γ, R(t), or h(t), based on each dataset in Table XVII, we shall highlight the superiority of the CUW model. Thus, we consider the other nine models in the literature as competitors, named as follows:

  1. Unit-Weibull [UW(α, γ)] by Mazucheli et al.23 

  2. Unit-log–log [ULL(α, γ)] by Korkmaz and Korkmaz.24 

  3. Unit-Birnbaum–Saunders [UBS(α, γ)] by Mazucheli et al.25 

  4. Unit-Gompertz [UGo(α, γ)] by Mazucheli et al.26 

  5. Unit-half-normal [UHN(γ)] by Bakouch et al.27 

  6. Unit-Lindley [UL(σ)] by Mazucheli et al.28 

  7. Topp–Leone [TL(γ)] by Topp and Leone.29 

  8. Kumaraswamy [K(α, γ)] by Mitnik and Baek.30 

  9. Beta [B(α, γ)] by Gupta and Nadarajah.31 

The comparison of fits of CUW and its competitive models is made using several measures, called log-likelihood (LL), Akaike (A), Bayesian (B), consistent Akaike (CA), Hannan–Quinn (HQ), Anderson–Darling (AD), Cramér–von Mises (CM), and Kolmogorov–Smirnov (KS) along its P-value. Using Table XVII, these goodness-of-fit measures are evaluated through the MLEs [along with associated standard-errors (St.Errs)] of α and γ; see Table XVIII. It indicates that the suggested CUW distribution has the lowest values of all the given measures except that it has the highest P-value. This conclusion implies that the proposed model in this study provides a better fit than others.

TABLE XVIII.

Fitting results of the CUW and its competitive models from three real datasets.

MLE(St.Err)
ModelασLLABCAHQADCMKS(P-value)
Data [1] 
CUW 1.7809 (0.1355) 0.4738 (0.0214) −27.504 −51.008 −45.662 −50.893 −48.841 0.613 0.095 0.066 (0.734) 
UW 0.9845 (0.1015) 1.5620 (0.1064) −16.921 −29.842 −24.497 −29.727 −27.675 2.425 0.396 0.121 (0.089) 
ULL 0.9670 (0.0678) 1.8782 (0.0941) −3.2152 −2.4305 −2.9152 −2.3151 −0.2634 4.183 0.722 0.175 (0.003) 
UBS 0.6546 (0.0447) 0.7408 (0.0444) −27.206 −50.412 −45.067 −50.297 −48.245 0.648 0.104 0.076 (0.558) 
UGo 2.1193 (0.8683) 0.3878 (0.1145) −5.4887 −6.9774 −1.6318 −6.8620 −4.8104 3.095 0.521 0.183 (0.001) 
UHN ⋯ 1.6343 (0.1117) −20.682 −39.364 −36.691 −39.326 −38.281 1.380 0.236 0.118 (0.101) 
UL ⋯ 1.2001 (0.0889) −25.380 −48.761 −45.088 −48.723 −47.677 0.620 0.104 0.110 (0.153) 
TL ⋯ 2.0802 (0.2011) −21.526 −41.052 −38.380 −41.014 −39.969 1.479 0.233 0.097 (0.264) 
2.1949 (0.2224) 3.4363 (0.5820) −25.395 −46.789 −41.444 −46.674 −44.622 1.009 0.156 0.076 (0.563) 
2.4125 (0.3145) 2.8297 (0.3744) −23.777 −43.554 −38.209 −43.439 −41.387 1.326 0.208 0.091 (0.338) 
Data [2] 
CUW 1.9360 (0.2228) 0.1565 (0.0123) −56.490 −108.98 −105.16 −108.72 −107.52 0.560 0.092 0.104 (0.656) 
UW 0.0876 (0.0288) 3.0516 (0.3079) −48.663 −93.325 −89.501 −93.070 −91.869 1.867 0.321 0.182 (0.073) 
ULL 1.9750 (0.2143) 1.1394 (0.0395) −43.133 −82.266 −78.442 −82.011 −80.810 2.632 0.458 0.219 (0.016) 
UBS 0.3120 (0.0312) 1.8948 (0.0826) −55.395 −106.79 −102.97 −106.53 −105.33 0.771 0.128 0.133 (0.341) 
UGo 40.689 (32.481) 0.0124 (0.0100) −14.462 −24.925 −21.101 −24.669 −23.468 1.039 0.173 0.436 (0.001) 
UHN ⋯ 0.2380 (0.0238) −53.778 −105.56 −103.64 −105.47 −104.83 0.626 0.264 0.193 (0.047) 
UL ⋯ 5.5829 (0.6977) −47.577 −93.155 −91.243 −93.071 −92.427 0.717 0.119 0.239 (0.007) 
TL ⋯ 0.7248 (0.1025) −28.408 −54.816 −52.904 −54.732 −54.088 0.992 0.165 0.362 (0.003) 
2.0751 (0.2542) 33.003 (13.832) −56.069 −108.14 −104.31 −107.88 −106.68 0.625 0.102 0.110 (0.576) 
2.6826 (0.5072) 13.8659 (2.8281) −54.607 −105.21 −101.39 −104.96 −103.76 0.893 0.148 0.141 (0.270) 
Data [3] 
CUW 3.8637 (0.7582) 0.7530 (0.0233) −16.024 −28.049 −26.057 −27.343 −27.660 0.922 0.144 0.168 (0.569) 
UW 4.2071 (1.1202) 1.5704 (0.2483) −10.957 −17.914 −15.922 −17.208 −17.525 1.874 0.332 0.242 (0.164) 
ULL 1.0375 (0.1846) 5.0257 (1.1194) −8.6847 −13.369 −11.378 −12.664 −12.981 2.220 0.405 0.274 (0.081) 
UBS 0.5782 (0.0914) 0.3052 (0.0378) −14.812 −25.624 −23.633 −24.918 −25.235 1.217 0.201 0.217 (0.262) 
UGo 1.6931 (1.6275) 1.1604 (0.7682) −8.8489 −13.698 −11.706 −12.992 −13.309 2.044 0.368 0.297 (0.047) 
UHN ⋯ 3.7200 (0.5882) −14.756 −27.512 −26.016 −27.289 −27.317 0.967 0.197 0.271 (0.087) 
UL ⋯ 0.4957 (0.0806) −13.827 −25.654 −24.659 −25.432 −25.460 0.983 0.199 0.242 (0.162) 
TL ⋯ 8.6668 (1.9379) −11.588 −21.175 −20.180 −20.953 −20.981 1.786 0.313 0.255 (0.124) 
6.3480 (1.5576) 4.4899 (2.0414) −13.475 −22.949 −20.958 −22.244 −22.561 1.424 0.241 0.221 (0.245) 
7.3155 (2.3181) 2.9098 (0.8754) −12.562 −21.124 −19.132 −20.418 −20.735 1.619 0.280 0.236 (0.183) 
MLE(St.Err)
ModelασLLABCAHQADCMKS(P-value)
Data [1] 
CUW 1.7809 (0.1355) 0.4738 (0.0214) −27.504 −51.008 −45.662 −50.893 −48.841 0.613 0.095 0.066 (0.734) 
UW 0.9845 (0.1015) 1.5620 (0.1064) −16.921 −29.842 −24.497 −29.727 −27.675 2.425 0.396 0.121 (0.089) 
ULL 0.9670 (0.0678) 1.8782 (0.0941) −3.2152 −2.4305 −2.9152 −2.3151 −0.2634 4.183 0.722 0.175 (0.003) 
UBS 0.6546 (0.0447) 0.7408 (0.0444) −27.206 −50.412 −45.067 −50.297 −48.245 0.648 0.104 0.076 (0.558) 
UGo 2.1193 (0.8683) 0.3878 (0.1145) −5.4887 −6.9774 −1.6318 −6.8620 −4.8104 3.095 0.521 0.183 (0.001) 
UHN ⋯ 1.6343 (0.1117) −20.682 −39.364 −36.691 −39.326 −38.281 1.380 0.236 0.118 (0.101) 
UL ⋯ 1.2001 (0.0889) −25.380 −48.761 −45.088 −48.723 −47.677 0.620 0.104 0.110 (0.153) 
TL ⋯ 2.0802 (0.2011) −21.526 −41.052 −38.380 −41.014 −39.969 1.479 0.233 0.097 (0.264) 
2.1949 (0.2224) 3.4363 (0.5820) −25.395 −46.789 −41.444 −46.674 −44.622 1.009 0.156 0.076 (0.563) 
2.4125 (0.3145) 2.8297 (0.3744) −23.777 −43.554 −38.209 −43.439 −41.387 1.326 0.208 0.091 (0.338) 
Data [2] 
CUW 1.9360 (0.2228) 0.1565 (0.0123) −56.490 −108.98 −105.16 −108.72 −107.52 0.560 0.092 0.104 (0.656) 
UW 0.0876 (0.0288) 3.0516 (0.3079) −48.663 −93.325 −89.501 −93.070 −91.869 1.867 0.321 0.182 (0.073) 
ULL 1.9750 (0.2143) 1.1394 (0.0395) −43.133 −82.266 −78.442 −82.011 −80.810 2.632 0.458 0.219 (0.016) 
UBS 0.3120 (0.0312) 1.8948 (0.0826) −55.395 −106.79 −102.97 −106.53 −105.33 0.771 0.128 0.133 (0.341) 
UGo 40.689 (32.481) 0.0124 (0.0100) −14.462 −24.925 −21.101 −24.669 −23.468 1.039 0.173 0.436 (0.001) 
UHN ⋯ 0.2380 (0.0238) −53.778 −105.56 −103.64 −105.47 −104.83 0.626 0.264 0.193 (0.047) 
UL ⋯ 5.5829 (0.6977) −47.577 −93.155 −91.243 −93.071 −92.427 0.717 0.119 0.239 (0.007) 
TL ⋯ 0.7248 (0.1025) −28.408 −54.816 −52.904 −54.732 −54.088 0.992 0.165 0.362 (0.003) 
2.0751 (0.2542) 33.003 (13.832) −56.069 −108.14 −104.31 −107.88 −106.68 0.625 0.102 0.110 (0.576) 
2.6826 (0.5072) 13.8659 (2.8281) −54.607 −105.21 −101.39 −104.96 −103.76 0.893 0.148 0.141 (0.270) 
Data [3] 
CUW 3.8637 (0.7582) 0.7530 (0.0233) −16.024 −28.049 −26.057 −27.343 −27.660 0.922 0.144 0.168 (0.569) 
UW 4.2071 (1.1202) 1.5704 (0.2483) −10.957 −17.914 −15.922 −17.208 −17.525 1.874 0.332 0.242 (0.164) 
ULL 1.0375 (0.1846) 5.0257 (1.1194) −8.6847 −13.369 −11.378 −12.664 −12.981 2.220 0.405 0.274 (0.081) 
UBS 0.5782 (0.0914) 0.3052 (0.0378) −14.812 −25.624 −23.633 −24.918 −25.235 1.217 0.201 0.217 (0.262) 
UGo 1.6931 (1.6275) 1.1604 (0.7682) −8.8489 −13.698 −11.706 −12.992 −13.309 2.044 0.368 0.297 (0.047) 
UHN ⋯ 3.7200 (0.5882) −14.756 −27.512 −26.016 −27.289 −27.317 0.967 0.197 0.271 (0.087) 
UL ⋯ 0.4957 (0.0806) −13.827 −25.654 −24.659 −25.432 −25.460 0.983 0.199 0.242 (0.162) 
TL ⋯ 8.6668 (1.9379) −11.588 −21.175 −20.180 −20.953 −20.981 1.786 0.313 0.255 (0.124) 
6.3480 (1.5576) 4.4899 (2.0414) −13.475 −22.949 −20.958 −22.244 −22.561 1.424 0.241 0.221 (0.245) 
7.3155 (2.3181) 2.9098 (0.8754) −12.562 −21.124 −19.132 −20.418 −20.735 1.619 0.280 0.236 (0.183) 

Furthermore, based on the entire datasets in Table XVII, the superiority of the CUW model is tested through various graphical drawing tools, namely, (i) probability–probability (PP), (ii) estimated PDFs, (iii) estimated/empirical RFs, (iv) fitted/empirical scaled total-time-on-test (TTT), and (v) contours; see Figs. 59. As we anticipated, the facts shown in Figs. 57 confirm the same results presented in Table XVIII. Figure 8 indicates that all given datasets have increasing failure rates. Lately, the natural logarithm of the likelihood function in Fig. 9 proves that the acquired MLEs of α and γ existed and are unique.

FIG. 5.

The PP plots from three real datasets. (a) Data [1]. (b) Data [2]. (c) Data [3].

FIG. 5.

The PP plots from three real datasets. (a) Data [1]. (b) Data [2]. (c) Data [3].

Close modal
FIG. 6.

The PDF plots with histograms from three real datasets. (a) Data [1]. (b) Data [2]. (c) Data [3].

FIG. 6.

The PDF plots with histograms from three real datasets. (a) Data [1]. (b) Data [2]. (c) Data [3].

Close modal
FIG. 7.

The RF plots from three real datasets. (a) Data [1]. (b) Data [2]. (c) Data [3].

FIG. 7.

The RF plots from three real datasets. (a) Data [1]. (b) Data [2]. (c) Data [3].

Close modal
FIG. 8.

The TTT plots from three real datasets. (a) Data [1]. (b) Data [2]. (c) Data [3].

FIG. 8.

The TTT plots from three real datasets. (a) Data [1]. (b) Data [2]. (c) Data [3].

Close modal
FIG. 9.

The contours of the log-likelihood of CUW(α, γ) from three real datasets. (a) Data [1]. (b) Data [2]. (c) Data [3].

FIG. 9.

The contours of the log-likelihood of CUW(α, γ) from three real datasets. (a) Data [1]. (b) Data [2]. (c) Data [3].

Close modal

From Table XVII, to obtain both point and interval estimates of α, γ, R(t), and h(t), different artificial APT2-HC samples based on various selects of T, n, m, and R are created; see Table XIX. For each artificial sample, the maximum likelihood and MCMC (with their St.Errs) estimates and 95% ACI/HPD estimates [with their interval lengths (ILs)] of α, γ, R(t), and h(t) (at t(=0.1, 0.6) for Data [1, 2] and [3]) are obtained; see Table XX. To carry out the Bayes evaluation for each unknown quantity, we generate 50 000 MCMC samples and then eliminate the first 10 000 samples (burn-in). Since prior information about the CUW parameters from all real datasets is not available, the Bayes’ estimates α, γ, R(t), and h(t) are developed based on improper priors. Point estimations in Table XX, due to a lack of prior knowledge, exhibited that the likelihood estimates for α, γ, R(t), and h(t) are quite approximate to those obtained through the Markov Chain Monte Carlo method. A similar conclusion is also observed when comparing the proposed interval estimations.

TABLE XIX.

Artificial APT2-HC samples from three real datasets.

Data (sample)SchemeT(d)R*Artificial data
Data[1](S1) (253, 0290.07 (2) 25 0.0168, 0.0671, 0.0854, 0.1131, 0.1525, 0.2160, 0.2361, 0.2605, 0.3134, 0.3175 
0.3188, 0.3259, 0.332 30.3406, 0.3635, 0.3821, 0.3891, 0.3906, 0.3945, 0.4049 
0.4151, 0.4260, 0.4530, 0.4553, 0.4564, 0.4612, 0.4741, 0.4752, 0.4800, 0.4990 
0.5150, 0.5349 
Data[1](S2) (014, 253, 0150.25 (15) 50 0.0168, 0.0609, 0.0650, 0.0671, 0.0776, 0.0854, 0.1131, 0.1167, 0.1479, 0.1525 
0.1546, 0.2160, 0.2303, 0.2356, 0.2361, 0.2605, 0.2747, 0.3175, 0.3323, 0.3383 
0.3627, 0.3635, 0.3906, 0.3945, 0.4049, 0.414 30.4260, 0.4332, 0.4470, 0.4530 
0.4612, 0.4741 
Data[1](S3) (029, 2530.40 (32) 0.0168, 0.0609, 0.0650, 0.0671, 0.0776, 0.0854, 0.1131, 0.1167, 0.1479, 0.1525 
0.1546, 0.2160, 0.2303, 0.2356, 0.2361, 0.2605, 0.2681, 0.2747, 0.3134, 0.3175 
0.3188, 0.3259, 0.3323, 0.3383, 0.3406, 0.3413, 0.3480, 0.3598, 0.3627, 0.3635 
0.3891, 0.3945 
Data[2](S1) (103, 0170.03 (2) 10 0.02, 0.02, 0.06, 0.06, 0.08, 0.08, 0.12, 0.12, 0.14, 0.14, 0.14, 0.14, 0.16, 0.16 
0.16, 0.16, 0.16, 0.16, 0.16, 0.18 
Data[2](S2) (08, 103, 090.09 (9) 20 0.02, 0.02, 0.04, 0.04, 0.04, 0.06, 0.06, 0.06, 0.08, 0.12, 0.12, 0.14, 0.14, 0.16 
0.16, 0.16, 0.16, 0.16, 0.18, 0.22 
Data[2](S3) (017, 1030.25 (20) 0.02, 0.02, 0.04, 0.04, 0.04, 0.06, 0.06, 0.06, 0.08, 0.08, 0.08, 0.08, 0.12, 0.12 
0.12, 0.12, 0.14, 0.14, 0.22, 0.24 
Data[3](S1) (52, 080.34 (1) 0.338 936, 0.431 915, 0.580 194, 0.695 970, 0.742 563, 0.757 583, 0.759 932, 0.768 007 
0.783 660, 0.785 339 
Data[3](S2) (04, 52, 040.72 (5) 0.338 936, 0.430 681, 0.431 915, 0.580 194, 0.695 970, 0.742 563, 0.757 583, 0.768 007 
0.783 660, 0.811 556 
Data[3](S3) (08, 520.82 (10) 0.338 936, 0.430 681, 0.431 915, 0.580 194, 0.695 970, 0.724 626, 0.742 563, 0.757 583 
0.759 932, 0.815 627 
Data (sample)SchemeT(d)R*Artificial data
Data[1](S1) (253, 0290.07 (2) 25 0.0168, 0.0671, 0.0854, 0.1131, 0.1525, 0.2160, 0.2361, 0.2605, 0.3134, 0.3175 
0.3188, 0.3259, 0.332 30.3406, 0.3635, 0.3821, 0.3891, 0.3906, 0.3945, 0.4049 
0.4151, 0.4260, 0.4530, 0.4553, 0.4564, 0.4612, 0.4741, 0.4752, 0.4800, 0.4990 
0.5150, 0.5349 
Data[1](S2) (014, 253, 0150.25 (15) 50 0.0168, 0.0609, 0.0650, 0.0671, 0.0776, 0.0854, 0.1131, 0.1167, 0.1479, 0.1525 
0.1546, 0.2160, 0.2303, 0.2356, 0.2361, 0.2605, 0.2747, 0.3175, 0.3323, 0.3383 
0.3627, 0.3635, 0.3906, 0.3945, 0.4049, 0.414 30.4260, 0.4332, 0.4470, 0.4530 
0.4612, 0.4741 
Data[1](S3) (029, 2530.40 (32) 0.0168, 0.0609, 0.0650, 0.0671, 0.0776, 0.0854, 0.1131, 0.1167, 0.1479, 0.1525 
0.1546, 0.2160, 0.2303, 0.2356, 0.2361, 0.2605, 0.2681, 0.2747, 0.3134, 0.3175 
0.3188, 0.3259, 0.3323, 0.3383, 0.3406, 0.3413, 0.3480, 0.3598, 0.3627, 0.3635 
0.3891, 0.3945 
Data[2](S1) (103, 0170.03 (2) 10 0.02, 0.02, 0.06, 0.06, 0.08, 0.08, 0.12, 0.12, 0.14, 0.14, 0.14, 0.14, 0.16, 0.16 
0.16, 0.16, 0.16, 0.16, 0.16, 0.18 
Data[2](S2) (08, 103, 090.09 (9) 20 0.02, 0.02, 0.04, 0.04, 0.04, 0.06, 0.06, 0.06, 0.08, 0.12, 0.12, 0.14, 0.14, 0.16 
0.16, 0.16, 0.16, 0.16, 0.18, 0.22 
Data[2](S3) (017, 1030.25 (20) 0.02, 0.02, 0.04, 0.04, 0.04, 0.06, 0.06, 0.06, 0.08, 0.08, 0.08, 0.08, 0.12, 0.12 
0.12, 0.12, 0.14, 0.14, 0.22, 0.24 
Data[3](S1) (52, 080.34 (1) 0.338 936, 0.431 915, 0.580 194, 0.695 970, 0.742 563, 0.757 583, 0.759 932, 0.768 007 
0.783 660, 0.785 339 
Data[3](S2) (04, 52, 040.72 (5) 0.338 936, 0.430 681, 0.431 915, 0.580 194, 0.695 970, 0.742 563, 0.757 583, 0.768 007 
0.783 660, 0.811 556 
Data[3](S3) (08, 520.82 (10) 0.338 936, 0.430 681, 0.431 915, 0.580 194, 0.695 970, 0.724 626, 0.742 563, 0.757 583 
0.759 932, 0.815 627 
TABLE XX.

Estimates of α, γ, R(t), and h(t) from three real datasets.

Data (sample)MLEMCMC95% ACI95% HPD
ParEstSEEstSELowerUpperILLowerUpperIL
Data[1](S1) α 1.6923 0.2564 1.6821 0.0940 1.1897 2.1949 1.0051 1.4953 1.8619 0.3666 
γ 0.5039 0.0365 0.5054 0.0348 0.4324 0.5754 0.1430 0.4406 0.5749 0.1343 
R(0.1) 0.9723 0.0067 0.9711 0.0070 0.9592 0.9854 0.0262 0.9577 0.9837 0.0260 
h(0.1) 0.5006 0.1821 0.5148 0.1059 0.1438 0.8575 0.7136 0.3231 0.7262 0.4031 
Data[1](S2) α 1.2907 0.2073 1.2835 0.0895 0.8843 1.6971 0.8127 1.1109 1.4614 0.3505 
γ 0.5984 0.0590 0.5997 0.0458 0.4828 0.7141 0.2313 0.5127 0.6922 0.1794 
R(0.1) 0.9582 0.0134 0.9566 0.0095 0.9318 0.9845 0.0527 0.9382 0.9739 0.0357 
h(0.1) 0.5817 0.1457 0.5949 0.1062 0.2960 0.8674 0.5713 0.3900 0.7969 0.4069 
Data[1](S3) α 1.9265 0.2942 1.9229 0.0497 1.3499 2.5031 1.1532 1.8271 2.0210 0.1938 
γ 0.2305 0.0217 0.2319 0.0178 0.1880 0.2731 0.0851 0.1989 0.2676 0.0688 
R(0.1) 0.8871 0.0199 0.8861 0.0190 0.8480 0.9261 0.0781 0.8496 0.9231 0.0735 
h(0.1) 2.4341 0.6944 2.4521 0.4176 1.0732 3.7950 2.7218 1.6627 3.2712 1.6085 
Data[2](S1) α 2.1961 0.4279 2.1961 0.0099 1.3575 3.0347 1.6772 2.1768 2.2157 0.0390 
γ 0.1540 0.0145 0.1539 0.0082 0.1256 0.1824 0.0567 0.1378 0.1699 0.0321 
R(0.1) 0.7778 0.0437 0.7757 0.0253 0.6921 0.8635 0.1713 0.7241 0.8222 0.0982 
h(0.1) 5.8191 1.3090 5.8948 0.7605 3.2535 8.3847 5.1312 4.4322 7.3677 2.9355 
Data[2](S2) α 1.2845 0.2535 1.2844 0.0099 0.7877 1.7813 0.9936 1.2651 1.3041 0.0390 
γ 0.2280 0.0363 0.2277 0.0096 0.1569 0.2992 0.1423 0.2090 0.2468 0.0378 
R(0.1) 0.8037 0.0434 0.8028 0.0111 0.7186 0.8889 0.1703 0.7804 0.8237 0.0433 
h(0.1) 2.9597 0.6705 2.9759 0.1861 1.6456 4.2738 2.6283 2.6282 3.3521 0.7239 
Data[2](S3) α 0.9905 0.2005 0.9903 0.0099 0.5976 1.3833 0.7858 0.9709 1.0099 0.0390 
γ 0.2294 0.0458 0.2290 0.0098 0.1396 0.3192 0.1796 0.2101 0.2486 0.0384 
R(0.1) 0.7537 0.0496 0.7529 0.0106 0.6566 0.8509 0.1943 0.7318 0.7730 0.0412 
h(0.1) 2.9529 0.6792 2.9648 0.1446 1.6218 4.2841 2.6624 2.6908 3.2538 0.5630 
Data[3](S1) α 3.3874 0.9640 3.3858 0.0500 1.4980 5.2769 3.7789 3.2858 3.4822 0.1965 
γ 0.7553 0.0323 0.7560 0.0268 0.6919 0.8187 0.1268 0.7032 0.8081 0.1049 
R(0.6) 0.8506 0.0745 0.8479 0.0367 0.7045 0.9966 0.2920 0.7740 0.9150 0.1409 
h(0.6) 1.4959 0.5328 1.5321 0.4038 0.4516 2.5401 2.0886 0.7799 2.3182 1.5382 
Data[3](S2) α 2.6083 0.7407 2.6069 0.0498 1.1567 4.0600 2.9034 2.5116 2.7069 0.1954 
γ 0.7841 0.0404 0.7833 0.0310 0.7050 0.8632 0.1582 0.7248 0.8460 0.1212 
R(0.6) 0.8343 0.0882 0.8305 0.0378 0.6614 0.9989 0.3375 0.7537 0.8988 0.1451 
h(0.6) 1.2892 0.4157 1.3282 0.3271 0.4744 2.1040 1.6296 0.7471 1.9905 1.2434 
Data[3](S3) α 2.3749 0.6922 2.3733 0.0498 1.0183 3.7315 2.7132 2.2778 2.4734 0.1956 
γ 0.7976 0.0435 0.7964 0.0323 0.7123 0.8829 0.1706 0.7337 0.8599 0.1262 
R(0.6) 0.8310 0.0934 0.8270 0.0375 0.6479 0.9991 0.3512 0.7529 0.8972 0.1443 
h(0.6) 1.1996 0.3834 1.2363 0.2959 0.4481 1.9510 1.5029 0.6906 1.8191 1.1285 
Data (sample)MLEMCMC95% ACI95% HPD
ParEstSEEstSELowerUpperILLowerUpperIL
Data[1](S1) α 1.6923 0.2564 1.6821 0.0940 1.1897 2.1949 1.0051 1.4953 1.8619 0.3666 
γ 0.5039 0.0365 0.5054 0.0348 0.4324 0.5754 0.1430 0.4406 0.5749 0.1343 
R(0.1) 0.9723 0.0067 0.9711 0.0070 0.9592 0.9854 0.0262 0.9577 0.9837 0.0260 
h(0.1) 0.5006 0.1821 0.5148 0.1059 0.1438 0.8575 0.7136 0.3231 0.7262 0.4031 
Data[1](S2) α 1.2907 0.2073 1.2835 0.0895 0.8843 1.6971 0.8127 1.1109 1.4614 0.3505 
γ 0.5984 0.0590 0.5997 0.0458 0.4828 0.7141 0.2313 0.5127 0.6922 0.1794 
R(0.1) 0.9582 0.0134 0.9566 0.0095 0.9318 0.9845 0.0527 0.9382 0.9739 0.0357 
h(0.1) 0.5817 0.1457 0.5949 0.1062 0.2960 0.8674 0.5713 0.3900 0.7969 0.4069 
Data[1](S3) α 1.9265 0.2942 1.9229 0.0497 1.3499 2.5031 1.1532 1.8271 2.0210 0.1938 
γ 0.2305 0.0217 0.2319 0.0178 0.1880 0.2731 0.0851 0.1989 0.2676 0.0688 
R(0.1) 0.8871 0.0199 0.8861 0.0190 0.8480 0.9261 0.0781 0.8496 0.9231 0.0735 
h(0.1) 2.4341 0.6944 2.4521 0.4176 1.0732 3.7950 2.7218 1.6627 3.2712 1.6085 
Data[2](S1) α 2.1961 0.4279 2.1961 0.0099 1.3575 3.0347 1.6772 2.1768 2.2157 0.0390 
γ 0.1540 0.0145 0.1539 0.0082 0.1256 0.1824 0.0567 0.1378 0.1699 0.0321 
R(0.1) 0.7778 0.0437 0.7757 0.0253 0.6921 0.8635 0.1713 0.7241 0.8222 0.0982 
h(0.1) 5.8191 1.3090 5.8948 0.7605 3.2535 8.3847 5.1312 4.4322 7.3677 2.9355 
Data[2](S2) α 1.2845 0.2535 1.2844 0.0099 0.7877 1.7813 0.9936 1.2651 1.3041 0.0390 
γ 0.2280 0.0363 0.2277 0.0096 0.1569 0.2992 0.1423 0.2090 0.2468 0.0378 
R(0.1) 0.8037 0.0434 0.8028 0.0111 0.7186 0.8889 0.1703 0.7804 0.8237 0.0433 
h(0.1) 2.9597 0.6705 2.9759 0.1861 1.6456 4.2738 2.6283 2.6282 3.3521 0.7239 
Data[2](S3) α 0.9905 0.2005 0.9903 0.0099 0.5976 1.3833 0.7858 0.9709 1.0099 0.0390 
γ 0.2294 0.0458 0.2290 0.0098 0.1396 0.3192 0.1796 0.2101 0.2486 0.0384 
R(0.1) 0.7537 0.0496 0.7529 0.0106 0.6566 0.8509 0.1943 0.7318 0.7730 0.0412 
h(0.1) 2.9529 0.6792 2.9648 0.1446 1.6218 4.2841 2.6624 2.6908 3.2538 0.5630 
Data[3](S1) α 3.3874 0.9640 3.3858 0.0500 1.4980 5.2769 3.7789 3.2858 3.4822 0.1965 
γ 0.7553 0.0323 0.7560 0.0268 0.6919 0.8187 0.1268 0.7032 0.8081 0.1049 
R(0.6) 0.8506 0.0745 0.8479 0.0367 0.7045 0.9966 0.2920 0.7740 0.9150 0.1409 
h(0.6) 1.4959 0.5328 1.5321 0.4038 0.4516 2.5401 2.0886 0.7799 2.3182 1.5382 
Data[3](S2) α 2.6083 0.7407 2.6069 0.0498 1.1567 4.0600 2.9034 2.5116 2.7069 0.1954 
γ 0.7841 0.0404 0.7833 0.0310 0.7050 0.8632 0.1582 0.7248 0.8460 0.1212 
R(0.6) 0.8343 0.0882 0.8305 0.0378 0.6614 0.9989 0.3375 0.7537 0.8988 0.1451 
h(0.6) 1.2892 0.4157 1.3282 0.3271 0.4744 2.1040 1.6296 0.7471 1.9905 1.2434 
Data[3](S3) α 2.3749 0.6922 2.3733 0.0498 1.0183 3.7315 2.7132 2.2778 2.4734 0.1956 
γ 0.7976 0.0435 0.7964 0.0323 0.7123 0.8829 0.1706 0.7337 0.8599 0.1262 
R(0.6) 0.8310 0.0934 0.8270 0.0375 0.6479 0.9991 0.3512 0.7529 0.8972 0.1443 
h(0.6) 1.1996 0.3834 1.2363 0.2959 0.4481 1.9510 1.5029 0.6906 1.8191 1.1285 

To check the existence and uniqueness of the acquired MLEs of α̂ and γ̂ calculated based on data[i](Si) for i = 1, 2, 3, Fig. 10 states that all estimates of α or γ exist and are unique, as well as supporting all classical results listed in Table XX.

FIG. 10.

Profile log-likelihoods of α (left) and γ (right) from three datasets.

FIG. 10.

Profile log-likelihoods of α (left) and γ (right) from three datasets.

Close modal

To show the efficiency of the MCMC results of α, γ, R(t), and h(t), density (along its Gaussian kernel) and trace plots based on their staying 40 000 iterations from samples Data[i](S1) for i = 1, 2, 3 (as an example) are shown in Fig. 11. For brevity, other density and trace plots of the same parameters based on the other artificial samples are presented in the supplementary material. To distinguish, in Fig. 11, the sample mean and two bounds of 95% HPD intervals are shown via solid and dashed lines, respectively. It demonstrates that after 40 000 MCMC iterations, the convergence is satisfactory; indicating that discarding the initial 10 000 iterations is effective in removing the influence of the initial values. It is also evident that we have the following:

  • From data[1](S1): Estimates of α are fairly symmetrical, and those of γ and h(t) are positively skewed, while those of R(t) are negatively skewed.

  • From data[i](S1) for i = 2, 3: Estimates of α and γ are fairly symmetrical, and those of R(t) are negatively skewed, while those of h(t) are positively skewed.

FIG. 11.

Density (left) and trace (right) plots of α, γ, R(t), and h(t) from three real datasets.

FIG. 11.

Density (left) and trace (right) plots of α, γ, R(t), and h(t) from three real datasets.

Close modal

Moreover, in Table XXI, several properties, such as the mean, median, mode, first quartile (Q1), third quartile (Q3), standard deviation (St.D), and skewness of α, γ, R(t), and h(t), are calculated based on their saying 40 000 MCMC iterations. All results in Table XXI support the same findings shown in Table XX.

TABLE XXI.

Several statistics of α, γ, R(t), and h(t) from three real datasets.

Data (sample)ParMeanModeQ1MedianQ3St.DSkew
Data[1](S1) α 1.682 11 1.720 75 1.617 67 1.680 47 1.746 40 0.093 43 0.035 62 
γ 0.505 41 0.520 36 0.481 26 0.504 02 0.528 06 0.034 76 0.244 71 
R(0.1) 0.971 15 0.975 78 0.966 94 0.971 82 0.976 10 0.006 85 −0.575 15 
h(0.1) 0.514 84 0.444 91 0.440 53 0.505 97 0.581 76 0.104 95 0.447 61 
Data[1](S2) α 1.283 50 1.301 34 1.222 54 1.283 26 1.343 11 0.089 23 0.030 76 
γ 0.599 65 0.599 66 0.568 45 0.598 05 0.629 30 0.045 79 0.188 81 
R(0.1) 0.956 59 0.959 27 0.950 70 0.957 43 0.963 36 0.009 35 −0.515 18 
h(0.1) 0.594 87 0.570 68 0.519 65 0.587 94 0.663 17 0.105 43 0.369 18 
Data[1](S3) α 1.922 88 1.855 76 1.889 20 1.922 30 1.956 16 0.049 56 0.006 59 
γ 0.231 91 0.228 67 0.219 55 0.230 94 0.243 36 0.017 76 0.304 23 
R(0.1) 0.886 14 0.878 09 0.873 74 0.887 24 0.899 50 0.018 97 −0.318 57 
h(0.1) 2.452 11 2.239 08 2.156 43 2.427 99 2.718 98 0.417 23 0.355 99 
Data[2](S1) α 2.196 06 2.198 03 2.189 26 2.196 06 2.202 76 0.009 95 0.012 87 
γ 0.153 90 0.152 64 0.148 25 0.153 79 0.159 42 0.008 21 0.100 54 
R(0.1) 0.775 69 0.773 79 0.759 49 0.777 24 0.793 47 0.025 20 −0.312 01 
h(0.1) 5.894 82 5.944 63 5.357 59 5.838 98 6.372 17 0.756 73 0.409 58 
Data[2](S2) α 1.284 41 1.278 91 1.277 64 1.284 41 1.291 08 0.009 93 0.010 85 
γ 0.227 67 0.218 00 0.221 11 0.227 63 0.234 17 0.009 64 0.039 56 
R(0.1) 0.802 82 0.792 57 0.795 58 0.803 23 0.810 50 0.011 08 −0.205 78 
h(0.1) 2.975 88 3.056 83 2.846 44 2.967 48 3.096 42 0.185 42 0.251 58 
Data[2](S3) α 0.990 35 0.968 61 0.983 58 0.990 35 0.997 01 0.009 92 0.010 83 
γ 0.228 97 0.217 01 0.222 31 0.228 95 0.235 58 0.009 80 0.033 79 
R(0.1) 0.752 91 0.739 78 0.745 97 0.753 22 0.760 17 0.010 56 −0.166 53 
h(0.1) 2.964 82 2.846 83 2.864 97 2.959 22 3.058 77 0.144 12 0.217 97 
Data[3](S1) α 3.385 83 3.385 52 3.351 78 3.385 45 3.419 31 0.049 95 0.038 36 
γ 0.755 96 0.754 12 0.737 40 0.755 15 0.773 75 0.026 83 0.130 13 
R(0.6) 0.847 93 0.848 84 0.824 61 0.850 26 0.874 06 0.036 60 −0.416 71 
h(0.6) 1.532 11 1.513 78 1.244 34 1.498 79 1.781 57 0.402 15 0.552 95 
Data[3](S2) α 2.606 93 2.617 95 2.572 92 2.607 05 2.640 26 0.049 78 0.014 60 
γ 0.783 27 0.777 21 0.762 50 0.783 13 0.804 12 0.030 96 0.026 93 
R(0.6) 0.830 46 0.826 77 0.806 87 0.833 08 0.856 82 0.037 65 −0.420 83 
h(0.6) 1.328 21 1.358 78 1.097 62 1.298 31 1.523 75 0.324 73 0.568 40 
Data[3](S3) α 2.373 30 2.384 52 2.339 19 2.373 46 2.406 70 0.049 82 0.017 73 
γ 0.796 36 0.790 76 0.774 69 0.796 37 0.818 14 0.032 28 0.007 83 
R(0.6) 0.826 96 0.823 98 0.803 43 0.829 39 0.853 03 0.037 31 −0.389 56 
h(0.6) 1.236 29 1.259 63 1.028 03 1.210 53 1.414 55 0.293 58 0.532 78 
Data (sample)ParMeanModeQ1MedianQ3St.DSkew
Data[1](S1) α 1.682 11 1.720 75 1.617 67 1.680 47 1.746 40 0.093 43 0.035 62 
γ 0.505 41 0.520 36 0.481 26 0.504 02 0.528 06 0.034 76 0.244 71 
R(0.1) 0.971 15 0.975 78 0.966 94 0.971 82 0.976 10 0.006 85 −0.575 15 
h(0.1) 0.514 84 0.444 91 0.440 53 0.505 97 0.581 76 0.104 95 0.447 61 
Data[1](S2) α 1.283 50 1.301 34 1.222 54 1.283 26 1.343 11 0.089 23 0.030 76 
γ 0.599 65 0.599 66 0.568 45 0.598 05 0.629 30 0.045 79 0.188 81 
R(0.1) 0.956 59 0.959 27 0.950 70 0.957 43 0.963 36 0.009 35 −0.515 18 
h(0.1) 0.594 87 0.570 68 0.519 65 0.587 94 0.663 17 0.105 43 0.369 18 
Data[1](S3) α 1.922 88 1.855 76 1.889 20 1.922 30 1.956 16 0.049 56 0.006 59 
γ 0.231 91 0.228 67 0.219 55 0.230 94 0.243 36 0.017 76 0.304 23 
R(0.1) 0.886 14 0.878 09 0.873 74 0.887 24 0.899 50 0.018 97 −0.318 57 
h(0.1) 2.452 11 2.239 08 2.156 43 2.427 99 2.718 98 0.417 23 0.355 99 
Data[2](S1) α 2.196 06 2.198 03 2.189 26 2.196 06 2.202 76 0.009 95 0.012 87 
γ 0.153 90 0.152 64 0.148 25 0.153 79 0.159 42 0.008 21 0.100 54 
R(0.1) 0.775 69 0.773 79 0.759 49 0.777 24 0.793 47 0.025 20 −0.312 01 
h(0.1) 5.894 82 5.944 63 5.357 59 5.838 98 6.372 17 0.756 73 0.409 58 
Data[2](S2) α 1.284 41 1.278 91 1.277 64 1.284 41 1.291 08 0.009 93 0.010 85 
γ 0.227 67 0.218 00 0.221 11 0.227 63 0.234 17 0.009 64 0.039 56 
R(0.1) 0.802 82 0.792 57 0.795 58 0.803 23 0.810 50 0.011 08 −0.205 78 
h(0.1) 2.975 88 3.056 83 2.846 44 2.967 48 3.096 42 0.185 42 0.251 58 
Data[2](S3) α 0.990 35 0.968 61 0.983 58 0.990 35 0.997 01 0.009 92 0.010 83 
γ 0.228 97 0.217 01 0.222 31 0.228 95 0.235 58 0.009 80 0.033 79 
R(0.1) 0.752 91 0.739 78 0.745 97 0.753 22 0.760 17 0.010 56 −0.166 53 
h(0.1) 2.964 82 2.846 83 2.864 97 2.959 22 3.058 77 0.144 12 0.217 97 
Data[3](S1) α 3.385 83 3.385 52 3.351 78 3.385 45 3.419 31 0.049 95 0.038 36 
γ 0.755 96 0.754 12 0.737 40 0.755 15 0.773 75 0.026 83 0.130 13 
R(0.6) 0.847 93 0.848 84 0.824 61 0.850 26 0.874 06 0.036 60 −0.416 71 
h(0.6) 1.532 11 1.513 78 1.244 34 1.498 79 1.781 57 0.402 15 0.552 95 
Data[3](S2) α 2.606 93 2.617 95 2.572 92 2.607 05 2.640 26 0.049 78 0.014 60 
γ 0.783 27 0.777 21 0.762 50 0.783 13 0.804 12 0.030 96 0.026 93 
R(0.6) 0.830 46 0.826 77 0.806 87 0.833 08 0.856 82 0.037 65 −0.420 83 
h(0.6) 1.328 21 1.358 78 1.097 62 1.298 31 1.523 75 0.324 73 0.568 40 
Data[3](S3) α 2.373 30 2.384 52 2.339 19 2.373 46 2.406 70 0.049 82 0.017 73 
γ 0.796 36 0.790 76 0.774 69 0.796 37 0.818 14 0.032 28 0.007 83 
R(0.6) 0.826 96 0.823 98 0.803 43 0.829 39 0.853 03 0.037 31 −0.389 56 
h(0.6) 1.236 29 1.259 63 1.028 03 1.210 53 1.414 55 0.293 58 0.532 78 

To illustrate the concept of selecting the ideal PT2C from the given real datasets, all optimum criteria Oi for i = 1, 2, 3, 4 are evaluated through the artificial samples data[i](Si) for i = 1, 2, 3, reported in Table XVII; see Table XXII. Results in Table XXII show the following:

  • For data[1]: The right censoring used in S3 is the best based on Oi for i = 1, 3, 4, while the middle censoring used in S2 is the best based on O2 compared to others.

  • For data[2]: The left censoring used in S1 is the best based on Oi for i = 1, 3, 4, while the right censoring used in S3 is the best based on O2 compared to others.

  • For data[3]: The left censoring used in S1 is the best based on Oi for i = 1, 4, while the right censoring used in S3 is the best based on Oi for i = 2, 3 compared to others.

TABLE XXII.

Optimal PT2C mechanisms from three real datasets. Note: The ideal PT2C is highlighted in bold.

Data (sample)O1O2O3O4
p0.30.60.9
Data[1](S1) 773.0954 0.067 08 0.000 09 0.001 16 0.001 59 0.002 90 
Data[1](S2) 429.3099 0.04647 0.000 11 0.001 87 0.004 40 0.003 86 
Data[1](S3) 2936.890 0.087 02 0.00003 0.00048 0.00048 0.00089 
Data[2](S1) 4823.754 0.183 27 0.00004 0.00018 0.00026 0.00092 
Data[2](S2) 843.4995 0.065 57 0.000 08 0.000 57 0.002 14 0.010 02 
Data[2](S3) 512.0544 0.04228 0.000 08 0.000 84 0.003 53 0.017 11 
Data[3](S1) 968.1136 0.930 37 0.000 96 0.00173 0.00098 0.00123 
Data[3](S2) 622.8582 0.550 22 0.000 88 0.002 39 0.001 59 0.001 62 
Data[3](S3) 541.6503 0.48097 0.00085 0.002 81 0.001 83 0.001 60 
Data (sample)O1O2O3O4
p0.30.60.9
Data[1](S1) 773.0954 0.067 08 0.000 09 0.001 16 0.001 59 0.002 90 
Data[1](S2) 429.3099 0.04647 0.000 11 0.001 87 0.004 40 0.003 86 
Data[1](S3) 2936.890 0.087 02 0.00003 0.00048 0.00048 0.00089 
Data[2](S1) 4823.754 0.183 27 0.00004 0.00018 0.00026 0.00092 
Data[2](S2) 843.4995 0.065 57 0.000 08 0.000 57 0.002 14 0.010 02 
Data[2](S3) 512.0544 0.04228 0.000 08 0.000 84 0.003 53 0.017 11 
Data[3](S1) 968.1136 0.930 37 0.000 96 0.00173 0.00098 0.00123 
Data[3](S2) 622.8582 0.550 22 0.000 88 0.002 39 0.001 59 0.001 62 
Data[3](S3) 541.6503 0.48097 0.00085 0.002 81 0.001 83 0.001 60 

It is useful to point out here that the recommended optimal right (or left) censoring based on the three real datasets has been previously recommended in Monte Carlo simulations. Overall, the proposed estimation results using comprehensive datasets collected from the veterinary, industrial, and physics fields provide a good illustration of the proposed complementary unitary Weibull model. In both real data applications, despite the fact that the classical and Bayesian estimates are approximately equal, the Bayes MCMC estimates still outperform the classical ones in terms of minimum SE and IL values.

We investigate adaptive progressive type-II hybrid censoring on complementary unit Weibull distribution with shape and median parameters in this research. The maximum likelihood estimator of the median parameter can be derived expressly as a function of the shape parameter. The shape parameter, on the other hand, can be produced via any iterative process. The approximate confidence intervals for the unknown parameters, as well as the reliability and hazard rate functions, are all taken into account. Bayesian analysis is performed with a gamma prior for the shape parameter and a beta prior for the median parameter. The Bayes estimates for the individual parameters are obtained using the symmetric squared error loss function. The posterior distribution, as expected, is difficult to handle due to its intricate form. To get the point and highest posterior density credible interval estimations, we used the MCMC technique to simulate samples from the posterior distribution. Since the estimates cannot be theoretically compared, simulation analysis is used to measure its statistical performance under various conditions of experimentation. We also offered the optimal sample strategies based on a variety of parameters. Three examples of veterinary, industrial, and physical datasets are examined to show how the proposed approaches can be used in practical circumstances. In future work, one can use the maximum product of the spacing approach and its extension in a Bayesian setup to generate estimators for the suggested distribution using the provided censoring scheme and compare the findings to those gathered in current work.

See the supplementary material for Fig. S1: density (left) and trace (right) plots of α, γ, R(t), and h(t) using S2 from three real datasets; Fig. S2: density (left) and trace (right) plots of α, γ, R(t), and h(t) using S3 from three real datasets.

Princess Nourah bint Abdulrahman University Researchers Supporting Project number (Grant No. PNURSP2024R50), Princess Nourah bint Abdulrahman University, Riyadh, Saudi Arabia.

Funding: This research was funded by the Princess Nourah bint Abdulrahman University Researchers Supporting Project number (Grant No. PNURSP2024R50), Princess Nourah bint Abdulrahman University, Riyadh, Saudi Arabia.

The authors have no conflicts to disclose.

Refah Alotaibi: Conceptualization (equal); Methodology (equal); Writing – original draft (equal). Mazen Nassar: Data curation (equal); Methodology (equal); Writing – original draft (equal). Ahmed Elshahhat: Software (equal); Validation (equal); Writing – review & editing (equal).

The data that support the findings of this study are available within the article and its supplementary material.

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