Classical and Bayesian inference of inverted modified Lindley distribution based on progressive type-II censoring for modeling engineering data

Experiments frequently incorporate censoring schemes that conclude testing before all units fail due to the unavailability of information or data. In essence, censoring implies that only a partial record of the exact failure times of observed units is available in studies. Two common censoring systems are Type-I censoring, in which the experiment ends after a certain number of failures, and Type-II censoring, in which the test ends after a predetermined period. However, these conventional censoring schemes may occasionally extend in duration due to the product’s characteristic of high reliability and a long lifetime.

Hence, to identify a more effective approach for gathering failure information in lifetime studies, numerous alternative censoring schemes have been proposed, with the progressive Type-II censoring scheme (Prog-II CS) emerging as the most widely employed in practical applications. The following is an outline of the detailed events in this scheme: Assume put n independent, identical components through a life test where the censoring scheme R = (R₁, R₂, …, R_m) has been identified, and only m(≤n) failures are observed.¹ 

Upon the occurrence of the initial failure time, indicated as x_1:m:n, R₁ survival components are randomly removed from the test. Subsequently, when the second failure time, x_2:m:n, occurs, R₂ survival components are also randomly withdrawn. This sequence continues until the mth failure, x_m:m:n, unfolds. At this juncture, the remaining R_m survival components are withdrawn, leading to the conclusion of the experiment. Therefore, x_1:m:n ≤ x_1:m:n ≤⋯≤ x_m:m:n are denoted as Prog-II CS within the censored scheme R = (R₁, R₂, …, R_m). It is acknowledged that $n=m+∑i=1mRi$⁠, and several censoring schemes, including Type-II censoring and the complete sample, are considered its specific instances.² 

A helpful description and well-crafted summary of Prog-II CS are available in references such as Refs. 3–15.

In recent times, numerous scholars have contributed to the statistical inferences of the IMLD. For example, Kumar et al.¹⁷ analyzed the IML distribution using dual generalized order statistics, while Kumar et al.¹⁸ explored the inverted modified Lindley distribution with classical inferences of order statistics and practical applications.

Figures 1–4 showcase the plots for the PDF, CDF, survival function, and hazard rate function, each depicted with varying parameter values. These visual representations illustrate that the IML distribution exhibits an unimodal distribution and can demonstrate right skewness. The hazard rate function is also unimodal in (x), demonstrating significant flexibility.

In this paper, our goal is to devise an estimation method for IML distribution parameters in the presence of Prog-II CS. We derive maximum likelihood estimates (MLEs) and corresponding approximate confidence intervals (ACIs). Furthermore, Bayesian point and interval estimates are obtained based on MLE under squared error (SE) and linear exponential (LINEX) loss functions. To validate the proposed approaches, we conduct a simulation study and apply them to two real datasets.

The remaining sections of the paper are structured as follows: Sec. II presents the MLEs for the unknown parameters, S(x) and h(x), along with their associated ACIs. In Sec. III, the use of Markov chain Monte Carlo (MCMC) for Bayesian estimation is discussed. Section IV delves into two applications using real-world datasets. Section V provides an extensive summary of the simulation study’s results. Section VI concludes with some results.

In addition to obtaining a singular estimate for the unknown parameter, it is crucial to establish a range of values that, with a specified level of confidence, can encompass the true parameter. This statistical inference process is known as interval estimation. In this context, we suggest using the asymptotic normality of the MLE to construct the ACI for α.

Thus, the (1-α)% ACI for the parameter α is given by $α̂±zη/2Var̂(α̂)$⁠, where z_η/2 represents the upper percentile of the standard normal distribution. Furthermore, to establish the ACIs for S(t) and h(t) both functions contingent on the parameter α, we must determine their variances. To approximate the variance estimates for $Ŝ(t)$ in Eq. (9) and $ĥ(t)$ in Eq. (10), we employ the delta method. This method, a general approach for calculating ACIs for functions of MLE estimates, is detailed by Greene.¹⁹ 

Consequently, the variances of $Ŝ(t)$ and $ĥ(t)$ are expressed as follows:

$σ̂S(t)2=∇Ŝ(t)TV̂∇Ŝ(t)$ and $σ̂h(t)2=∇ĥ(t)TV̂∇ĥ(t)$⁠; here, $∇Ŝ(t)$ and $∇ĥ(t)$ represent the gradients of $Ŝ(t)$ and $ĥ(t)$⁠, respectively, with respect to α, and $V̂=I−1(α)$⁠.

Thus, the (1 − η)100% ACIs for S(t) and h(t) are obtained as $Ŝ(t)±Zη/2σ̂S(t)2$ and $ĥ(t)±Zη/2σ̂h(t)2$⁠.

The conditional distribution for α does not conform to any familiar density, as indicated in Eq. (14). To generate samples from Eq. (14) using a normal proposal distribution, the Metropolis–Hastings (M–H) technique is implemented in this situation. The subsequent steps are utilized to generate samples through the M–H technique and derive Bayesian estimates

1. Define the initial value for α as α₀ = $α̂$⁠.

2. Set j = 1.

3. Generate α^j using Eq. (14) with a normal proposal distribution through M–H steps.

4. For a given value of (t), calculate S^(j)(t) in Eq. (15) and h^(j)(t) in Eq. (16) as

   $S(j)(t)=1−1+αjt−1e−αjt−1(1+αj)e−αjt−1$

   (15)

   and

   $h(j)(t)=αjt−2e−2αjt−1(1+αj)(1+αj)eαjt−1+2αjt−1−11−1+αjt−1e−αjt−1(1+αj)e−αjt−1.$

   (16)

5. Set j = j + 1.

6. Repeat Steps 3 to 5, M times to acquire

   $(α(1),S(1)(t),h(1)(t)),……,(α(m),S(m)(t),h(m)(t)),j=1,…,m.$
7. Calculate the Bayesian estimates after discarding z samples as a burn-in period, as outlined below:

   $α̂=∑j=Z+1mα(j)m−z,Ŝ(t)=∑j=Z+1mS(j)(t)m−z,andĥ(t)=∑j=Z+1mh(j)(t)m−z.$
8. To determine the CRIs, arrange α^(j), S^(j)(t), and h^(j)(t) for j = z + 1, …, m. Subsequently, one can adopt the approach proposed by Chen and Shao²⁰ to derive the necessary interval for the parameter α, as outlined below:

   $α(z+1),α(z+1)+(1−α)(m−z),$

   the identical procedure can be employed to derive the CRIs for S(t) and h(t).

The initial dataset originates from the Open University (1993) and pertains to the prices of 31 different children’s wooden toys available in a Suffolk craft shop in April 1991. Shafiei et al.²¹ have conducted an analysis of this dataset. Chesneau et al.¹⁶ utilized this data to examine the IML distribution. The P-value stands at 0.7589, and the Kolmogorov–Smirnov (K–S) distance between the empirical distribution of failure data and the CDF of the IML distribution is 0.1225. According to Fig. 5, the IML distribution corresponds well with the provided data. Afterward, a progressive censoring method using R = (10, 10, 0, 0, 0, 0, 0, 0, 0, 0) was utilized, and a Prog-II censored sample with a sample size of m = 20 was chosen randomly from 30 failed observations, as shown in Table I.

Table II displays the MLEs for the parameters $α̂$ and the reliability functions $S(t)̂$ and $h(t)̂$⁠, which were obtained from the Prog-II failure data given in Table I. Furthermore, for different values of the shape parameter a in the LINEX loss function, which correspond to the parameters α and the functions S(t) and h(t), Bayes estimates are calculated. The results are also shown in Table II for the parameter and the reliability functions S(t) and h(t). These Bayes estimates are computed taking into account both the SE and LINEX loss functions. The reliability functions via MLE and the probability profile of the IML parameter were covered in Fig. 6. When we observe that the profile likelihood has a single maximum and is symmetric and unimodal, it usually indicates a well-behaved MLE. Figure 7 displays trace plots of the MCMC results to show how the method’s convergence is graphically represented.

The 95% ACIs and CRIs are used to calculate the parameters α and the reliability functions S(t) and h(t). Table III presents the obtained results.

The time between failures for repairable objects, which was obtained from Murthy et al.,²² made up the second dataset. This dataset was used by Chesneau et al.¹⁶ to examine the IML distribution. The Kolmogorov–Smirnov (K–S) distance between the CDF of the IML distribution and the empirical distribution of failure data is 0.1394, and the P-value is 0.6043. The IML distribution fits the given data very well, as Fig. 8 illustrates. Afterward, from 30 failed observations, a Prog-II censored sample with an effective size of m = 15 was randomly chosen using a progressive censored procedure with R = (0, 0, 0, 0, 0, 0, 0, 5, 10, 0, 0, 0), in Table IV.

Table V displays the MLEs for the parameter $α̂$ and the reliability functions $S(t)̂$ and $h(t)̂$⁠, which were obtained from the Prog-II failure data given in Table IV. Furthermore, for different values of the shape parameter a in the LINEX loss function, which relate to the parameters α and the functions S(t) and h(t), Bayes estimates are calculated. The results are also shown in Table V for the parameter and the reliability functions S(t) and h(t). These Bayes estimates are computed taking into account both the SE and LINEX loss functions. The reliability functions via MLE and the probability profile of the IML parameter were covered in Fig. 9. When we observe that the profile likelihood has a single, symmetric and unimodal point, it usually indicates a well-behaved MLE. Figure 10 presents trace plots of the MCMC findings to visually represent the MCMC algorithm’s convergence.

The 95% ACIs and CRIs are used to calculate the parameters α and the reliability functions S(t) and h(t). Table VI presents the obtained results.

Modern statistical problems demonstrate that comprehending statistical inferential methods necessitates a blend of theoretical and applied skills. This section focuses on the numerical technique, while the previous sections addressed the analytical one, through a simulation study. For each simulation, 1000 Prog-II censored samples were used in a simulation analysis to compare the IML distribution’s parameter estimators and particular lifetime parameters.

Prog-II censored samples were generated using the IML distribution with initial values of α_◦ = 0.6, S(t) = 0.2, and h(t) = 1.5. The MSE obtained by the resulting estimators of α, S(t), and h(t) considered comparisons between various approaches for k = 1, 2, 3 (μ₁ = α, μ₂ = S(t), μ₃ = h(t)), as $MSE(μk)=11000∑i=11000μ̂ki−μk2$⁠. Another criterion was applied to compare the 95% CIs by using asymptotic distributions of the MLEs and CRIs. For the comparison, coverage probability (CP) and ACLs were employed. In this analysis, the progressive systems listed below were taken into consideration:

• Scheme A: R₁ = n − m, R_i = 0 for i ≠ 1.

• Scheme B: $Rm2=Rm2+1=n−m2,;Ri=0$ for $i≠m2$ and $i≠m2+1$⁠.

• Scheme C: R_m = n − m, R_i = 0 for i ≠ m. Tables VII–IX display the estimated parameter values together with their MSEs, and Table X presents the results of the ACL and CP of the 95% CIs.

We note the following patterns from the outcomes:

1. According to Tables VII–IX, as sample sizes increase, MSEs get smaller, and BE have the smallest MSEs for the parameters α, S(t) and h(t). Therefore, when all variables are considered, Bayes estimation outperforms MLEs methods.

2. The estimates from Bayes are better for α, S(t) and h(t) in that the MSEs are smaller.

3. For smaller MSEs with c = −2.0, the LINEX estimates with c = 2.0 are greater estimates.

4. The performance of Scheme A is greater than Scheme B and C due to its smaller MSEs for fixed-value samples n and m failure time intervals.

5. The results from the CRIs are very perfect than those from the ACIs for identified failures, approaches, and size of the sample, as shown in Table X.

In this investigation, employing a Prog-II CS, we considered both Bayesian and non-Bayesian estimations for the parameters and survival functions of the IMLD. Alongside obtaining the MLEs and associated ACIs, we also introduced Bayesian estimations for the unknown parameters. It should be noted that although explicit formulation of Bayes estimators is not possible, they can be obtained by numerical integration. Consequently, we utilized the MCMC method to obtain point estimates and the CRIs. Two real datasets were utilized to demonstrate the effectiveness of the proposed estimators. We also examined the efficiency of alternative strategies using a simulated analysis that included a range of sample sizes, effective sample sizes, and sampling methods. Future studies might investigate toward statistics applied to the IML distribution. In future works, we will use Joint Prog-II censored data to estimate the IML distribution’s parameters and compare it to all other censored algorithms that we will use.

This research project was supported by the Researchers Supporting Project No. (RSP2024R488), King Saud University, Riyadh, Saudi Arabia.

The authors have no conflicts to disclose.

Mustafa M. Hasaballah: Data curation (equal); Methodology (equal); Writing – original draft (equal); Writing – review & editing (equal). Yusra A. Tashkandy: Funding acquisition (equal); Writing – review & editing (equal). M. E. Bakr: Data curation (equal); Project administration (equal); Validation (equal). Oluwafemi Samson Balogun: Investigation (equal); Supervision (equal); Validation (equal). Dina A. Ramadan: Conceptualization (equal); Formal analysis (equal); Software (equal).

The data that support the findings of this study are available within the article.