This paper discussed gull alpha power Weibull distribution with a three-parameter. Different statistical inference methods of Gull Alpha Power Weibull distribution parameters have been obtained, estimated, and evaluated. Then, the results are compared to find a suitable model. The unknown parameters of the published Gull Alpha Power Weibull distribution are analyzed. Seven estimation methods are maximum likelihood, Anderson–Darling, right-tail Anderson–Darling, Cramér–von Mises, ordinary least-squares, weighted least-squares, and maximum product of spacing. In addition, the performance of this distribution is computed using the Monte Carlo method, and the limited sample features of parameter estimates for the proposed distribution are analyzed. In light of the importance of heavy-tailed distributions, actuarial approaches are employed. Applying actuarial criteria such as value at risk and tail value at risk to the suggested distribution shows that the model under study has a larger tail than the Weibull distribution. Two real-world COVID-19 infection datasets are used to evaluate the distribution. We analyze the existence and uniqueness of the log-probability roots to establish that they represent the global maximum. We conclude by summarizing the outcomes reported in this study.

Topics
Data processing, Statistical models, Computer simulation, Probability theory, Diseases and conditions, Coronaviruses, Statistical analysis

Typically, scientists want a straightforward approach to fitting data and calculating prediction parameters. Even though several statistical methods are available and may be used to reconstruct data, some are intricate and difficult. Alternatively, essential solutions may compete and are straightforward to apply. Statistical theory usually plays a primary role in real-world modeling investigations. Many accounts of these theories start from observation, probability distribution, and analysis—see Ref. 1 for more details. Uncertain and complex natural phenomena are modeled by using probability distributions. However, many identified probability distributions do not usually accurately represent natural event data.

Due to the lack of the usual distributions’ representation ability, researchers generalize probability distributions by adding one or more additional parameters, which causes an improvement in the suitability of these distributions for real-world data modeling and the precision of the distribution tail shape description. Note that this procedure makes the form more complicated, and the power of distribution rises.

Several generalized distribution approaches have been created to increase the original model’s flexibility by increasing the parameters. Further information can be found in Refs. 2–4, which can be used in many fields, like health, engineering, and economics. Giving distributions greater flexibility is critical for modeling data.

Creating new families of continuous distributions during the last few years has spurred many statisticians to build innovative models. The exponential and Weibull distributions and their extensions are the types of constant distributions commonly used; these distributions have many merits, such as great superiority for fitting different kinds of data.

It is essential to study pandemics and illnesses to understand viruses’ spread and impact on humanity. The COVID-19 virus has emerged in the past several years and has become a significant threat to human health. As a result of the coronavirus epidemic, experts throughout the globe are modeling daily mortality in certain nations.

The growth of the COVID-19 outbreak over 2021 has caused massive devastation and economic instability globally. A comprehensive scientific examination of this phenomenon is now ongoing. However, obtaining the correct data and numbers is essential to take all necessary precautions to prevent the COVID-19 infection from growing larger. It is always desirable to offer the most accurate data description to understand and apply extensive data studies. New studies have demonstrated statistical distributions for modeling data in applied disciplines, including medicine. More papers discussed COVID-19 with different models.5–9 

Analyzing COVID-19 data using statistical distributions has been essential for gaining insights into the virus’s spread and impact. Statistical distributions, such as the normal, binomial, and Poisson distributions, have been utilized to model various aspects of the pandemic. For instance, the Poisson distribution is often applied to model the number of new cases or deaths within a specific time frame, assuming these events occur constantly. This allows researchers to estimate the likelihood of different numbers of daily cases and assess the effectiveness of interventions. The normal distribution can approximate the spread of variables like incubation periods and recovery times, providing a clearer picture of the pandemic’s dynamics. Additionally, the use of statistical distributions aids in understanding the variability and uncertainty inherent in COVID-19 data. By fitting data to appropriate distributions, researchers can quantify the uncertainty in their estimates and make probabilistic forecasts. For example, the gamma distribution is frequently employed to model the time between infection and symptom onset, as it accommodates the right-skewed nature of this duration. This helps create more accurate disease transmission models and evaluate the timing of public health measures. Furthermore, analyzing the distribution of age-specific case fatality rates can inform targeted healthcare responses, ensuring that high-risk groups receive appropriate attention and resources. Statistical distributions are crucial in hypothesis testing and confidence interval estimation in COVID-19 research. By comparing observed data against theoretical distributions, scientists can test hypotheses about the virus’s behavior and the effectiveness of interventions.

We continue this study in this article by presenting the best estimation method for a new extension to the Weibull distribution that has been presented before and has many applications in various fields. Weibull distribution has high applicability in various domains, particularly biomedical sciences, life testing issues, chemical data, and health sciences. Many families of distributions have been constructed by augmenting well-known distributions with one or more parameters. These extended distribution classes allow for more modeling flexibility in medical data, manufacturing, lifespan analysis, health coverage, marketing, and earth sciences.

While heavy-tailed distributions are critical in analyzing risk management data, the number of probability distributions suggested to describe such data is limited. Financial analysts are always looking for novel and suitable statistical models to handle data about risk issues. However, very few probability distributions account for some or all of the characteristics mentioned earlier in the literature. Therefore, it is critical to create models, whether from current distributions or a new family of models, to account for insurance loss data and financial returns, among other things. Because models with heavy-tailed distributions are critical in actuarial science for giving acceptable descriptions of claim size distributions, there has been a surge of interest in these areas during the last decade. For example, see Refs. 10–12.

Possible problems or limitations with existing (classical) distributions could include the following:

  • The classical distributions may not fit the observed data well, especially if the data exhibit complex patterns or behaviors not well-modeled by the classical distribution.

  • The classical distributions have specific few parameters that govern their form, which may not always capture the variability or nuances present in the data.

  • The classical distributions assume certain underlying data characteristics, such as independence of observations and a specific type of hazard function, which may not hold true in all cases.

  • Classical distributions may not be suitable for modeling certain types of data or phenomena, particularly those with heavy-tailed or asymmetric distributions.

We aim to address these problems or perceived limitations by offering a better estimation method for a new extension of the Weibull distribution made by the Gull Alpha Power family that provides improved flexibility, better fit to the data, or additional features to capture properties of the data under study better.

Ijaz et al.13 presented a new method for generating distributions by adding an extra shape parameter to well-known baseline distributions. This method is called the Gull Alpha Power family (GAPF). The cumulative distribution function CDF of GAPF is defined as follows:
(1)
and its corresponding probability density function pdf is given by
(2)
where G(x) is the baseline distribution, and when α = 1, we have G(x).

Different seven estimation methods are considered to ensure robustness, accuracy, and reliability to estimate the unknown parameters of the gull alpha power Weibull distribution, which are as follows: maximum likelihood, Anderson–Darling, right-tail Anderson–Darling, Cramér–von Mises, ordinary least-squares, weighted least-squares, and maximum product of spacing. Each estimation method brings its own strengths and limitations, and considering a variety of approaches allows for a comprehensive assessment of the distribution’s characteristics. Recent papers that used this method are Refs. 14–17. The practical implications of using seven different estimation methods include the following:

  • By employing various estimation methods of gull alpha power Weibull parameters, researchers can cross-validate their results and ensure that the estimated parameters are robust and reliable given the smallest mean square error and its bias nearly to zero, even in the presence of different data characteristics or modeling assumptions.

  • Different estimation methods may yield different models with the best parameter estimators. Comparing these results allows researchers to identify the most suitable model that best fits the data, leading to better model selection and interpretation for different real datasets.

  • Accurate parameter estimation is crucial for risk assessment and decision-making processes in fields such as finance, insurance, and public health. By employing robust estimation methods, researchers can make more informed decisions based on reliable statistical models.

The study of the GAPW distribution is motivated by the following:

  • To introduce estimation of a GAPW distribution that can be used as a butter substitute for other classical distributions or other extension distributions in modeling lifetime datasets.

  • Actuarial approaches are employed for a proposed model, such as value at risk and tail value at risk.

  • Seven estimation methods are maximum likelihood, Anderson–Darling, right-tail Anderson–Darling, Cramér–von Mises, ordinary least-squares, weighted least-squares, and maximum product of spacing.

  • Analyzing the limited sample features of parameter estimates for the proposed distribution by using Monte Carlo simulation.

The paper works on estimating the Gull Alpha Weibull distribution parameters in Sec. II. We used COVID-19 datasets for the distribution to check its applicability. Section III discusses many classical methods to evaluate the estimates of the proposed distribution’s parameters, applying some simulations to measure some numerical values for the estimates in Sec. IV. Section V is devoted to recording the results of the actuarial measures. Finally, the main findings and the conclusion of the whole paper are represented in Sec. VI.

If a random variable X has a Weibull distribution with parameters α, β > 0, then the cdf of the GAPW distribution is defined as follows:
(3)
and its corresponding pdf is given by
(4)
Following are definitions of the suggested model’s survival function (SRF) and hazard rate function (HZF):
(5)
(6)
Plots of pdf and HZF, as in Eqs. (4) and (6), respectively, are shown in Figs. 1 and 2 in Ref. 13 for several parameter values of the proposed model. Different statistical properties of this model were introduced in the main article.

This section covers the estimation (E) of GAPW distribution parameters using various classical estimation methods, presented in the subsections below. We made a classical inference on the published distribution; see Ijaz et al.13 Different estimation methods in statistics have various properties that make them suitable for different scenarios. Here are some key properties of common estimation methods.

Maximum likelihood and maximum product of spacing: Consistency, efficiency, asymptotic normality, and invariance. Ordinary least square and weighted least square: computational efficiency and robustness to outliers. Meanwhile, Anderson–Darling, right-tail Anderson–Darling, and Cramér–von Mises are associated with goodness-of-fit tests rather than direct parameter estimation. However, these tests can be used as part of the estimation process or to assess the fit of a given distribution to a dataset. Here are the properties of these tests.

Anderson–Darling (AD) Test: Sensitivity to tail behavior, weighted test, and consistency. Right-tail Anderson–Darling (RTAD) Test: focuses on the right tail, sensitivity to extreme values, and is useful for heavy-tailed distributions. Cramér–von Mises (CVM) Test: Overall goodness-of-fit, uniform power, and sensitivity to central tendency.

Let x1, x2, …, xn be a random sample of size n from the pdf of the proposed model, then the log-likelihood function takes the form
(7)
By differentiating Eq. (7) with respect to α, a and b, respectively, we have
∂L∂α=∑i=1ne−axibα+∑i=1n1α−eaxibαlog(α)−(log(α)−1)eaxib,
∂L∂a=∑i=1n−(log(α)−1)xibeaxiblog(α)−(log(α)−1)eaxib+∑i=1nlog(α)xib−e−axib−2∑i=1nxib+na,
∂L∂b=∑i=1n−a(log(α)−1)xib⁡logxieaxiblog(α)−(log(α)−1)eaxib+∑i=1n−a⁡log(α)xib⁡logxie−axib−2a∑i=1nxib⁡logxi+∑i=1nlogxi+nb,
to get the estimated values of the three parameters α, a, and b, we will set the above three equations equal to zero; then, we will have MLE of the three parameters.

The asymptotic efficiency of MLE refers to its property of achieving the smallest possible variance among all unbiased estimators as the sample size approaches infinity. While MLE is indeed asymptotically efficient under certain regularity conditions, there are several reasons why this may not be the case in a specific study or dataset: Finite sample size, model misspecification, complexity of the model, non-iid data, numerical optimization, presence of outliers, and influential observations. In this paper, the model is complex and has nonlineal equations; this is a sufficient reason to have alternative methods to arrive at the best way to estimate features under different circumstances.

Suppose we have an ordered random sample as follows: x(1), x(2), …, x(n) from F(x) of the (PB?) distribution. By minimizing the ADE of the parameters of this model, they are estimated as follows:
A(α,a,b)=−n−1n∑i=1n(2i−1)[log⁡F(xi)+log⁡S(xi)].
The ADE can also be derived by solving the next non-linear equation
∑i=1n(2i−1)Δs(xi)F(xi)−Δi(xn+1−i)S(xn+1−i)=0,s=1,2,3,
where
(8)
By minimizing the following equation with respect to α, a, and b, we can find the RADEs
R(α,a,b)=n2−2∑i=1nFxi:n|α,a,b−1n∑i=1n2i−1×log⁡Sxn+1−i:n|α,a,b.
Additionally, the RADE may be obtained analytically from the following non-linear equations:
−2∑i=1nΔsxi:n|α,a,b+1n∑i=1n2i−1Δsxn+1−i:n|α,a,bSxn+1−i:n|α,a,b=0,s=1,2,
where Δ1⋅|α,a,b⁠, Δ2⋅|α,a,b⁠, and Δ3⋅|α,a,b are defined in Eq. (8).
The CDF estimate is compared to the empirical CDF, and the difference between the two is used to calculate the CVME. Finding the CVME of the provided model parameters is as simple as minimizing the following:
C(α,a,b)=−112n+∑i=1nF(xi|α,a,b)−2i−12n2,
with respect to α, a, and b. The CVME is calculated by solving the next non-linear equation
∑i=1nF(xi|α,a,b)−2i−12nΔs(xi|α,a,b)=0,s=1,2,3,
where Δ1⋅|α,a,b⁠, Δ2⋅|α,a,b⁠, and Δ3⋅|α,a,b are defined in Eq. (8).
Calculating the OLSE of the parameters of the suggested model requires minimizing the following function with respect to α, a, and b. This will result in the calculation of the OLSE,
V(α,a,b)=∑i=1nF(xi|α,a,b)−in+12.
In addition, the OLSE may be computed by finding the solution to the following non-linear equation:
∑i=1nF(xi|α,a,b)−in+1Δs(xi|α,a,b)=0,s=1,2,3,
where Δ1⋅|α,a,b⁠, Δ2⋅|α,a,b⁠, and Δ3⋅|α,a,b are defined in Eq. (8).
TABLE I.

(Continued.)

nEst.Est. Par.MLEADECVMEMPSELSERTADEWLSE
150 BIAS α̂ 0.007 578{7} 0.004 314{5} 0.002 688{3} 0.004 072{4} 0.001 336{1} 0.001 763{2} 0.006 292{6} 
â 0.000 315{4} 0.000 617{7} 0.000 273{2} 0.000 321{5} 0.000 152{1} 0.000 303{3} 0.000 473{6} 
b̂ 0.002 458{5} 0.005 111{7} 0.001 899{3} 0.001 641{2} 0.000 848{1} 0.002 222{4} 0.002 689{6} 
MSE α̂ 0.010 608{7} 0.002 419{3} 0.002 609{4} 0.004 877{5} 0.000 681{2} 0.000 666{1} 0.007 678{6} 
â 3.1e − 05{5} 5e − 05{7} 2.1e − 05{3} 2.3e − 05{4} 8e − 06{1} 1.9e − 05{2} 4.1e − 05{6} 
b̂ 0.001 008{3} 0.004 285{7} 0.001 249{6} 0.000 618{2} 0.000 242{1} 0.001 117{4} 0.001 227{5} 
MRE α̂ 0.015 156{7} 0.008 628{5} 0.005 376{3} 0.008 143{4} 0.002 673{1} 0.003 525{2} 0.012 584{6} 
 â 0.001 259{4} 0.002 466{7} 0.001 09{2} 0.001 284{5} 0.000 61{1} 0.001 211{3} 0.001 89{6} 
 b̂ 0.003 278{5} 0.006 814{7} 0.002 532{3} 0.002 187{2} 0.001 131{1} 0.002 963{4} 0.003 586{6} 
∑Ranks  46{5} 54{7} 28{3} 32{4} 16{1} 24{2} 52{6} 
nEst.Est. Par.MLEADECVMEMPSELSERTADEWLSE
150 BIAS α̂ 0.007 578{7} 0.004 314{5} 0.002 688{3} 0.004 072{4} 0.001 336{1} 0.001 763{2} 0.006 292{6} 
â 0.000 315{4} 0.000 617{7} 0.000 273{2} 0.000 321{5} 0.000 152{1} 0.000 303{3} 0.000 473{6} 
b̂ 0.002 458{5} 0.005 111{7} 0.001 899{3} 0.001 641{2} 0.000 848{1} 0.002 222{4} 0.002 689{6} 
MSE α̂ 0.010 608{7} 0.002 419{3} 0.002 609{4} 0.004 877{5} 0.000 681{2} 0.000 666{1} 0.007 678{6} 
â 3.1e − 05{5} 5e − 05{7} 2.1e − 05{3} 2.3e − 05{4} 8e − 06{1} 1.9e − 05{2} 4.1e − 05{6} 
b̂ 0.001 008{3} 0.004 285{7} 0.001 249{6} 0.000 618{2} 0.000 242{1} 0.001 117{4} 0.001 227{5} 
MRE α̂ 0.015 156{7} 0.008 628{5} 0.005 376{3} 0.008 143{4} 0.002 673{1} 0.003 525{2} 0.012 584{6} 
 â 0.001 259{4} 0.002 466{7} 0.001 09{2} 0.001 284{5} 0.000 61{1} 0.001 211{3} 0.001 89{6} 
 b̂ 0.003 278{5} 0.006 814{7} 0.002 532{3} 0.002 187{2} 0.001 131{1} 0.002 963{4} 0.003 586{6} 
∑Ranks  46{5} 54{7} 28{3} 32{4} 16{1} 24{2} 52{6} 
View Large
TABLE II.

(Continued.)

nEst.Est. Par.MLEADECVMEMPSELSERTADEWLSE
120 BIAS α̂ 0.476 707{1} 0.573 18{3} 0.575 386{4} 0.528 613{2} 0.589 648{7} 0.579 243{5} 0.583 323{6} 
â 0.231 363{7} 0.076 11{3} 0.083 744{5} 0.067 504{1} 0.086 041{6} 0.069 977{2} 0.077 535{4} 
b̂ 0.061 239{1} 0.284 492{4} 0.287 763{5} 0.249 549{2} 0.295 69{7} 0.262 042{3} 0.294 015{6} 
MSE α̂ 0.324 088{1} 0.440 214{5} 0.432 437{4} 0.430 652{3} 0.449 986{7} 0.425 028{2} 0.446 31{6} 
â 0.095 463{7} 0.009 797{3} 0.010 938{5} 0.008 405{2} 0.011 895{6} 0.007 771{1} 0.010 348{4} 
b̂ 0.006 369{1} 0.144 238{4} 0.149 091{5} 0.115 37{3} 0.153 942{7} 0.107 138{2} 0.151 708{6} 
MRE α̂ 0.317 804{1} 0.382 12{3} 0.383 591{4} 0.352 409{2} 0.393 098{7} 0.386 162{5} 0.388 882{6} 
â 0.308 484{7} 0.101 48{3} 0.111 659{5} 0.090 005{1} 0.114 722{6} 0.093 302{2} 0.103 38{4} 
b̂ 0.081 651{1} 0.379 322{4} 0.383 684{5} 0.332 732{2} 0.394 254{7} 0.349 389{3} 0.392 02{6} 
∑Ranks  27{3} 32{4} 42{5} 18{1} 60{7} 25{2} 48{6} 
150 BIAS α̂ 0.463 428{2} 0.555 37{6} 0.544 149{3} 0.456 065{1} 0.575 713{7} 0.554 548{5} 0.549 379{4} 
â 0.219 144{7} 0.070 369{3} 0.075 921{5} 0.057 423{1} 0.077 41{6} 0.064 968{2} 0.071 684{4} 
b̂ 0.056 216{1} 0.273 922{6} 0.268 69{4} 0.218 459{2} 0.280 401{7} 0.249 499{3} 0.272 292{5} 
MSE α̂ 0.312 031{1} 0.423 442{6} 0.388 656{3} 0.354 296{2} 0.429 411{7} 0.395 937{4} 0.418 228{5} 
â 0.086 08{7} 0.009 076{3} 0.009 22{5} 0.006 313{1} 0.009 89{6} 0.006 818{2} 0.009 184{4} 
b̂ 0.005 455{1} 0.140 436{7} 0.124 064{4} 0.093 171{2} 0.137 81{5} 0.100 466{3} 0.138 678{6} 
MRE α̂ 0.308 952{2} 0.370 246{6} 0.362 766{3} 0.304 043{1} 0.383 808{7} 0.369 699{5} 0.366 253{4} 
â 0.292 192{7} 0.093 825{3} 0.101 228{5} 0.076 563{1} 0.103 213{6} 0.086 624{2} 0.095 578{4} 
b̂ 0.074 955{1} 0.365 23{6} 0.358 253{4} 0.291 278{2} 0.373 868{7} 0.332 666{3} 0.363 057{5} 
∑Ranks  29{2.5} 46{6} 36{4} 13{1} 58{7} 29{2.5} 41{5} 
nEst.Est. Par.MLEADECVMEMPSELSERTADEWLSE
120 BIAS α̂ 0.476 707{1} 0.573 18{3} 0.575 386{4} 0.528 613{2} 0.589 648{7} 0.579 243{5} 0.583 323{6} 
â 0.231 363{7} 0.076 11{3} 0.083 744{5} 0.067 504{1} 0.086 041{6} 0.069 977{2} 0.077 535{4} 
b̂ 0.061 239{1} 0.284 492{4} 0.287 763{5} 0.249 549{2} 0.295 69{7} 0.262 042{3} 0.294 015{6} 
MSE α̂ 0.324 088{1} 0.440 214{5} 0.432 437{4} 0.430 652{3} 0.449 986{7} 0.425 028{2} 0.446 31{6} 
â 0.095 463{7} 0.009 797{3} 0.010 938{5} 0.008 405{2} 0.011 895{6} 0.007 771{1} 0.010 348{4} 
b̂ 0.006 369{1} 0.144 238{4} 0.149 091{5} 0.115 37{3} 0.153 942{7} 0.107 138{2} 0.151 708{6} 
MRE α̂ 0.317 804{1} 0.382 12{3} 0.383 591{4} 0.352 409{2} 0.393 098{7} 0.386 162{5} 0.388 882{6} 
â 0.308 484{7} 0.101 48{3} 0.111 659{5} 0.090 005{1} 0.114 722{6} 0.093 302{2} 0.103 38{4} 
b̂ 0.081 651{1} 0.379 322{4} 0.383 684{5} 0.332 732{2} 0.394 254{7} 0.349 389{3} 0.392 02{6} 
∑Ranks  27{3} 32{4} 42{5} 18{1} 60{7} 25{2} 48{6} 
150 BIAS α̂ 0.463 428{2} 0.555 37{6} 0.544 149{3} 0.456 065{1} 0.575 713{7} 0.554 548{5} 0.549 379{4} 
â 0.219 144{7} 0.070 369{3} 0.075 921{5} 0.057 423{1} 0.077 41{6} 0.064 968{2} 0.071 684{4} 
b̂ 0.056 216{1} 0.273 922{6} 0.268 69{4} 0.218 459{2} 0.280 401{7} 0.249 499{3} 0.272 292{5} 
MSE α̂ 0.312 031{1} 0.423 442{6} 0.388 656{3} 0.354 296{2} 0.429 411{7} 0.395 937{4} 0.418 228{5} 
â 0.086 08{7} 0.009 076{3} 0.009 22{5} 0.006 313{1} 0.009 89{6} 0.006 818{2} 0.009 184{4} 
b̂ 0.005 455{1} 0.140 436{7} 0.124 064{4} 0.093 171{2} 0.137 81{5} 0.100 466{3} 0.138 678{6} 
MRE α̂ 0.308 952{2} 0.370 246{6} 0.362 766{3} 0.304 043{1} 0.383 808{7} 0.369 699{5} 0.366 253{4} 
â 0.292 192{7} 0.093 825{3} 0.101 228{5} 0.076 563{1} 0.103 213{6} 0.086 624{2} 0.095 578{4} 
b̂ 0.074 955{1} 0.365 23{6} 0.358 253{4} 0.291 278{2} 0.373 868{7} 0.332 666{3} 0.363 057{5} 
∑Ranks  29{2.5} 46{6} 36{4} 13{1} 58{7} 29{2.5} 41{5} 
View Large
TABLE III.

(Continued.)

nEst.Est. Par.MLEADECVMEMPSELSERTADEWLSE
80 BIAS α̂ 0.041 309{2} 0.051 789{3} 0.052 18{4} 0.034 827{1} 0.060 656{6} 0.071 481{7} 0.056 063{5} 
â 0.185 069{7} 0.006 705{4} 0.008 156{6} 0.004 754{1} 0.006 282{3} 0.007 07{5} 0.005 226{2} 
b̂ 0.157 195{7} 0.047 339{2} 0.055 803{5} 0.032 737{1} 0.053 317{4} 0.059 512{6} 0.048 404{3} 
MSE α̂ 0.016 643{2} 0.037 248{3} 0.039 84{4} 0.014 772{1} 0.060 102{7} 0.055 819{6} 0.047 48{5} 
â 0.226 754{7} 0.000 442{4} 0.000 661{6} 0.000 273{1} 0.000 413{3} 0.000 477{5} 0.000 315{2} 
b̂ 0.151 88{7} 0.025 922{2} 0.030 539{4} 0.012 259{1} 0.030 765{5} 0.038 864{6} 0.029 631{3} 
MRE α̂ 0.020 654{2} 0.025 894{3} 0.026 09{4} 0.017 413{1} 0.030 328{6} 0.035 741{7} 0.028 031{5} 
â 0.370 139{7} 0.013 41{4} 0.016 312{6} 0.009 508{1} 0.012 563{3} 0.014 141{5} 0.010 452{2} 
b̂ 0.104 797{7} 0.031 56{2} 0.037 202{5} 0.021 825{1} 0.035 545{4} 0.039 675{6} 0.032 269{3} 
∑Ranks  48{6} 27{2} 44{5} 9{1} 41{4} 53{7} 30{3} 
120 BIAS α̂ 0.011 545{1} 0.013 36{2} 0.018 925{5} 0.014 751{3} 0.021 08{6} 0.028 539{7} 0.015 579{4} 
â 0.055 903{7} 0.001 932{2} 0.002 563{4} 0.001 718{1} 0.002 232{3} 0.002 632{5} 0.002 984{6} 
b̂ 0.050 469{7} 0.016 656{3} 0.016 157{1} 0.016 297{2} 0.018 107{4} 0.023 743{6} 0.018 797{5} 
MSE α̂ 0.003 914{1} 0.005 997{3} 0.016 488{5} 0.004 83{2} 0.016 884{6} 0.022 611{7} 0.007 987{4} 
â 0.064 308{7} 0.000 105{2} 0.000 187{5.5} 7e − 05{1} 0.000 13{3} 0.000 157{4} 0.000 187{5.5} 
b̂ 0.049 066{7} 0.007 427{2} 0.010 204{5} 0.006 062{1} 0.009 239{4} 0.018 15{6} 0.008 392{3} 
MRE α̂ 0.005 772{1} 0.006 68{2} 0.009 462{5} 0.007 376{3} 0.010 54{6} 0.014 27{7} 0.007 79{4} 
â 0.111 807{7} 0.003 865{2} 0.005 126{4} 0.003 437{1} 0.004 464{3} 0.005 265{5} 0.005 969{6} 
b̂ 0.033 646{7} 0.011 104{3} 0.010 772{1} 0.010 865{2} 0.012 071{4} 0.015 829{6} 0.012 532{5} 
∑Ranks  44{6} 20{1} 34.5{3} 22{2} 38{4} 52{7} 41.5{5} 
150 BIAS α̂ 0.006 579{4} 0.004 386{1} 0.012 804{7} 0.007 906{5} 0.006 332{3} 0.010 966{6} 0.005 342{2} 
â 0.025 866{7} 0.000 641{1} 0.001 181{4} 0.000 767{2} 0.001 081{3} 0.001 239{5.5} 0.001 239{5.5} 
b̂ 0.025 212{7} 0.003 157{1} 0.010 213{6} 0.006 962{3} 0.005 899{2} 0.009 68{5} 0.007 425{4} 
MSE α̂ 0.002 116{3} 0.001 569{1} 0.010 816{7} 0.003 391{4} 0.003 853{5} 0.007 297{6} 0.001 767{2} 
â 0.027 339{7} 3.1e − 05{1} 5.3e − 05{3} 3.8e − 05{2} 7.7e − 05{6} 7.5e − 05{5} 6.6e − 05{4} 
b̂ 0.024 463{7} 0.000 783{1} 0.005 431{5} 0.002 576{3} 0.002 574{2} 0.005 866{6} 0.002 75{4} 
MRE α̂ 0.003 289{4} 0.002 193{1} 0.006 402{7} 0.003 953{5} 0.003 166{3} 0.005 483{6} 0.002 671{2} 
â 0.051 733{7} 0.001 282{1} 0.002 362{4} 0.001 534{2} 0.002 163{3} 0.002 477{5} 0.002 478{6} 
b̂ 0.016 808{7} 0.002 105{1} 0.006 809{6} 0.004 641{3} 0.003 932{2} 0.006 453{5} 0.004 95{4} 
∑Ranks  47{5} 10{1} 50{6} 30{2.5} 30{2.5} 50.5{7} 34.5{4} 
nEst.Est. Par.MLEADECVMEMPSELSERTADEWLSE
80 BIAS α̂ 0.041 309{2} 0.051 789{3} 0.052 18{4} 0.034 827{1} 0.060 656{6} 0.071 481{7} 0.056 063{5} 
â 0.185 069{7} 0.006 705{4} 0.008 156{6} 0.004 754{1} 0.006 282{3} 0.007 07{5} 0.005 226{2} 
b̂ 0.157 195{7} 0.047 339{2} 0.055 803{5} 0.032 737{1} 0.053 317{4} 0.059 512{6} 0.048 404{3} 
MSE α̂ 0.016 643{2} 0.037 248{3} 0.039 84{4} 0.014 772{1} 0.060 102{7} 0.055 819{6} 0.047 48{5} 
â 0.226 754{7} 0.000 442{4} 0.000 661{6} 0.000 273{1} 0.000 413{3} 0.000 477{5} 0.000 315{2} 
b̂ 0.151 88{7} 0.025 922{2} 0.030 539{4} 0.012 259{1} 0.030 765{5} 0.038 864{6} 0.029 631{3} 
MRE α̂ 0.020 654{2} 0.025 894{3} 0.026 09{4} 0.017 413{1} 0.030 328{6} 0.035 741{7} 0.028 031{5} 
â 0.370 139{7} 0.013 41{4} 0.016 312{6} 0.009 508{1} 0.012 563{3} 0.014 141{5} 0.010 452{2} 
b̂ 0.104 797{7} 0.031 56{2} 0.037 202{5} 0.021 825{1} 0.035 545{4} 0.039 675{6} 0.032 269{3} 
∑Ranks  48{6} 27{2} 44{5} 9{1} 41{4} 53{7} 30{3} 
120 BIAS α̂ 0.011 545{1} 0.013 36{2} 0.018 925{5} 0.014 751{3} 0.021 08{6} 0.028 539{7} 0.015 579{4} 
â 0.055 903{7} 0.001 932{2} 0.002 563{4} 0.001 718{1} 0.002 232{3} 0.002 632{5} 0.002 984{6} 
b̂ 0.050 469{7} 0.016 656{3} 0.016 157{1} 0.016 297{2} 0.018 107{4} 0.023 743{6} 0.018 797{5} 
MSE α̂ 0.003 914{1} 0.005 997{3} 0.016 488{5} 0.004 83{2} 0.016 884{6} 0.022 611{7} 0.007 987{4} 
â 0.064 308{7} 0.000 105{2} 0.000 187{5.5} 7e − 05{1} 0.000 13{3} 0.000 157{4} 0.000 187{5.5} 
b̂ 0.049 066{7} 0.007 427{2} 0.010 204{5} 0.006 062{1} 0.009 239{4} 0.018 15{6} 0.008 392{3} 
MRE α̂ 0.005 772{1} 0.006 68{2} 0.009 462{5} 0.007 376{3} 0.010 54{6} 0.014 27{7} 0.007 79{4} 
â 0.111 807{7} 0.003 865{2} 0.005 126{4} 0.003 437{1} 0.004 464{3} 0.005 265{5} 0.005 969{6} 
b̂ 0.033 646{7} 0.011 104{3} 0.010 772{1} 0.010 865{2} 0.012 071{4} 0.015 829{6} 0.012 532{5} 
∑Ranks  44{6} 20{1} 34.5{3} 22{2} 38{4} 52{7} 41.5{5} 
150 BIAS α̂ 0.006 579{4} 0.004 386{1} 0.012 804{7} 0.007 906{5} 0.006 332{3} 0.010 966{6} 0.005 342{2} 
â 0.025 866{7} 0.000 641{1} 0.001 181{4} 0.000 767{2} 0.001 081{3} 0.001 239{5.5} 0.001 239{5.5} 
b̂ 0.025 212{7} 0.003 157{1} 0.010 213{6} 0.006 962{3} 0.005 899{2} 0.009 68{5} 0.007 425{4} 
MSE α̂ 0.002 116{3} 0.001 569{1} 0.010 816{7} 0.003 391{4} 0.003 853{5} 0.007 297{6} 0.001 767{2} 
â 0.027 339{7} 3.1e − 05{1} 5.3e − 05{3} 3.8e − 05{2} 7.7e − 05{6} 7.5e − 05{5} 6.6e − 05{4} 
b̂ 0.024 463{7} 0.000 783{1} 0.005 431{5} 0.002 576{3} 0.002 574{2} 0.005 866{6} 0.002 75{4} 
MRE α̂ 0.003 289{4} 0.002 193{1} 0.006 402{7} 0.003 953{5} 0.003 166{3} 0.005 483{6} 0.002 671{2} 
â 0.051 733{7} 0.001 282{1} 0.002 362{4} 0.001 534{2} 0.002 163{3} 0.002 477{5} 0.002 478{6} 
b̂ 0.016 808{7} 0.002 105{1} 0.006 809{6} 0.004 641{3} 0.003 932{2} 0.006 453{5} 0.004 95{4} 
∑Ranks  47{5} 10{1} 50{6} 30{2.5} 30{2.5} 50.5{7} 34.5{4} 
View Large
TABLE IV.

(Continued.)

nEst.Est. Par.MLEADECVMEMPSELSERTADEWLSE
  â 0.903 49{7} 0.016 535{1} 0.021 926{4} 0.018 837{2} 0.024 376{6} 0.020 05{3} 0.023 133{5} 
  b̂ 0.257 368{2} 0.305 459{3} 0.391 061{7} 0.228 349{1} 0.385 918{5} 0.364 449{4} 0.388 049{6} 
 MRE α̂ 0.207 529{3} 0.238 091{7} 0.232 403{5} 0.105 844{1} 0.206 883{2} 0.234 142{6} 0.225 608{4} 
 â 0.498 548{7} 0.057 85{1} 0.074 624{5} 0.067 288{2} 0.077 433{6} 0.071 757{3} 0.073 539{4} 
 b̂ 0.159 513{1} 0.237 823{7} 0.237 388{6} 0.164 133{2} 0.235 342{4} 0.231 49{3} 0.236 148{5} 
 ∑Ranks  34{3} 43{6} 48{7} 12{1} 41{4.5} 33{2} 41{4.5} 
nEst.Est. Par.MLEADECVMEMPSELSERTADEWLSE
  â 0.903 49{7} 0.016 535{1} 0.021 926{4} 0.018 837{2} 0.024 376{6} 0.020 05{3} 0.023 133{5} 
  b̂ 0.257 368{2} 0.305 459{3} 0.391 061{7} 0.228 349{1} 0.385 918{5} 0.364 449{4} 0.388 049{6} 
 MRE α̂ 0.207 529{3} 0.238 091{7} 0.232 403{5} 0.105 844{1} 0.206 883{2} 0.234 142{6} 0.225 608{4} 
 â 0.498 548{7} 0.057 85{1} 0.074 624{5} 0.067 288{2} 0.077 433{6} 0.071 757{3} 0.073 539{4} 
 b̂ 0.159 513{1} 0.237 823{7} 0.237 388{6} 0.164 133{2} 0.235 342{4} 0.231 49{3} 0.236 148{5} 
 ∑Ranks  34{3} 43{6} 48{7} 12{1} 41{4.5} 33{2} 41{4.5} 
View Large
TABLE V.

(Continued.)

nEst.Est. Par.MLEADECVMEMPSELSERTADEWLSE
 MRE α̂ 0.335 378{1} 0.391 61{5} 0.380 429{2} 0.382 935{3} 0.401 054{7} 0.384 708{4} 0.396 514{6} 
 â 0.268 243{7} 0.108 043{4} 0.111 425{5} 0.104 231{2} 0.117 107{6} 0.097 799{1} 0.106 259{3} 
 b̂ 0.088 569{1} 0.307 048{4} 0.3004{3} 0.310 943{5} 0.318 181{7} 0.294 306{2} 0.313 113{6} 
 ∑Ranks  27{2} 37{4} 30{3} 39{5} 58{7} 19{1} 42{6} 
150 BIAS α̂ 0.487 262{1} 0.565 174{4} 0.554{3} 0.534 335{2} 0.593 447{7} 0.568 326{5} 0.588 011{6} 
â 0.400 499{7} 0.145 432{3} 0.154 239{5} 0.144 107{2} 0.164 45{6} 0.136 428{1} 0.152 753{4} 
b̂ 0.128 161{1} 0.442 269{5} 0.426 227{2} 0.430 13{3} 0.464 91{7} 0.431 775{4} 0.462 946{6} 
MSE α̂ 0.351 315{1} 0.435 618{4} 0.408 473{2} 0.457 044{5} 0.457 342{6} 0.417 949{3} 0.458 448{7} 
â 0.271 506{7} 0.038 845{3} 0.038 383{2} 0.041 745{4} 0.044 307{6} 0.030 573{1} 0.041 869{5} 
b̂ 0.031 044{1} 0.320 352{5} 0.295 478{3} 0.320 172{4} 0.335 538{7} 0.271 343{2} 0.334 633{6} 
MRE α̂ 0.324 842{1} 0.376 783{4} 0.369 334{3} 0.356 223{2} 0.395 632{7} 0.378 884{5} 0.392 007{6} 
â 0.266 999{7} 0.096 954{3} 0.102 826{5} 0.096 071{2} 0.109 633{6} 0.090 952{1} 0.101 835{4} 
b̂ 0.085 441{1} 0.294 846{5} 0.284 151{2} 0.286 753{3} 0.309 94{7} 0.287 85{4} 0.308 631{6} 
∑Ranks  27{3} 36{5} 27{3} 27{3} 59{7} 26{1} 50{6} 
nEst.Est. Par.MLEADECVMEMPSELSERTADEWLSE
 MRE α̂ 0.335 378{1} 0.391 61{5} 0.380 429{2} 0.382 935{3} 0.401 054{7} 0.384 708{4} 0.396 514{6} 
 â 0.268 243{7} 0.108 043{4} 0.111 425{5} 0.104 231{2} 0.117 107{6} 0.097 799{1} 0.106 259{3} 
 b̂ 0.088 569{1} 0.307 048{4} 0.3004{3} 0.310 943{5} 0.318 181{7} 0.294 306{2} 0.313 113{6} 
 ∑Ranks  27{2} 37{4} 30{3} 39{5} 58{7} 19{1} 42{6} 
150 BIAS α̂ 0.487 262{1} 0.565 174{4} 0.554{3} 0.534 335{2} 0.593 447{7} 0.568 326{5} 0.588 011{6} 
â 0.400 499{7} 0.145 432{3} 0.154 239{5} 0.144 107{2} 0.164 45{6} 0.136 428{1} 0.152 753{4} 
b̂ 0.128 161{1} 0.442 269{5} 0.426 227{2} 0.430 13{3} 0.464 91{7} 0.431 775{4} 0.462 946{6} 
MSE α̂ 0.351 315{1} 0.435 618{4} 0.408 473{2} 0.457 044{5} 0.457 342{6} 0.417 949{3} 0.458 448{7} 
â 0.271 506{7} 0.038 845{3} 0.038 383{2} 0.041 745{4} 0.044 307{6} 0.030 573{1} 0.041 869{5} 
b̂ 0.031 044{1} 0.320 352{5} 0.295 478{3} 0.320 172{4} 0.335 538{7} 0.271 343{2} 0.334 633{6} 
MRE α̂ 0.324 842{1} 0.376 783{4} 0.369 334{3} 0.356 223{2} 0.395 632{7} 0.378 884{5} 0.392 007{6} 
â 0.266 999{7} 0.096 954{3} 0.102 826{5} 0.096 071{2} 0.109 633{6} 0.090 952{1} 0.101 835{4} 
b̂ 0.085 441{1} 0.294 846{5} 0.284 151{2} 0.286 753{3} 0.309 94{7} 0.287 85{4} 0.308 631{6} 
∑Ranks  27{3} 36{5} 27{3} 27{3} 59{7} 26{1} 50{6} 
View Large
TABLE VI.

(Continued.)

nEst.Est. Par.MLEADECVMEMPSELSERTADEWLSE
 MRE α̂ 0.654 475{2} 0.720 248{6} 0.748 355{7} 0.652 47{1} 0.701 946{5} 0.688 919{3} 0.697 981{4} 
 â 0.440 137{7} 0.161 651{3} 0.164 136{4} 0.171 516{5} 0.157 177{2} 0.182 703{6} 0.157 042{1} 
 b̂ 0.891 898{7} 0.434 351{3} 0.459 912{5} 0.430 881{2} 0.435 721{4} 0.527 512{6} 0.402 949{1} 
 ∑Ranks  47{7} 33{3} 46{6} 26{2} 36{4} 44{5} 20{1} 
120 BIAS α̂ 0.327 134{3} 0.346 721{6} 0.366 296{7} 0.302 721{1} 0.337 851{4} 0.326 485{2} 0.340 321{5} 
â 0.612 279{7} 0.2309{3} 0.232 91{4} 0.240 715{5} 0.225 106{2} 0.258 773{6} 0.221 068{1} 
b̂ 0.644 244{7} 0.308 289{3} 0.309 522{5} 0.295 106{2} 0.309 166{4} 0.351 743{6} 0.280 253{1} 
MSE α̂ 0.129 908{1} 0.212 042{4} 0.261 343{7} 0.170 393{3} 0.229 933{6} 0.149 651{2} 0.222 206{5} 
â 0.525 27{7} 0.081 13{2} 0.088 351{5} 0.085 11{4} 0.082 991{3} 0.101 17{6} 0.076 226{1} 
b̂ 0.532 891{7} 0.171 696{3} 0.190 328{4} 0.151 473{2} 0.194 854{5} 0.231 707{6} 0.143 807{1} 
MRE α̂ 0.654 267{3} 0.693 442{6} 0.732 592{7} 0.605 442{1} 0.675 702{4} 0.652 971{2} 0.680 642{5} 
â 0.434 853{7} 0.153 933{3} 0.155 273{4} 0.160 477{5} 0.150 071{2} 0.172 515{6} 0.147 379{1} 
b̂ 0.825 658{7} 0.411 052{3} 0.412 696{5} 0.393 475{2} 0.412 221{4} 0.468 99{6} 0.373 67{1} 
∑Ranks  49{7} 33{3} 48{6} 25{2} 34{4} 42{5} 21{1} 
150 BIAS α̂ 0.319 608{5} 0.313 508{3} 0.336 786{7} 0.271 649{1} 0.330 241{6} 0.316 143{4} 0.303 052{2} 
â 0.607 219{7} 0.211 309{2} 0.216 532{3} 0.219 708{5} 0.210 553{1} 0.240 573{6} 0.217 045{4} 
b̂ 0.610 892{7} 0.270 928{1} 0.282 907{4} 0.272 805{2} 0.283 64{5} 0.313 165{6} 0.276 246{3} 
MSE α̂ 0.113 492{1} 0.182 963{5} 0.222 571{7} 0.127 506{2} 0.218 531{6} 0.139 295{3} 0.153 587{4} 
â 0.470 046{7} 0.069 467{1} 0.075 624{5} 0.073 384{3} 0.073 934{4} 0.085 068{6} 0.072 713{2} 
b̂ 0.506 707{7} 0.133 616{2} 0.155 965{4} 0.132 46{1} 0.157 436{5} 0.170 078{6} 0.137 017{3} 
MRE α̂ 0.639 216{5} 0.627 015{3} 0.673 573{7} 0.543 297{1} 0.660 483{6} 0.632 287{4} 0.546 104{2} 
â 0.388 146{7} 0.140 873{2} 0.144 355{3} 0.146 472{5} 0.140 369{1} 0.160 382{6} 0.144 697{4} 
b̂ 0.734 523{7} 0.361 237{1} 0.377 21{4} 0.363 74{2} 0.378 186{5} 0.417 554{6} 0.368 328{3} 
∑Ranks  53{7} 20{1} 44{5} 22{2} 39{4} 47{6} 27{3} 
nEst.Est. Par.MLEADECVMEMPSELSERTADEWLSE
 MRE α̂ 0.654 475{2} 0.720 248{6} 0.748 355{7} 0.652 47{1} 0.701 946{5} 0.688 919{3} 0.697 981{4} 
 â 0.440 137{7} 0.161 651{3} 0.164 136{4} 0.171 516{5} 0.157 177{2} 0.182 703{6} 0.157 042{1} 
 b̂ 0.891 898{7} 0.434 351{3} 0.459 912{5} 0.430 881{2} 0.435 721{4} 0.527 512{6} 0.402 949{1} 
 ∑Ranks  47{7} 33{3} 46{6} 26{2} 36{4} 44{5} 20{1} 
120 BIAS α̂ 0.327 134{3} 0.346 721{6} 0.366 296{7} 0.302 721{1} 0.337 851{4} 0.326 485{2} 0.340 321{5} 
â 0.612 279{7} 0.2309{3} 0.232 91{4} 0.240 715{5} 0.225 106{2} 0.258 773{6} 0.221 068{1} 
b̂ 0.644 244{7} 0.308 289{3} 0.309 522{5} 0.295 106{2} 0.309 166{4} 0.351 743{6} 0.280 253{1} 
MSE α̂ 0.129 908{1} 0.212 042{4} 0.261 343{7} 0.170 393{3} 0.229 933{6} 0.149 651{2} 0.222 206{5} 
â 0.525 27{7} 0.081 13{2} 0.088 351{5} 0.085 11{4} 0.082 991{3} 0.101 17{6} 0.076 226{1} 
b̂ 0.532 891{7} 0.171 696{3} 0.190 328{4} 0.151 473{2} 0.194 854{5} 0.231 707{6} 0.143 807{1} 
MRE α̂ 0.654 267{3} 0.693 442{6} 0.732 592{7} 0.605 442{1} 0.675 702{4} 0.652 971{2} 0.680 642{5} 
â 0.434 853{7} 0.153 933{3} 0.155 273{4} 0.160 477{5} 0.150 071{2} 0.172 515{6} 0.147 379{1} 
b̂ 0.825 658{7} 0.411 052{3} 0.412 696{5} 0.393 475{2} 0.412 221{4} 0.468 99{6} 0.373 67{1} 
∑Ranks  49{7} 33{3} 48{6} 25{2} 34{4} 42{5} 21{1} 
150 BIAS α̂ 0.319 608{5} 0.313 508{3} 0.336 786{7} 0.271 649{1} 0.330 241{6} 0.316 143{4} 0.303 052{2} 
â 0.607 219{7} 0.211 309{2} 0.216 532{3} 0.219 708{5} 0.210 553{1} 0.240 573{6} 0.217 045{4} 
b̂ 0.610 892{7} 0.270 928{1} 0.282 907{4} 0.272 805{2} 0.283 64{5} 0.313 165{6} 0.276 246{3} 
MSE α̂ 0.113 492{1} 0.182 963{5} 0.222 571{7} 0.127 506{2} 0.218 531{6} 0.139 295{3} 0.153 587{4} 
â 0.470 046{7} 0.069 467{1} 0.075 624{5} 0.073 384{3} 0.073 934{4} 0.085 068{6} 0.072 713{2} 
b̂ 0.506 707{7} 0.133 616{2} 0.155 965{4} 0.132 46{1} 0.157 436{5} 0.170 078{6} 0.137 017{3} 
MRE α̂ 0.639 216{5} 0.627 015{3} 0.673 573{7} 0.543 297{1} 0.660 483{6} 0.632 287{4} 0.546 104{2} 
â 0.388 146{7} 0.140 873{2} 0.144 355{3} 0.146 472{5} 0.140 369{1} 0.160 382{6} 0.144 697{4} 
b̂ 0.734 523{7} 0.361 237{1} 0.377 21{4} 0.363 74{2} 0.378 186{5} 0.417 554{6} 0.368 328{3} 
∑Ranks  53{7} 20{1} 44{5} 22{2} 39{4} 47{6} 27{3} 
View Large
It is possible to determine the WLSE of the suggested model’s parameters by the process of minimizing the following function:
W(α,a,b)=∑i=1n(n+1)2(n+2)i(n−i+1)F(xi|α,a,b)−in+12,
with regard to the variables α, a, and b. In addition, the WLSE may be determined by finding the solution to the following non-linear equation:
∑i=1n(n+1)2(n+2)i(n−i+1)F(xi|α,a,b)−in+1Δs(xi)=0,s=1,2,3,
where Δ1⋅|α,a,b⁠, Δ2⋅|α,a,b⁠, and Δ3⋅|α,a,b are defined in Eq. (8).
The MPS method is a traditional technique of estimate, and it is a good alternative to the MLE method because of its capability to approximate the Kullback–Leibler information measure. This ability makes the MPS method a suitable alternative to the MLE approach. Suppose that
Di(α,a,b)= Fx(i)|α,a,b−Fx(i−1)|α,a,b,for i=1,2,…,n+1,
be the uniform spacing of a random sample from the proposed model, where
Fx(0)|α,a,b=0,Fx(n+1)|α,a,b=1and ∑i=1n+1Di(α,a,b)=1.
The MPSE for α̂MPSE⁠, âMPSE⁠, and b̂MPSE can be found by maximizing the geometric mean of the spacing,
Gα,a,b=∏i=1n+1Di(α,a,b)1n+1,
with respect to α, a, and b, or, equivalently, by maximizing the logarithm of the geometric mean of sample spacings,
Hα,a,b=1n+1∑i=1n+1logDi(α,a,b).
The MPSE of the suggested model parameters may then be obtained analytically from the subsequent non-linear equations stated by
1n+1∑i=1n+11Di(α,a,b)Δs(x(i)|α,a,b)−Δs(x(i−1)|α,a,b)=0,s=1,2,3,
where Δ1⋅|α,a,b⁠, Δ2⋅|α,a,b⁠,and Δ3⋅|α,a,b are defined in Eq. (8).
We will use the previously described estimate techniques in several simulations to evaluate the suggested model’s performance. We use n = {30, 50, 80, 120, 150} sample sizes to generate N = 5000 random samples from the quantile function of the proposed distribution. Then, we calculate the average values of absolute biases (|BIAS|), mean square errors (MSEs), and point relative errors (MREs) using the following equations:
|BIAS|=1N∑i=1N|ϑ̂−ϑ|,MSEs=1N∑i=1N(ϑ̂−ϑ)2,MREs=1N∑i=1N|ϑ̂−ϑ|/ϑ,
where ϑ = α, a, b. The R programming language is used to perform all calculations. To select the initial values, we can use the information of parameters as a range of parameter values and select the set of initial values that maximize the likelihood function. This method is called ”Grid Search.” The simulation results of various values of the GAPW model parameters were presented in Tables I–VI. As the sample size increases, all calculated measures decrease, implying that all estimation methods perform admirably for estimating GAPW model parameters. From Table VII, we may determine that MPSE is superior (overall score of 91). The following Table VII summarizes the overall rank of the other estimation methods.
TABLE I.

This table records the values deduced from the simulation study for the estimated BIAS, MSE, and MRE by using initial values for the parameters (α = 0.5, a = 0.25, b = 0.75).

nEst.Est. Par.MLEADECVMEMPSELSERTADEWLSE
30 BIAS α̂ 0.295 145{7} 0.195 917{5} 0.165 901{2} 0.182 274{4} 0.167 36{3} 0.150 698{1} 0.199 188{6} 
â 0.049 501{7} 0.021 936{3} 0.019 961{1} 0.023 762{5} 0.021 591{2} 0.024 93{6} 0.022 155{4} 
b̂ 0.134 993{1} 0.181 028{4} 0.152 387{2} 0.184 439{6} 0.178 767{3} 0.202 985{7} 0.182 01{5} 
MSE α̂ 0.336 649{7} 0.171 944{5} 0.149 696{4} 0.144 144{3} 0.134 184{2} 0.085 252{1} 0.183 595{6} 
â 0.018 992{7} 0.001 961{4} 0.001 792{1} 0.002 221{5} 0.001 957{3} 0.002 432{6} 0.001 928{2} 
b̂ 0.055 59{1} 0.158 037{5} 0.127 001{2} 0.159 883{6} 0.152 625{4} 0.200 033{7} 0.151 493{3} 
MRE α̂ 0.590 289{7} 0.391 834{5} 0.331 803{2} 0.364 548{4} 0.334 72{3} 0.301 395{1} 0.398 376{6} 
â 0.198 004{7} 0.087 744{3} 0.079 843{1} 0.095 047{5} 0.086 365{2} 0.099 721{6} 0.088 62{4} 
b̂ 0.179 991{1} 0.241 371{4} 0.203 182{2} 0.245 918{6} 0.238 356{3} 0.270 646{7} 0.242 68{5} 
∑Ranks  45{7} 38{3} 17{1} 44{6} 25{2} 42{5} 41{4} 
50 BIAS α̂ 0.188 375{7} 0.099 985{5} 0.098 879{4} 0.093 964{3} 0.086 644{2} 0.077 389{1} 0.121 397{6} 
â 0.023 034{7} 0.011 329{2} 0.011 628{4} 0.011 364{3} 0.009 42{1} 0.012 353{5} 0.012 412{6} 
b̂ 0.076 847{1} 0.104 694{7} 0.092 22{4} 0.087 165{3} 0.081 851{2} 0.104 416{6} 0.093 139{5} 
MSE α̂ 0.233 196{7} 0.084 338{4} 0.088 942{5} 0.083 19{3} 0.073 749{2} 0.039 707{1} 0.119 723{6} 
â 0.005 727{7} 0.000 936{2} 0.001 059{5} 0.001 023{3} 0.000 718{1} 0.001 064{6} 0.001 033{4} 
b̂ 0.031 601{1} 0.091 388{6} 0.077 293{5} 0.0755{4} 0.066 447{2} 0.0945{7} 0.066 55{3} 
MRE α̂ 0.376 75{7} 0.199 971{5} 0.197 757{4} 0.187 928{3} 0.173 288{2} 0.154 778{1} 0.242 795{6} 
â 0.092 135{7} 0.045 317{2} 0.046 511{4} 0.045 456{3} 0.037 678{1} 0.049 411{5} 0.049 647{6} 
b̂ 0.102 463{1} 0.139 592{7} 0.122 959{4} 0.116 219{3} 0.109 134{2} 0.139 222{6} 0.124 186{5} 
∑Ranks  45{6} 40{5} 39{4} 28{2} 15{1} 38{3} 47{7} 
80 BIAS α̂ 0.063 337{7} 0.039 087{5} 0.036 312{4} 0.039 822{6} 0.034 614{2} 0.022 682{1} 0.035 916{3} 
â 0.006 44{7} 0.004 204{3} 0.004 08{2} 0.004 998{6} 0.004 79{5} 0.003 608{1} 0.004 516{4} 
b̂ 0.025 935{1} 0.035 682{5} 0.029 361{2} 0.040 225{7} 0.039 546{6} 0.030 727{3} 0.034 089{4} 
MSE α̂ 0.0764{7} 0.033 615{6} 0.030 581{4} 0.033 319{5} 0.024 162{2} 0.010 805{1} 0.030 252{3} 
â 0.001 183{7} 0.000 349{3} 0.000 303{2} 0.000 446{6} 0.000 415{5} 0.000 277{1} 0.000 399{4} 
b̂ 0.010 736{1} 0.030 545{5} 0.019 328{2} 0.036 394{7} 0.035 962{6} 0.027 219{3} 0.028 638{4} 
MRE α̂ 0.126 673{7} 0.078 175{5} 0.072 624{4} 0.079 643{6} 0.069 228{2} 0.045 365{1} 0.071 832{3} 
â 0.025 76{7} 0.016 818{3} 0.016 32{2} 0.019 992{6} 0.019 159{5} 0.014 433{1} 0.018 066{4} 
b̂ 0.034 58{1} 0.047 576{5} 0.039 148{2} 0.053 633{7} 0.052 728{6} 0.040 969{3} 0.045 452{4} 
∑Ranks  45{6} 40{5} 24{2} 56{7} 39{4} 15{1} 33{3} 
120 BIAS α̂ 0.016 967{7} 0.012 47{4} 0.012 27{3} 0.016 781{6} 0.007 314{1} 0.007 944{2} 0.015 029{5} 
â 0.001 437{6} 0.001 33{4} 0.001 175{3} 0.001 656{7} 0.000 902{1} 0.001 048{2} 0.001 41{5} 
b̂ 0.007 002{2} 0.009 782{6} 0.008 644{3} 0.009 563{5} 0.006 608{1} 0.008 758{4} 0.009 874{7} 
MSE α̂ 0.019 22{7} 0.010 614{3} 0.012 797{4} 0.017 883{6} 0.005 312{2} 0.003 769{1} 0.014 627{5} 
â 0.000 185{7} 0.000 113{4} 9e − 05{3} 0.000 138{6} 7.1e − 05{1.5} 7.1e − 05{1.5} 0.000 116{5} 
b̂ 0.002 893{1} 0.007 068{7} 0.005 919{5} 0.005 837{3} 0.004 483{2} 0.005 916{4} 0.006 526{6} 
MRE α̂ 0.033 933{7} 0.024 939{4} 0.024 54{3} 0.033 562{6} 0.014 627{1} 0.015 889{2} 0.030 057{5} 
â 0.005 749{6} 0.005 321{4} 0.004 699{3} 0.006 625{7} 0.003 606{1} 0.004 193{2} 0.005 642{5} 
b̂ 0.009 336{2} 0.013 042{6} 0.011 526{3} 0.012 75{5} 0.008 811{1} 0.011 678{4} 0.013 165{7} 
∑Ranks  42{5} 39{4} 34{3} 48{7} 15.5{1} 26.5{2} 47{6} 
nEst.Est. Par.MLEADECVMEMPSELSERTADEWLSE
30 BIAS α̂ 0.295 145{7} 0.195 917{5} 0.165 901{2} 0.182 274{4} 0.167 36{3} 0.150 698{1} 0.199 188{6} 
â 0.049 501{7} 0.021 936{3} 0.019 961{1} 0.023 762{5} 0.021 591{2} 0.024 93{6} 0.022 155{4} 
b̂ 0.134 993{1} 0.181 028{4} 0.152 387{2} 0.184 439{6} 0.178 767{3} 0.202 985{7} 0.182 01{5} 
MSE α̂ 0.336 649{7} 0.171 944{5} 0.149 696{4} 0.144 144{3} 0.134 184{2} 0.085 252{1} 0.183 595{6} 
â 0.018 992{7} 0.001 961{4} 0.001 792{1} 0.002 221{5} 0.001 957{3} 0.002 432{6} 0.001 928{2} 
b̂ 0.055 59{1} 0.158 037{5} 0.127 001{2} 0.159 883{6} 0.152 625{4} 0.200 033{7} 0.151 493{3} 
MRE α̂ 0.590 289{7} 0.391 834{5} 0.331 803{2} 0.364 548{4} 0.334 72{3} 0.301 395{1} 0.398 376{6} 
â 0.198 004{7} 0.087 744{3} 0.079 843{1} 0.095 047{5} 0.086 365{2} 0.099 721{6} 0.088 62{4} 
b̂ 0.179 991{1} 0.241 371{4} 0.203 182{2} 0.245 918{6} 0.238 356{3} 0.270 646{7} 0.242 68{5} 
∑Ranks  45{7} 38{3} 17{1} 44{6} 25{2} 42{5} 41{4} 
50 BIAS α̂ 0.188 375{7} 0.099 985{5} 0.098 879{4} 0.093 964{3} 0.086 644{2} 0.077 389{1} 0.121 397{6} 
â 0.023 034{7} 0.011 329{2} 0.011 628{4} 0.011 364{3} 0.009 42{1} 0.012 353{5} 0.012 412{6} 
b̂ 0.076 847{1} 0.104 694{7} 0.092 22{4} 0.087 165{3} 0.081 851{2} 0.104 416{6} 0.093 139{5} 
MSE α̂ 0.233 196{7} 0.084 338{4} 0.088 942{5} 0.083 19{3} 0.073 749{2} 0.039 707{1} 0.119 723{6} 
â 0.005 727{7} 0.000 936{2} 0.001 059{5} 0.001 023{3} 0.000 718{1} 0.001 064{6} 0.001 033{4} 
b̂ 0.031 601{1} 0.091 388{6} 0.077 293{5} 0.0755{4} 0.066 447{2} 0.0945{7} 0.066 55{3} 
MRE α̂ 0.376 75{7} 0.199 971{5} 0.197 757{4} 0.187 928{3} 0.173 288{2} 0.154 778{1} 0.242 795{6} 
â 0.092 135{7} 0.045 317{2} 0.046 511{4} 0.045 456{3} 0.037 678{1} 0.049 411{5} 0.049 647{6} 
b̂ 0.102 463{1} 0.139 592{7} 0.122 959{4} 0.116 219{3} 0.109 134{2} 0.139 222{6} 0.124 186{5} 
∑Ranks  45{6} 40{5} 39{4} 28{2} 15{1} 38{3} 47{7} 
80 BIAS α̂ 0.063 337{7} 0.039 087{5} 0.036 312{4} 0.039 822{6} 0.034 614{2} 0.022 682{1} 0.035 916{3} 
â 0.006 44{7} 0.004 204{3} 0.004 08{2} 0.004 998{6} 0.004 79{5} 0.003 608{1} 0.004 516{4} 
b̂ 0.025 935{1} 0.035 682{5} 0.029 361{2} 0.040 225{7} 0.039 546{6} 0.030 727{3} 0.034 089{4} 
MSE α̂ 0.0764{7} 0.033 615{6} 0.030 581{4} 0.033 319{5} 0.024 162{2} 0.010 805{1} 0.030 252{3} 
â 0.001 183{7} 0.000 349{3} 0.000 303{2} 0.000 446{6} 0.000 415{5} 0.000 277{1} 0.000 399{4} 
b̂ 0.010 736{1} 0.030 545{5} 0.019 328{2} 0.036 394{7} 0.035 962{6} 0.027 219{3} 0.028 638{4} 
MRE α̂ 0.126 673{7} 0.078 175{5} 0.072 624{4} 0.079 643{6} 0.069 228{2} 0.045 365{1} 0.071 832{3} 
â 0.025 76{7} 0.016 818{3} 0.016 32{2} 0.019 992{6} 0.019 159{5} 0.014 433{1} 0.018 066{4} 
b̂ 0.034 58{1} 0.047 576{5} 0.039 148{2} 0.053 633{7} 0.052 728{6} 0.040 969{3} 0.045 452{4} 
∑Ranks  45{6} 40{5} 24{2} 56{7} 39{4} 15{1} 33{3} 
120 BIAS α̂ 0.016 967{7} 0.012 47{4} 0.012 27{3} 0.016 781{6} 0.007 314{1} 0.007 944{2} 0.015 029{5} 
â 0.001 437{6} 0.001 33{4} 0.001 175{3} 0.001 656{7} 0.000 902{1} 0.001 048{2} 0.001 41{5} 
b̂ 0.007 002{2} 0.009 782{6} 0.008 644{3} 0.009 563{5} 0.006 608{1} 0.008 758{4} 0.009 874{7} 
MSE α̂ 0.019 22{7} 0.010 614{3} 0.012 797{4} 0.017 883{6} 0.005 312{2} 0.003 769{1} 0.014 627{5} 
â 0.000 185{7} 0.000 113{4} 9e − 05{3} 0.000 138{6} 7.1e − 05{1.5} 7.1e − 05{1.5} 0.000 116{5} 
b̂ 0.002 893{1} 0.007 068{7} 0.005 919{5} 0.005 837{3} 0.004 483{2} 0.005 916{4} 0.006 526{6} 
MRE α̂ 0.033 933{7} 0.024 939{4} 0.024 54{3} 0.033 562{6} 0.014 627{1} 0.015 889{2} 0.030 057{5} 
â 0.005 749{6} 0.005 321{4} 0.004 699{3} 0.006 625{7} 0.003 606{1} 0.004 193{2} 0.005 642{5} 
b̂ 0.009 336{2} 0.013 042{6} 0.011 526{3} 0.012 75{5} 0.008 811{1} 0.011 678{4} 0.013 165{7} 
∑Ranks  42{5} 39{4} 34{3} 48{7} 15.5{1} 26.5{2} 47{6} 
View Large
TABLE II.

The results of the record the values deduced from the simulation study for the estimated BIAS, MSE, and MRE by using initial values for the parameters (α = 1.5, a = 0.75, b = 0.75).

nEst.Est. Par.MLEADECVMEMPSELSERTADEWLSE
30 BIAS α̂ 0.514 397{1} 0.670 879{5} 0.625 645{2} 0.644 594{4} 0.676 246{7} 0.630 039{3} 0.675 912{6} 
â 0.299 085{7} 0.121 116{2} 0.125 317{4} 0.124 234{3} 0.132 427{6} 0.108 467{1} 0.127 164{5} 
b̂ 0.102 282{1} 0.362 166{6} 0.322 167{3} 0.366 355{7} 0.349 446{4} 0.291 96{2} 0.356 59{5} 
MSE α̂ 0.373 108{1} 0.567 374{7} 0.484 71{3} 0.557 078{5} 0.555 33{4} 0.468 922{2} 0.559 806{6} 
â 0.160 21{7} 0.022 013{2} 0.026 358{6} 0.023 068{3} 0.025 756{5} 0.019 035{1} 0.024 358{4} 
b̂ 0.016 749{1} 0.239 626{7} 0.181 211{3} 0.232 051{6} 0.207 419{4} 0.135 663{2} 0.215 07{5} 
MRE α̂ 0.342 932{1} 0.447 253{5} 0.417 096{2} 0.429 729{4} 0.450 831{7} 0.420 026{3} 0.450 608{6} 
â 0.398 78{7} 0.161 489{2} 0.167 089{4} 0.165 646{3} 0.176 569{6} 0.144 623{1} 0.169 552{5} 
b̂ 0.136 377{1} 0.482 888{6} 0.429 556{3} 0.488 473{7} 0.465 928{4} 0.389 28{2} 0.475 454{5} 
∑Ranks  27{2} 42{4.5} 30{3} 42{4.5} 47{6.5} 17{1} 47{6.5} 
50 BIAS α̂ 0.501 382{1} 0.634 422{4} 0.615 106{2} 0.641 188{6} 0.636 179{5} 0.627 769{3} 0.668 103{7} 
â 0.258 14{7} 0.100 533{2} 0.102 189{3} 0.109 357{5} 0.112 719{6} 0.088 228{1} 0.106 704{4} 
b̂ 0.084 024{1} 0.330 648{5} 0.323 649{3} 0.362 477{7} 0.323 868{4} 0.287 589{2} 0.338 176{6} 
MSE α̂ 0.355 922{1} 0.515 578{5} 0.481 955{3} 0.576 649{7} 0.504 875{4} 0.478 138{2} 0.550 624{6} 
â 0.120 395{7} 0.015 523{2} 0.016 078{3} 0.019 208{6} 0.018 802{5} 0.011 939{1} 0.017 827{4} 
b̂ 0.011 059{1} 0.193 674{5} 0.179 36{4} 0.236 158{7} 0.176 588{3} 0.128 533{2} 0.194 041{6} 
MRE α̂ 0.334 255{1} 0.422 948{4} 0.410 071{2} 0.427 459{6} 0.424 119{5} 0.418 513{3} 0.445 402{7} 
â 0.344 187{7} 0.134 044{2} 0.136 252{3} 0.145 81{5} 0.150 292{6} 0.117 638{1} 0.142 272{4} 
b̂ 0.112 032{1} 0.440 864{5} 0.431 532{3} 0.483 303{7} 0.431 824{4} 0.383 451{2} 0.450 902{6} 
∑Ranks  27{3} 34{4} 26{2} 56{7} 42{5} 17{1} 50{6} 
80 BIAS α̂ 0.488 067{1} 0.600 918{4} 0.590 32{3} 0.569 606{2} 0.614 122{6} 0.602 222{5} 0.617 184{7} 
â 0.247 388{7} 0.085 186{3} 0.089 986{5} 0.082 204{2} 0.096 185{6} 0.077 335{1} 0.088 193{4} 
b̂ 0.069 381{1} 0.300 418{5} 0.295 715{4} 0.282 278{3} 0.306 717{6} 0.272 955{2} 0.308 737{7} 
MSE α̂ 0.341 307{1} 0.473 511{4} 0.450 916{3} 0.477 998{5} 0.481 81{6} 0.447 401{2} 0.490 044{7} 
â 0.108 415{7} 0.011 532{3} 0.012 488{4} 0.011 516{2} 0.014 32{6} 0.009 28{1} 0.012 998{5} 
b̂ 0.007 732{1} 0.160 767{5} 0.157 988{4} 0.150 79{3} 0.162 445{6} 0.112 864{2} 0.166 998{7} 
MRE α̂ 0.325 378{1} 0.400 612{4} 0.393 546{3} 0.379 737{2} 0.409 415{6} 0.401 481{5} 0.411 456{7} 
â 0.329 85{7} 0.113 581{3} 0.119 981{5} 0.109 605{2} 0.128 247{6} 0.103 114{1} 0.117 591{4} 
b̂ 0.092 509{1} 0.400 558{5} 0.394 286{4} 0.376 371{3} 0.408 956{6} 0.363 94{2} 0.411 649{7} 
∑Ranks  27{3} 36{5} 35{4} 24{2} 54{6} 21{1} 55{7} 
nEst.Est. Par.MLEADECVMEMPSELSERTADEWLSE
30 BIAS α̂ 0.514 397{1} 0.670 879{5} 0.625 645{2} 0.644 594{4} 0.676 246{7} 0.630 039{3} 0.675 912{6} 
â 0.299 085{7} 0.121 116{2} 0.125 317{4} 0.124 234{3} 0.132 427{6} 0.108 467{1} 0.127 164{5} 
b̂ 0.102 282{1} 0.362 166{6} 0.322 167{3} 0.366 355{7} 0.349 446{4} 0.291 96{2} 0.356 59{5} 
MSE α̂ 0.373 108{1} 0.567 374{7} 0.484 71{3} 0.557 078{5} 0.555 33{4} 0.468 922{2} 0.559 806{6} 
â 0.160 21{7} 0.022 013{2} 0.026 358{6} 0.023 068{3} 0.025 756{5} 0.019 035{1} 0.024 358{4} 
b̂ 0.016 749{1} 0.239 626{7} 0.181 211{3} 0.232 051{6} 0.207 419{4} 0.135 663{2} 0.215 07{5} 
MRE α̂ 0.342 932{1} 0.447 253{5} 0.417 096{2} 0.429 729{4} 0.450 831{7} 0.420 026{3} 0.450 608{6} 
â 0.398 78{7} 0.161 489{2} 0.167 089{4} 0.165 646{3} 0.176 569{6} 0.144 623{1} 0.169 552{5} 
b̂ 0.136 377{1} 0.482 888{6} 0.429 556{3} 0.488 473{7} 0.465 928{4} 0.389 28{2} 0.475 454{5} 
∑Ranks  27{2} 42{4.5} 30{3} 42{4.5} 47{6.5} 17{1} 47{6.5} 
50 BIAS α̂ 0.501 382{1} 0.634 422{4} 0.615 106{2} 0.641 188{6} 0.636 179{5} 0.627 769{3} 0.668 103{7} 
â 0.258 14{7} 0.100 533{2} 0.102 189{3} 0.109 357{5} 0.112 719{6} 0.088 228{1} 0.106 704{4} 
b̂ 0.084 024{1} 0.330 648{5} 0.323 649{3} 0.362 477{7} 0.323 868{4} 0.287 589{2} 0.338 176{6} 
MSE α̂ 0.355 922{1} 0.515 578{5} 0.481 955{3} 0.576 649{7} 0.504 875{4} 0.478 138{2} 0.550 624{6} 
â 0.120 395{7} 0.015 523{2} 0.016 078{3} 0.019 208{6} 0.018 802{5} 0.011 939{1} 0.017 827{4} 
b̂ 0.011 059{1} 0.193 674{5} 0.179 36{4} 0.236 158{7} 0.176 588{3} 0.128 533{2} 0.194 041{6} 
MRE α̂ 0.334 255{1} 0.422 948{4} 0.410 071{2} 0.427 459{6} 0.424 119{5} 0.418 513{3} 0.445 402{7} 
â 0.344 187{7} 0.134 044{2} 0.136 252{3} 0.145 81{5} 0.150 292{6} 0.117 638{1} 0.142 272{4} 
b̂ 0.112 032{1} 0.440 864{5} 0.431 532{3} 0.483 303{7} 0.431 824{4} 0.383 451{2} 0.450 902{6} 
∑Ranks  27{3} 34{4} 26{2} 56{7} 42{5} 17{1} 50{6} 
80 BIAS α̂ 0.488 067{1} 0.600 918{4} 0.590 32{3} 0.569 606{2} 0.614 122{6} 0.602 222{5} 0.617 184{7} 
â 0.247 388{7} 0.085 186{3} 0.089 986{5} 0.082 204{2} 0.096 185{6} 0.077 335{1} 0.088 193{4} 
b̂ 0.069 381{1} 0.300 418{5} 0.295 715{4} 0.282 278{3} 0.306 717{6} 0.272 955{2} 0.308 737{7} 
MSE α̂ 0.341 307{1} 0.473 511{4} 0.450 916{3} 0.477 998{5} 0.481 81{6} 0.447 401{2} 0.490 044{7} 
â 0.108 415{7} 0.011 532{3} 0.012 488{4} 0.011 516{2} 0.014 32{6} 0.009 28{1} 0.012 998{5} 
b̂ 0.007 732{1} 0.160 767{5} 0.157 988{4} 0.150 79{3} 0.162 445{6} 0.112 864{2} 0.166 998{7} 
MRE α̂ 0.325 378{1} 0.400 612{4} 0.393 546{3} 0.379 737{2} 0.409 415{6} 0.401 481{5} 0.411 456{7} 
â 0.329 85{7} 0.113 581{3} 0.119 981{5} 0.109 605{2} 0.128 247{6} 0.103 114{1} 0.117 591{4} 
b̂ 0.092 509{1} 0.400 558{5} 0.394 286{4} 0.376 371{3} 0.408 956{6} 0.363 94{2} 0.411 649{7} 
∑Ranks  27{3} 36{5} 35{4} 24{2} 54{6} 21{1} 55{7} 
View Large
TABLE III.

This table records the values deduced from the simulation study for the estimated BIAS, MSE, and MRE by using initial values for the parameters (α = 2, a = 0.5, b = 1.5).

nEst.Est. Par.MLEADECVMEMPSELSERTADEWLSE
30 BIAS α̂ 0.123 296{1} 0.232 145{3} 0.441 899{7} 0.181 039{2} 0.361 298{6} 0.284 259{5} 0.241 874{4} 
â 0.469 914{7} 0.032 091{2} 0.040 063{6} 0.027 335{1} 0.035 933{5} 0.033 798{4} 0.032 676{3} 
b̂ 0.400 684{7} 0.221 948{3} 0.247 833{6} 0.169 775{1} 0.222 068{4} 0.220 809{2} 0.222 485{5} 
MSE α̂ 0.060 663{1} 0.249 848{3} 15.820 559{7} 0.209 421{2} 14.152 989{6} 0.777 966{5} 0.349 29{4} 
â 0.605 871{7} 0.003 414{2} 0.005 602{6} 0.002 478{1} 0.004 222{5} 0.003 797{4} 0.003 628{3} 
b̂ 0.388 844{7} 0.174 365{4} 0.258 599{6} 0.104 08{1} 0.167 78{3} 0.163 314{2} 0.175 695{5} 
MRE α̂ 0.061 648{1} 0.116 073{3} 0.220 95{7} 0.090 519{2} 0.180 649{6} 0.142 13{5} 0.120 937{4} 
â 0.939 828{7} 0.064 181{2} 0.080 125{6} 0.054 67{1} 0.071 867{5} 0.067 596{4} 0.065 352{3} 
b̂ 0.267 123{7} 0.147 965{3} 0.165 222{6} 0.113 184{1} 0.148 045{4} 0.147 206{2} 0.148 323{5} 
∑Ranks  45{6} 25{2} 57{7} 12{1} 44{5} 33{3} 36{4} 
50 BIAS α̂ 0.073 924{1} 0.110 572{4} 0.146 165{6} 0.097 104{2} 0.126 056{5} 0.159 05{7} 0.101 196{3} 
â 0.318 05{7} 0.016 054{4} 0.020 216{6} 0.013 524{1} 0.019 646{5} 0.015 473{3} 0.015 262{2} 
b̂ 0.271 754{7} 0.113 793{3} 0.141 891{6} 0.095 079{1} 0.117 897{4} 0.129 821{5} 0.102 406{2} 
MSE α̂ 0.031 871{1} 0.087 355{4} 0.176 673{7} 0.057 837{2} 0.129 536{5} 0.153 037{6} 0.085 931{3} 
â 0.391 567{7} 0.001 366{3} 0.001 968{6} 0.000 938{1} 0.001 946{5} 0.001 269{2} 0.001 396{4} 
b̂ 0.263 608{7} 0.073 818{4} 0.101 12{6} 0.047 969{1} 0.071 776{3} 0.093 265{5} 0.064 14{2} 
MRE α̂ 0.036 962{1} 0.055 286{4} 0.073 083{6} 0.048 552{2} 0.063 028{5} 0.079 525{7} 0.050 598{3} 
â 0.636 099{7} 0.032 109{4} 0.040 432{6} 0.027 048{1} 0.039 293{5} 0.030 946{3} 0.030 524{2} 
b̂ 0.181 169{7} 0.075 862{3} 0.094 594{6} 0.063 386{1} 0.078 598{4} 0.086 547{5} 0.068 271{2} 
∑Ranks  45{6} 33{3} 55{7} 12{1} 41{4} 43{5} 23{2} 
nEst.Est. Par.MLEADECVMEMPSELSERTADEWLSE
30 BIAS α̂ 0.123 296{1} 0.232 145{3} 0.441 899{7} 0.181 039{2} 0.361 298{6} 0.284 259{5} 0.241 874{4} 
â 0.469 914{7} 0.032 091{2} 0.040 063{6} 0.027 335{1} 0.035 933{5} 0.033 798{4} 0.032 676{3} 
b̂ 0.400 684{7} 0.221 948{3} 0.247 833{6} 0.169 775{1} 0.222 068{4} 0.220 809{2} 0.222 485{5} 
MSE α̂ 0.060 663{1} 0.249 848{3} 15.820 559{7} 0.209 421{2} 14.152 989{6} 0.777 966{5} 0.349 29{4} 
â 0.605 871{7} 0.003 414{2} 0.005 602{6} 0.002 478{1} 0.004 222{5} 0.003 797{4} 0.003 628{3} 
b̂ 0.388 844{7} 0.174 365{4} 0.258 599{6} 0.104 08{1} 0.167 78{3} 0.163 314{2} 0.175 695{5} 
MRE α̂ 0.061 648{1} 0.116 073{3} 0.220 95{7} 0.090 519{2} 0.180 649{6} 0.142 13{5} 0.120 937{4} 
â 0.939 828{7} 0.064 181{2} 0.080 125{6} 0.054 67{1} 0.071 867{5} 0.067 596{4} 0.065 352{3} 
b̂ 0.267 123{7} 0.147 965{3} 0.165 222{6} 0.113 184{1} 0.148 045{4} 0.147 206{2} 0.148 323{5} 
∑Ranks  45{6} 25{2} 57{7} 12{1} 44{5} 33{3} 36{4} 
50 BIAS α̂ 0.073 924{1} 0.110 572{4} 0.146 165{6} 0.097 104{2} 0.126 056{5} 0.159 05{7} 0.101 196{3} 
â 0.318 05{7} 0.016 054{4} 0.020 216{6} 0.013 524{1} 0.019 646{5} 0.015 473{3} 0.015 262{2} 
b̂ 0.271 754{7} 0.113 793{3} 0.141 891{6} 0.095 079{1} 0.117 897{4} 0.129 821{5} 0.102 406{2} 
MSE α̂ 0.031 871{1} 0.087 355{4} 0.176 673{7} 0.057 837{2} 0.129 536{5} 0.153 037{6} 0.085 931{3} 
â 0.391 567{7} 0.001 366{3} 0.001 968{6} 0.000 938{1} 0.001 946{5} 0.001 269{2} 0.001 396{4} 
b̂ 0.263 608{7} 0.073 818{4} 0.101 12{6} 0.047 969{1} 0.071 776{3} 0.093 265{5} 0.064 14{2} 
MRE α̂ 0.036 962{1} 0.055 286{4} 0.073 083{6} 0.048 552{2} 0.063 028{5} 0.079 525{7} 0.050 598{3} 
â 0.636 099{7} 0.032 109{4} 0.040 432{6} 0.027 048{1} 0.039 293{5} 0.030 946{3} 0.030 524{2} 
b̂ 0.181 169{7} 0.075 862{3} 0.094 594{6} 0.063 386{1} 0.078 598{4} 0.086 547{5} 0.068 271{2} 
∑Ranks  45{6} 33{3} 55{7} 12{1} 41{4} 43{5} 23{2} 
View Large
TABLE IV.

This table records the values deduced from the simulation study for the estimated BIAS, MSE, and MRE by using initial values for the parameters (α = 1.5, a = 1.5, b = 1.5).

nEst.Est. Par.MLEADECVMEMPSELSERTADEWLSE
30 BIAS α̂ 0.463 652{1} 1.040 781{7} 0.844 141{6} 0.696 506{3} 0.629 154{2} 0.759 961{5} 0.719 935{4} 
â 0.923 314{7} 0.225 92{3} 0.245 417{5} 0.219 341{2} 0.245 26{4} 0.210 409{1} 0.245 575{6} 
b̂ 0.6512{5} 0.648 206{4} 0.747 939{7} 0.548 102{1} 0.625 821{3} 0.611 048{2} 0.685 528{6} 
MSE α̂ 0.414 29{1} 0.977 959{5} 0.834 283{2} 0.984 53{6} 1.260 925{7} 0.837 062{3} 0.866 767{4} 
â 1.233 413{7} 0.079 018{3} 0.101 885{6} 0.074 978{2} 0.092 51{4} 0.070 956{1} 0.092 719{5} 
b̂ 0.727 807{5} 0.701 124{3} 1.152 065{7} 0.488 171{1} 0.715 144{4} 0.601 067{2} 0.766 503{6} 
MRE α̂ 0.231 826{1} 0.520 391{7} 0.422 071{6} 0.348 253{3} 0.314 577{2} 0.379 981{5} 0.359 968{4} 
â 0.615 542{7} 0.150 614{3} 0.163 612{5} 0.146 227{2} 0.163 507{4} 0.140 273{1} 0.163 717{6} 
b̂ 0.4606{7} 0.324 103{4} 0.373 969{6} 0.274 051{1} 0.312 911{3} 0.305 524{2} 0.342 764{5} 
∑Ranks  41{5} 39{4} 50{7} 21{1} 33{3} 22{2} 46{6} 
50 BIAS α̂ 0.452 31{1} 0.591 904{6} 0.546 238{3} 0.535 858{2} 0.605 064{7} 0.570 836{5} 0.558 585{4} 
â 0.859 28{7} 0.181 215{3} 0.200 655{6} 0.165 787{1} 0.198 673{5} 0.168 014{2} 0.197 94{4} 
b̂ 0.534 427{2} 0.580 159{5} 0.594 841{7} 0.467 856{1} 0.572 27{4} 0.541 401{3} 0.584 829{6} 
MSE α̂ 0.385 974{1} 0.641 581{4} 0.637 613{3} 0.782 331{7} 0.777 007{6} 0.642 561{5} 0.625 857{2} 
â 1.122 064{7} 0.051 379{3} 0.063 751{5} 0.048 538{2} 0.062 153{4} 0.043 517{1} 0.064 034{6} 
b̂ 0.324 588{1} 0.535 408{5} 0.577 154{7} 0.407 616{2} 0.518 528{4} 0.466 304{3} 0.562 617{6} 
MRE α̂ 0.226 155{1} 0.295 952{6} 0.273 119{3} 0.267 929{2} 0.302 532{7} 0.285 418{5} 0.279 293{4} 
â 0.572 853{7} 0.120 81{3} 0.133 77{6} 0.110 525{1} 0.132 449{5} 0.112 009{2} 0.131 96{4} 
b̂ 357 213{7} 0.290 08{4} 0.297 421{6} 0.233 928{1} 0.286 135{3} 0.270 701{2} 0.292 414{5} 
∑Ranks  34{3} 39{4} 46{7} 19{1} 45{6} 28{2} 41{5} 
80 BIAS α̂ 0.450 286{2} 0.497 464{3} 0.502 684{5} 0.395 805{1} 0.533 487{7} 0.502 507{4} 0.520 104{6} 
â 0.839 544{7} 0.143 033{2} 0.154 93{4} 0.147 164{3} 0.167 925{6} 0.137 726{1} 0.157 007{5} 
b̂ 0.507 068{3} 0.530 744{7} 0.515 754{4} 0.406 677{1} 0.527 736{5} 0.489 166{2} 0.5296{6} 
MSE α̂ 0.353 774{2} 0.474 546{4} 0.522 37{6} 0.347 555{1} 0.577 921{7} 0.423 112{3} 0.512 508{5} 
â 1.119 532{7} 0.035 865{2} 0.039 415{4} 0.037 991{3} 0.046 756{6} 0.030 114{1} 0.041 681{5} 
b̂ 0.315 959{1} 0.464 282{6} 0.441 109{4} 0.316 297{2} 0.457 587{5} 0.400 796{3} 0.473 848{7} 
MRE α̂ 0.221 143{2} 0.248 732{3} 0.251 342{5} 0.197 902{1} 0.266 744{7} 0.251 254{4} 0.260 052{6} 
â 0.559 696{7} 0.095 355{2} 0.103 287{4} 0.098 11{3} 0.111 95{6} 0.091 817{1} 0.104 671{5} 
b̂ 0.268 534{7} 0.265 372{6} 0.257 877{3} 0.203 339{1} 0.263 868{4} 0.244 583{2} 0.2648{5} 
∑Ranks  38{4} 35{3} 39{5} 16{1} 53{7} 21{2} 50{6} 
120 BIAS α̂ 0.731 056{7} 0.477 663{4} 0.475 917{3} 0.309 191{1} 0.467 957{2} 0.481 203{5} 0.485 053{6} 
â 0.815 75{7} 0.117 715{2} 0.125 165{5} 0.108 687{1} 0.133 864{6} 0.123 262{4} 0.122 806{3} 
b̂ 0.484 261{5} 0.476 938{3} 0.483 144{4} 0.333 393{1} 0.474 686{2} 0.492 906{6} 0.504 852{7} 
MSE α̂ 0.334 789{2} 0.417 684{4} 0.467 215{7} 0.209 978{1} 0.456 085{6} 0.403 493{3} 0.451 363{5} 
â 1.035 494{7} 0.025 64{3} 0.026 055{4} 0.020 757{1} 0.029 583{6} 0.024 059{2} 0.028 193{5} 
b̂ 0.302 72{2} 0.397 89{6} 0.391 369{5} 0.238 037{1} 0.387 164{3} 0.390 541{4} 0.438 866{7} 
MRE α̂ 0.220 528{2} 0.238 832{5} 0.237 958{4} 0.154 595{1} 0.233 979{3} 0.240 601{6} 0.242 527{7} 
â 0.543 833{7} 0.078 477{2} 0.083 443{5} 0.072 458{1} 0.089 243{6} 0.082 175{4} 0.081 871{3} 
b̂ 0.207 131{2} 0.238 469{4} 0.241 572{6} 0.166 696{1} 0.237 343{3} 0.240 453{5} 0.252 426{7} 
∑Ranks  41{5} 33{2} 43{6} 9{1} 37{3} 39{4} 50{7} 
 BIAS α̂ 0.415 058{3} 0.476 183{7} 0.464 806{5} 0.211 688{1} 0.373 766{2} 0.468 283{6} 0.451 216{4} 
 â 0.747 822{7} 0.116 775{6} 0.111 936{4} 0.100 932{1} 0.116 15{5} 0.107 635{2} 0.110 309{3} 
 b̂ 0.419 027{2} 0.475 646{7} 0.474 776{6} 0.328 267{1} 0.470 685{4} 0.462 98{3} 0.472 296{5} 
150 MSE α̂ 0.305 741{2} 0.378 878{4} 0.435 429{6} 0.216 446{1} 0.453 969{7} 0.363 224{3} 0.385 15{5} 
nEst.Est. Par.MLEADECVMEMPSELSERTADEWLSE
30 BIAS α̂ 0.463 652{1} 1.040 781{7} 0.844 141{6} 0.696 506{3} 0.629 154{2} 0.759 961{5} 0.719 935{4} 
â 0.923 314{7} 0.225 92{3} 0.245 417{5} 0.219 341{2} 0.245 26{4} 0.210 409{1} 0.245 575{6} 
b̂ 0.6512{5} 0.648 206{4} 0.747 939{7} 0.548 102{1} 0.625 821{3} 0.611 048{2} 0.685 528{6} 
MSE α̂ 0.414 29{1} 0.977 959{5} 0.834 283{2} 0.984 53{6} 1.260 925{7} 0.837 062{3} 0.866 767{4} 
â 1.233 413{7} 0.079 018{3} 0.101 885{6} 0.074 978{2} 0.092 51{4} 0.070 956{1} 0.092 719{5} 
b̂ 0.727 807{5} 0.701 124{3} 1.152 065{7} 0.488 171{1} 0.715 144{4} 0.601 067{2} 0.766 503{6} 
MRE α̂ 0.231 826{1} 0.520 391{7} 0.422 071{6} 0.348 253{3} 0.314 577{2} 0.379 981{5} 0.359 968{4} 
â 0.615 542{7} 0.150 614{3} 0.163 612{5} 0.146 227{2} 0.163 507{4} 0.140 273{1} 0.163 717{6} 
b̂ 0.4606{7} 0.324 103{4} 0.373 969{6} 0.274 051{1} 0.312 911{3} 0.305 524{2} 0.342 764{5} 
∑Ranks  41{5} 39{4} 50{7} 21{1} 33{3} 22{2} 46{6} 
50 BIAS α̂ 0.452 31{1} 0.591 904{6} 0.546 238{3} 0.535 858{2} 0.605 064{7} 0.570 836{5} 0.558 585{4} 
â 0.859 28{7} 0.181 215{3} 0.200 655{6} 0.165 787{1} 0.198 673{5} 0.168 014{2} 0.197 94{4} 
b̂ 0.534 427{2} 0.580 159{5} 0.594 841{7} 0.467 856{1} 0.572 27{4} 0.541 401{3} 0.584 829{6} 
MSE α̂ 0.385 974{1} 0.641 581{4} 0.637 613{3} 0.782 331{7} 0.777 007{6} 0.642 561{5} 0.625 857{2} 
â 1.122 064{7} 0.051 379{3} 0.063 751{5} 0.048 538{2} 0.062 153{4} 0.043 517{1} 0.064 034{6} 
b̂ 0.324 588{1} 0.535 408{5} 0.577 154{7} 0.407 616{2} 0.518 528{4} 0.466 304{3} 0.562 617{6} 
MRE α̂ 0.226 155{1} 0.295 952{6} 0.273 119{3} 0.267 929{2} 0.302 532{7} 0.285 418{5} 0.279 293{4} 
â 0.572 853{7} 0.120 81{3} 0.133 77{6} 0.110 525{1} 0.132 449{5} 0.112 009{2} 0.131 96{4} 
b̂ 357 213{7} 0.290 08{4} 0.297 421{6} 0.233 928{1} 0.286 135{3} 0.270 701{2} 0.292 414{5} 
∑Ranks  34{3} 39{4} 46{7} 19{1} 45{6} 28{2} 41{5} 
80 BIAS α̂ 0.450 286{2} 0.497 464{3} 0.502 684{5} 0.395 805{1} 0.533 487{7} 0.502 507{4} 0.520 104{6} 
â 0.839 544{7} 0.143 033{2} 0.154 93{4} 0.147 164{3} 0.167 925{6} 0.137 726{1} 0.157 007{5} 
b̂ 0.507 068{3} 0.530 744{7} 0.515 754{4} 0.406 677{1} 0.527 736{5} 0.489 166{2} 0.5296{6} 
MSE α̂ 0.353 774{2} 0.474 546{4} 0.522 37{6} 0.347 555{1} 0.577 921{7} 0.423 112{3} 0.512 508{5} 
â 1.119 532{7} 0.035 865{2} 0.039 415{4} 0.037 991{3} 0.046 756{6} 0.030 114{1} 0.041 681{5} 
b̂ 0.315 959{1} 0.464 282{6} 0.441 109{4} 0.316 297{2} 0.457 587{5} 0.400 796{3} 0.473 848{7} 
MRE α̂ 0.221 143{2} 0.248 732{3} 0.251 342{5} 0.197 902{1} 0.266 744{7} 0.251 254{4} 0.260 052{6} 
â 0.559 696{7} 0.095 355{2} 0.103 287{4} 0.098 11{3} 0.111 95{6} 0.091 817{1} 0.104 671{5} 
b̂ 0.268 534{7} 0.265 372{6} 0.257 877{3} 0.203 339{1} 0.263 868{4} 0.244 583{2} 0.2648{5} 
∑Ranks  38{4} 35{3} 39{5} 16{1} 53{7} 21{2} 50{6} 
120 BIAS α̂ 0.731 056{7} 0.477 663{4} 0.475 917{3} 0.309 191{1} 0.467 957{2} 0.481 203{5} 0.485 053{6} 
â 0.815 75{7} 0.117 715{2} 0.125 165{5} 0.108 687{1} 0.133 864{6} 0.123 262{4} 0.122 806{3} 
b̂ 0.484 261{5} 0.476 938{3} 0.483 144{4} 0.333 393{1} 0.474 686{2} 0.492 906{6} 0.504 852{7} 
MSE α̂ 0.334 789{2} 0.417 684{4} 0.467 215{7} 0.209 978{1} 0.456 085{6} 0.403 493{3} 0.451 363{5} 
â 1.035 494{7} 0.025 64{3} 0.026 055{4} 0.020 757{1} 0.029 583{6} 0.024 059{2} 0.028 193{5} 
b̂ 0.302 72{2} 0.397 89{6} 0.391 369{5} 0.238 037{1} 0.387 164{3} 0.390 541{4} 0.438 866{7} 
MRE α̂ 0.220 528{2} 0.238 832{5} 0.237 958{4} 0.154 595{1} 0.233 979{3} 0.240 601{6} 0.242 527{7} 
â 0.543 833{7} 0.078 477{2} 0.083 443{5} 0.072 458{1} 0.089 243{6} 0.082 175{4} 0.081 871{3} 
b̂ 0.207 131{2} 0.238 469{4} 0.241 572{6} 0.166 696{1} 0.237 343{3} 0.240 453{5} 0.252 426{7} 
∑Ranks  41{5} 33{2} 43{6} 9{1} 37{3} 39{4} 50{7} 
 BIAS α̂ 0.415 058{3} 0.476 183{7} 0.464 806{5} 0.211 688{1} 0.373 766{2} 0.468 283{6} 0.451 216{4} 
 â 0.747 822{7} 0.116 775{6} 0.111 936{4} 0.100 932{1} 0.116 15{5} 0.107 635{2} 0.110 309{3} 
 b̂ 0.419 027{2} 0.475 646{7} 0.474 776{6} 0.328 267{1} 0.470 685{4} 0.462 98{3} 0.472 296{5} 
150 MSE α̂ 0.305 741{2} 0.378 878{4} 0.435 429{6} 0.216 446{1} 0.453 969{7} 0.363 224{3} 0.385 15{5} 
View Large
TABLE V.

This table records the values deduced from the simulation study for the estimated BIAS, MSE, and MRE by using initial values for the parameters (α = 2, a = 1.5, b = 0.25).

nEst.Est. Par.MLEADECVMEMPSELSERTADEWLSE
30 BIAS α̂ 0.549 487{1} 0.676 235{5} 0.638 01{2} 0.677 053{6} 0.660 144{4} 0.639 818{3} 0.724 696{7} 
â 0.560 607{7} 0.245 651{3} 0.244 46{2} 0.265 779{5} 0.2676{6} 0.210 17{1} 0.265 092{4} 
b̂ 0.208 988{1} 0.530 581{5} 0.498 159{3} 0.580 896{7} 0.517 443{4} 0.496 398{2} 0.556 136{6} 
MSE α̂ 0.425 422{1} 0.590 471{5} 0.524 489{3} 0.615 74{6} 0.539 837{4} 0.488 101{2} 0.626 295{7} 
â 0.548 686{7} 0.092 678{2} 0.098 876{3} 0.107 552{6} 0.102 386{4} 0.068 378{1} 0.107 445{5} 
b̂ 0.071 648{1} 0.449 667{5} 0.437 095{4} 0.526 387{7} 0.412 242{3} 0.386 075{2} 0.476 779{6} 
MRE α̂ 0.366 325{1} 0.450 823{5} 0.425 34{2} 0.451 368{6} 0.440 096{4} 0.426 546{3} 0.483 131{7} 
â 0.373 738{7} 0.163 768{3} 0.162 974{2} 0.177 186{5} 0.1784{6} 0.140 114{1} 0.176 728{4} 
b̂ 0.139 326{1} 0.353 721{5} 0.332 106{3} 0.387 264{7} 0.344 962{4} 0.330 932{2} 0.370 758{6} 
∑Ranks  27{3} 38{4} 24{2} 55{7} 39{5} 17{1} 52{6} 
50 BIAS α̂ 0.543 011{1} 0.653 143{5} 0.618 73{3} 0.658 076{6} 0.640 72{4} 0.609 274{2} 0.673 554{7} 
â 0.4891{7} 0.209 611{2} 0.214 765{4} 0.214 776{5} 0.227 046{6} 0.175 664{1} 0.211 619{3} 
b̂ 0.176 892{1} 0.516 894{6} 0.476 604{3} 0.532 948{7} 0.489 903{4} 0.458 172{2} 0.514 83{5} 
MSE α̂ 0.417 941{1} 0.534 76{5} 0.481 044{3} 0.584 347{7} 0.510 948{4} 0.4598{2} 0.548 246{6} 
â 0.416 99{7} 0.067 322{2} 0.068 812{3} 0.076 783{6} 0.076 075{5} 0.047 692{1} 0.070 791{4} 
b̂ 0.050 43{1} 0.415 321{6} 0.372 649{3} 0.438 956{7} 0.382 536{4} 0.319 791{2} 0.394 132{5} 
MRE α̂ 0.362 007{1} 0.435 429{5} 0.412 487{3} 0.438 718{6} 0.427 147{4} 0.406 183{2} 0.449 036{7} 
â 0.326 067{7} 0.139 74{2} 0.143 177{4} 0.143 184{5} 0.151 364{6} 0.117 109{1} 0.141 079{3} 
b̂ 0.117 928{1} 0.344 596{6} 0.317 736{3} 0.355 299{7} 0.326 602{4} 0.305 448{2} 0.343 22{5} 
∑Ranks  27{2} 39{4} 29{3} 56{7} 41{5} 15{1} 45{6} 
80 BIAS α̂ 0.535 316{1} 0.639 745{7} 0.586 261{2} 0.605 314{3} 0.614 739{4} 0.615 767{5} 0.618 906{6} 
â 0.457 814{7} 0.182 126{3} 0.176 132{2} 0.189 413{5} 0.195 899{6} 0.151 539{1} 0.182 178{4} 
b̂ 0.148 217{1} 0.504 613{7} 0.464 846{2} 0.491 212{6} 0.468 426{3} 0.469 634{4} 0.483 56{5} 
MSE α̂ 0.407 955{1} 0.527 698{6} 0.445 202{2} 0.530 167{7} 0.482 024{4} 0.469 202{3} 0.490 433{5} 
â 0.348 535{7} 0.054 084{4} 0.048 173{2} 0.061 235{6} 0.058 311{5} 0.036 163{1} 0.052 991{3} 
b̂ 0.038 365{1} 0.405 915{7} 0.359 149{4} 0.390 077{6} 0.345 355{3} 0.314 549{2} 0.361 656{5} 
MRE α̂ 0.356 877{1} 0.426 496{7} 0.390 841{2} 0.403 543{3} 0.409 826{4} 0.410 511{5} 0.412 604{6} 
â 0.305 21{7} 0.121 418{3} 0.117 421{2} 0.126 275{5} 0.1306{6} 0.101 026{1} 0.121 452{4} 
b̂ 0.098 811{1} 0.336 409{7} 0.309 897{2} 0.327 475{6} 0.312 284{3} 0.313 089{4} 0.322 373{5} 
∑Ranks  27{3} 51{7} 20{1} 47{6} 38{4} 26{2} 43{5} 
 BIAS α̂ 0.503 067{1} 0.587 414{5} 0.570 644{2} 0.574 402{3} 0.601 582{7} 0.577 062{4} 0.594 771{6} 
 â 0.402 365{7} 0.162 065{4} 0.167 137{5} 0.156 346{2} 0.175 661{6} 0.146 698{1} 0.159 388{3} 
 b̂ 0.132 854{1} 0.460 573{4} 0.4506{3} 0.466 415{5} 0.477 271{7} 0.441 46{2} 0.469 669{6} 
 MSE α̂ 0.368 865{1} 0.457 39{4} 0.432 271{3} 0.495 51{7} 0.466 224{6} 0.423 965{2} 0.464 957{5} 
120 â 0.279 446{7} 0.044 066{4} 0.043 771{3} 0.047 209{5} 0.048 294{6} 0.034 236{1} 0.043 07{2} 
 b̂ 0.032 712{1} 0.3352{3} 0.336 26{4} 0.362 166{7} 0.352 771{6} 0.288 864{2} 0.342 633{5} 
nEst.Est. Par.MLEADECVMEMPSELSERTADEWLSE
30 BIAS α̂ 0.549 487{1} 0.676 235{5} 0.638 01{2} 0.677 053{6} 0.660 144{4} 0.639 818{3} 0.724 696{7} 
â 0.560 607{7} 0.245 651{3} 0.244 46{2} 0.265 779{5} 0.2676{6} 0.210 17{1} 0.265 092{4} 
b̂ 0.208 988{1} 0.530 581{5} 0.498 159{3} 0.580 896{7} 0.517 443{4} 0.496 398{2} 0.556 136{6} 
MSE α̂ 0.425 422{1} 0.590 471{5} 0.524 489{3} 0.615 74{6} 0.539 837{4} 0.488 101{2} 0.626 295{7} 
â 0.548 686{7} 0.092 678{2} 0.098 876{3} 0.107 552{6} 0.102 386{4} 0.068 378{1} 0.107 445{5} 
b̂ 0.071 648{1} 0.449 667{5} 0.437 095{4} 0.526 387{7} 0.412 242{3} 0.386 075{2} 0.476 779{6} 
MRE α̂ 0.366 325{1} 0.450 823{5} 0.425 34{2} 0.451 368{6} 0.440 096{4} 0.426 546{3} 0.483 131{7} 
â 0.373 738{7} 0.163 768{3} 0.162 974{2} 0.177 186{5} 0.1784{6} 0.140 114{1} 0.176 728{4} 
b̂ 0.139 326{1} 0.353 721{5} 0.332 106{3} 0.387 264{7} 0.344 962{4} 0.330 932{2} 0.370 758{6} 
∑Ranks  27{3} 38{4} 24{2} 55{7} 39{5} 17{1} 52{6} 
50 BIAS α̂ 0.543 011{1} 0.653 143{5} 0.618 73{3} 0.658 076{6} 0.640 72{4} 0.609 274{2} 0.673 554{7} 
â 0.4891{7} 0.209 611{2} 0.214 765{4} 0.214 776{5} 0.227 046{6} 0.175 664{1} 0.211 619{3} 
b̂ 0.176 892{1} 0.516 894{6} 0.476 604{3} 0.532 948{7} 0.489 903{4} 0.458 172{2} 0.514 83{5} 
MSE α̂ 0.417 941{1} 0.534 76{5} 0.481 044{3} 0.584 347{7} 0.510 948{4} 0.4598{2} 0.548 246{6} 
â 0.416 99{7} 0.067 322{2} 0.068 812{3} 0.076 783{6} 0.076 075{5} 0.047 692{1} 0.070 791{4} 
b̂ 0.050 43{1} 0.415 321{6} 0.372 649{3} 0.438 956{7} 0.382 536{4} 0.319 791{2} 0.394 132{5} 
MRE α̂ 0.362 007{1} 0.435 429{5} 0.412 487{3} 0.438 718{6} 0.427 147{4} 0.406 183{2} 0.449 036{7} 
â 0.326 067{7} 0.139 74{2} 0.143 177{4} 0.143 184{5} 0.151 364{6} 0.117 109{1} 0.141 079{3} 
b̂ 0.117 928{1} 0.344 596{6} 0.317 736{3} 0.355 299{7} 0.326 602{4} 0.305 448{2} 0.343 22{5} 
∑Ranks  27{2} 39{4} 29{3} 56{7} 41{5} 15{1} 45{6} 
80 BIAS α̂ 0.535 316{1} 0.639 745{7} 0.586 261{2} 0.605 314{3} 0.614 739{4} 0.615 767{5} 0.618 906{6} 
â 0.457 814{7} 0.182 126{3} 0.176 132{2} 0.189 413{5} 0.195 899{6} 0.151 539{1} 0.182 178{4} 
b̂ 0.148 217{1} 0.504 613{7} 0.464 846{2} 0.491 212{6} 0.468 426{3} 0.469 634{4} 0.483 56{5} 
MSE α̂ 0.407 955{1} 0.527 698{6} 0.445 202{2} 0.530 167{7} 0.482 024{4} 0.469 202{3} 0.490 433{5} 
â 0.348 535{7} 0.054 084{4} 0.048 173{2} 0.061 235{6} 0.058 311{5} 0.036 163{1} 0.052 991{3} 
b̂ 0.038 365{1} 0.405 915{7} 0.359 149{4} 0.390 077{6} 0.345 355{3} 0.314 549{2} 0.361 656{5} 
MRE α̂ 0.356 877{1} 0.426 496{7} 0.390 841{2} 0.403 543{3} 0.409 826{4} 0.410 511{5} 0.412 604{6} 
â 0.305 21{7} 0.121 418{3} 0.117 421{2} 0.126 275{5} 0.1306{6} 0.101 026{1} 0.121 452{4} 
b̂ 0.098 811{1} 0.336 409{7} 0.309 897{2} 0.327 475{6} 0.312 284{3} 0.313 089{4} 0.322 373{5} 
∑Ranks  27{3} 51{7} 20{1} 47{6} 38{4} 26{2} 43{5} 
 BIAS α̂ 0.503 067{1} 0.587 414{5} 0.570 644{2} 0.574 402{3} 0.601 582{7} 0.577 062{4} 0.594 771{6} 
 â 0.402 365{7} 0.162 065{4} 0.167 137{5} 0.156 346{2} 0.175 661{6} 0.146 698{1} 0.159 388{3} 
 b̂ 0.132 854{1} 0.460 573{4} 0.4506{3} 0.466 415{5} 0.477 271{7} 0.441 46{2} 0.469 669{6} 
 MSE α̂ 0.368 865{1} 0.457 39{4} 0.432 271{3} 0.495 51{7} 0.466 224{6} 0.423 965{2} 0.464 957{5} 
120 â 0.279 446{7} 0.044 066{4} 0.043 771{3} 0.047 209{5} 0.048 294{6} 0.034 236{1} 0.043 07{2} 
 b̂ 0.032 712{1} 0.3352{3} 0.336 26{4} 0.362 166{7} 0.352 771{6} 0.288 864{2} 0.342 633{5} 
View Large
TABLE VI.

This table records the values deduced from the simulation study for the estimated BIAS, MSE, and MRE by using initial values for the parameters (α = 0.5, a = 1.5, b = 0.75).

nEst.Est. Par.MLEADECVMEMPSELSERTADEWLSE
30 BIAS α̂ 0.362 219{1} 0.435 625{7} 0.416 558{6} 0.370 812{2} 0.387 977{4} 0.382 998{3} 0.405 039{5} 
â 0.721 031{7} 0.293 528{1} 0.309 027{4} 0.313 367{5} 0.295 793{3} 0.359 458{6} 0.294 914{2} 
b̂ 0.714 42{7} 0.422 951{5} 0.412 001{4} 0.387 035{1} 0.395 348{2} 0.512 203{6} 0.401 23{3} 
MSE α̂ 0.179 459{1} 0.347 104{7} 0.333 877{5} 0.254 955{3} 0.302 138{4} 0.213 417{2} 0.340 213{6} 
â 0.620 359{7} 0.131 969{2} 0.145 872{5} 0.143 018{4} 0.133 752{3} 0.196 016{6} 0.131 201{1} 
b̂ 0.699 204{7} 0.331 529{4} 0.334 81{5} 0.269 16{1} 0.3161{3} 0.497 735{6} 0.306 618{2} 
MRE α̂ 0.724 437{1} 0.871 25{7} 0.833 117{6} 0.741 623{2} 0.775 955{4} 0.765 997{3} 0.810 079{5} 
â 0.480 688{7} 0.195 685{1} 0.206 018{4} 0.208 911{5} 0.197 195{3} 0.239 639{6} 0.196 609{2} 
b̂ 0.952 56{7} 0.563 934{5} 0.549 335{4} 0.516 046{1} 0.527 13{2} 0.682 938{6} 0.534 973{3} 
∑Ranks  45{7} 39{4} 43{5} 24{1} 28{2} 44{6} 29{3} 
50 BIAS α̂ 0.354 469{2} 0.390 629{7} 0.387 568{5} 0.352 266{1} 0.356 464{3} 0.362 786{4} 0.390 259{6} 
â 0.666 029{7} 0.265 703{2} 0.268 142{3} 0.287 471{5} 0.258 835{1} 0.317 81{6} 0.268 636{4} 
b̂ 0.673 356{7} 0.369 86{4} 0.370 198{5} 0.362 363{2} 0.348 243{1} 0.449 974{6} 0.366 979{3} 
MSE α̂ 0.168 558{1} 0.275 456{5} 0.290 335{6} 0.227 524{3} 0.269 194{4} 0.189 998{2} 0.306 734{7} 
â 0.549 735{7} 0.107 782{1} 0.113 255{4} 0.118 896{5} 0.107 872{2} 0.152 139{6} 0.110 883{3} 
b̂ 0.600 192{7} 0.256 007{3} 0.277 895{5} 0.229 045{1} 0.252 802{2} 0.380 327{6} 0.256 36{4} 
MRE α̂ 0.708 939{2} 0.781 259{7} 0.775 136{5} 0.704 532{1} 0.712 928{3} 0.725 571{4} 0.780 517{6} 
â 0.444 019{7} 0.177 135{2} 0.178 761{3} 0.191 648{5} 0.172 556{1} 0.211 873{6} 0.179 091{4} 
b̂ 0.897 808{7} 0.493 146{4} 0.493 597{5} 0.483 151{2} 0.464 325{1} 0.599 966{6} 0.489 305{3} 
∑Ranks  47{7} 35{3} 41{5} 25{2} 18{1} 46{6} 40{4} 
 BIAS α̂ 0.327 237{2} 0.360 124{6} 0.374 178{7} 0.326 235{1} 0.350 973{5} 0.344 459{3} 0.348 991{4} 
 â 0.639 205{7} 0.242 477{3} 0.246 203{4} 0.257 273{5} 0.235 765{2} 0.274 055{6} 0.235 564{1} 
 b̂ 0.668 923{7} 0.325 763{3} 0.344 934{5} 0.323 161{2} 0.326 791{4} 0.395 634{6} 0.302 211{1} 
 MSE α̂ 0.147 412{1} 0.233 667{4} 0.252 966{5} 0.196 238{3} 0.255 65{7} 0.173 965{2} 0.255 553{6} 
80 â 0.555 47{7} 0.092 37{2} 0.097 306{4} 0.097 554{5} 0.093 35{3} 0.115 979{6} 0.088 037{1} 
 b̂ 0.558 31{7} 0.192 594{3} 0.235 627{5} 0.179 392{2} 0.222 023{4} 0.295 817{6} 0.176 538{1} 
nEst.Est. Par.MLEADECVMEMPSELSERTADEWLSE
30 BIAS α̂ 0.362 219{1} 0.435 625{7} 0.416 558{6} 0.370 812{2} 0.387 977{4} 0.382 998{3} 0.405 039{5} 
â 0.721 031{7} 0.293 528{1} 0.309 027{4} 0.313 367{5} 0.295 793{3} 0.359 458{6} 0.294 914{2} 
b̂ 0.714 42{7} 0.422 951{5} 0.412 001{4} 0.387 035{1} 0.395 348{2} 0.512 203{6} 0.401 23{3} 
MSE α̂ 0.179 459{1} 0.347 104{7} 0.333 877{5} 0.254 955{3} 0.302 138{4} 0.213 417{2} 0.340 213{6} 
â 0.620 359{7} 0.131 969{2} 0.145 872{5} 0.143 018{4} 0.133 752{3} 0.196 016{6} 0.131 201{1} 
b̂ 0.699 204{7} 0.331 529{4} 0.334 81{5} 0.269 16{1} 0.3161{3} 0.497 735{6} 0.306 618{2} 
MRE α̂ 0.724 437{1} 0.871 25{7} 0.833 117{6} 0.741 623{2} 0.775 955{4} 0.765 997{3} 0.810 079{5} 
â 0.480 688{7} 0.195 685{1} 0.206 018{4} 0.208 911{5} 0.197 195{3} 0.239 639{6} 0.196 609{2} 
b̂ 0.952 56{7} 0.563 934{5} 0.549 335{4} 0.516 046{1} 0.527 13{2} 0.682 938{6} 0.534 973{3} 
∑Ranks  45{7} 39{4} 43{5} 24{1} 28{2} 44{6} 29{3} 
50 BIAS α̂ 0.354 469{2} 0.390 629{7} 0.387 568{5} 0.352 266{1} 0.356 464{3} 0.362 786{4} 0.390 259{6} 
â 0.666 029{7} 0.265 703{2} 0.268 142{3} 0.287 471{5} 0.258 835{1} 0.317 81{6} 0.268 636{4} 
b̂ 0.673 356{7} 0.369 86{4} 0.370 198{5} 0.362 363{2} 0.348 243{1} 0.449 974{6} 0.366 979{3} 
MSE α̂ 0.168 558{1} 0.275 456{5} 0.290 335{6} 0.227 524{3} 0.269 194{4} 0.189 998{2} 0.306 734{7} 
â 0.549 735{7} 0.107 782{1} 0.113 255{4} 0.118 896{5} 0.107 872{2} 0.152 139{6} 0.110 883{3} 
b̂ 0.600 192{7} 0.256 007{3} 0.277 895{5} 0.229 045{1} 0.252 802{2} 0.380 327{6} 0.256 36{4} 
MRE α̂ 0.708 939{2} 0.781 259{7} 0.775 136{5} 0.704 532{1} 0.712 928{3} 0.725 571{4} 0.780 517{6} 
â 0.444 019{7} 0.177 135{2} 0.178 761{3} 0.191 648{5} 0.172 556{1} 0.211 873{6} 0.179 091{4} 
b̂ 0.897 808{7} 0.493 146{4} 0.493 597{5} 0.483 151{2} 0.464 325{1} 0.599 966{6} 0.489 305{3} 
∑Ranks  47{7} 35{3} 41{5} 25{2} 18{1} 46{6} 40{4} 
 BIAS α̂ 0.327 237{2} 0.360 124{6} 0.374 178{7} 0.326 235{1} 0.350 973{5} 0.344 459{3} 0.348 991{4} 
 â 0.639 205{7} 0.242 477{3} 0.246 203{4} 0.257 273{5} 0.235 765{2} 0.274 055{6} 0.235 564{1} 
 b̂ 0.668 923{7} 0.325 763{3} 0.344 934{5} 0.323 161{2} 0.326 791{4} 0.395 634{6} 0.302 211{1} 
 MSE α̂ 0.147 412{1} 0.233 667{4} 0.252 966{5} 0.196 238{3} 0.255 65{7} 0.173 965{2} 0.255 553{6} 
80 â 0.555 47{7} 0.092 37{2} 0.097 306{4} 0.097 554{5} 0.093 35{3} 0.115 979{6} 0.088 037{1} 
 b̂ 0.558 31{7} 0.192 594{3} 0.235 627{5} 0.179 392{2} 0.222 023{4} 0.295 817{6} 0.176 538{1} 
View Large
TABLE VII.

Ranks of estimation methods for the GAPW distribution.

ParameternMLEADECVMEMPSEOLSERTADEWLSE
α = 0.5, a = 0.25, b = 0.75 30 7 3 1 6 2 5 4 
50 6 5 4 2 1 3 7 
80 6 5 2 7 4 1 3 
120 5 4 3 7 1 2 6 
150 5 7 3 4 1 2 6 
α = 1.5, a = 0.75, b = 0.75 30 2 4.5 3 4.5 6.5 1 6.5 
50 3 4 2 7 5 1 6 
80 3 5 4 2 6 1 7 
120 3 4 5 1 7 2 6 
150 2.5 6 4 1 7 2.5 5 
α = 2, a = 0.5, b = 1.5 30 6 2 7 1 5 3 4 
50 6 3 7 1 4 5 2 
80 6 2 5 1 4 7 3 
120 6 1 3 2 4 7 5 
150 5 1 6 2.5 2.5 7 4 
α = 1.5, a = 1.5, b = 1.5 30 5 4 7 1 3 2 6 
50 3 4 7 1 6 2 5 
80 4 3 5 1 7 2 6 
120 5 2 6 1 3 4 7 
150 3 6 7 1 4.5 2 4.5 
α = 2, a = 1.5, b = 0.25 30 3 4 2 7 5 1 6 
50 2 4 3 7 5 1 6 
80 3 7 1 6 4 2 5 
120 2 4 3 5 7 1 6 
150 3 5 3 3 7 1 6 
α = 0.5, a = 1.5, b = 0.75 30 7 4 5 1 2 6 3 
50 7 3 5 2 1 6 4 
80 7 3 6 2 4 5 1 
120 7 3 6 2 4 5 1 
150 7 1 5 2 4 6 3 
∑ ranks  139.5 113.5 130.0 91.0 126.5 95.5 144.0 
Overall rank  6 3 5 1 4 2 7 
ParameternMLEADECVMEMPSEOLSERTADEWLSE
α = 0.5, a = 0.25, b = 0.75 30 7 3 1 6 2 5 4 
50 6 5 4 2 1 3 7 
80 6 5 2 7 4 1 3 
120 5 4 3 7 1 2 6 
150 5 7 3 4 1 2 6 
α = 1.5, a = 0.75, b = 0.75 30 2 4.5 3 4.5 6.5 1 6.5 
50 3 4 2 7 5 1 6 
80 3 5 4 2 6 1 7 
120 3 4 5 1 7 2 6 
150 2.5 6 4 1 7 2.5 5 
α = 2, a = 0.5, b = 1.5 30 6 2 7 1 5 3 4 
50 6 3 7 1 4 5 2 
80 6 2 5 1 4 7 3 
120 6 1 3 2 4 7 5 
150 5 1 6 2.5 2.5 7 4 
α = 1.5, a = 1.5, b = 1.5 30 5 4 7 1 3 2 6 
50 3 4 7 1 6 2 5 
80 4 3 5 1 7 2 6 
120 5 2 6 1 3 4 7 
150 3 6 7 1 4.5 2 4.5 
α = 2, a = 1.5, b = 0.25 30 3 4 2 7 5 1 6 
50 2 4 3 7 5 1 6 
80 3 7 1 6 4 2 5 
120 2 4 3 5 7 1 6 
150 3 5 3 3 7 1 6 
α = 0.5, a = 1.5, b = 0.75 30 7 4 5 1 2 6 3 
50 7 3 5 2 1 6 4 
80 7 3 6 2 4 5 1 
120 7 3 6 2 4 5 1 
150 7 1 5 2 4 6 3 
∑ ranks  139.5 113.5 130.0 91.0 126.5 95.5 144.0 
Overall rank  6 3 5 1 4 2 7 
View Large

This section will describe the mathematical and computational characteristics of the suggested distribution’s value at risk (VaR) and tail value at risk (TVaR). These are two critical actuarial indicators for optimizing portfolios in uncertainty. Probability models may be used to quantify risk exposure, where the degree of risk exposure is decided by crucial risk indicators, which are numerical values that reflect model functions. Risk managers and actuaries often use key indicators to determine how much their firms are exposed to a particular risk.

The value at risk, also known as the quantile risk measure or the idea of quantile premium, is characterized by a degree of certainty when defining its parameters, say q (typically 90%, 95%, or 99%). In comparison, VaR is a quantitative sum of the accumulated loss distribution. Risk management is also interested in the chance of an unfavorable outcome, which can be represented at a specific likelihood level using the VaR. In contrast, value at risk (VaR) refers to a quantitative total of the cumulative loss distribution. VaR allows for the chance of an unpleasant event to be expressed at a certain probability level. This is relevant to risk management, which is likewise concerned with the possibility of an unfavorable outcome. The VaR of a random variable X is the qth quantile of its CDF, denoted by VaRq, and it is defined by VaRq = Q(q); see Artzner.18 

Another important indication is the conditional TVaR, which goes by the titles conditional tail-expectation and conditional VaR. These names have recently been given to the TVaR. The TVaR is used to quantify the potential worth of the loss if an incident that falls outside of the specified probability level has occurred. A random variable, denoted by X, has a TVaR that is represented by
TVaRq=1(1−q)∫VaRq∞xf(x)dx.

It shows numerical examples of the VaR and TVaR risk measures for the GAPW and W distributions and how they compare to other measures. We take a random sample (n = 100) from the GAPW and W distributions and use the ML method to determine the parameters. After 1000 runs, the two risk measures for the examined distributions are calculated. The numerical results of the two risk measures for the GAPW and W distributions are reported in Tables VIII and IX. We graphically display the results in the two tables in Figs. 1 and 2 for visual comparisons.

TABLE VIII.

The GAPW and W distributions’ two risk measures’ simulation outcomes.

DistributionParametersSignificance levelVaRTVaR
GAPW α = 0.5, a = 0.25, b = 0.75 0.60 8.909 88 276.513 33 
0.65 13.779 23 314.295 07 
0.70 22.262 23 363.233 90 
0.75 37.876 29 428.745 16 
0.80 69.843 22 519.903 45 
0.85 145.944 72 653.355 33 
0.90 380.533 45 865.334 12 
W a = 02.5, b = 0.75 0.60 2.404 23 203.391 79 
0.65 4.095 85 231.994 17 
0.70 7.017 11 269.756 36 
0.75 12.236 45 321.834 29 
0.80 22.091 34 398.138 37 
0.85 42.468 02 520.530 22 
0.90 92.028 93 749.194 59 
DistributionParametersSignificance levelVaRTVaR
GAPW α = 0.5, a = 0.25, b = 0.75 0.60 8.909 88 276.513 33 
0.65 13.779 23 314.295 07 
0.70 22.262 23 363.233 90 
0.75 37.876 29 428.745 16 
0.80 69.843 22 519.903 45 
0.85 145.944 72 653.355 33 
0.90 380.533 45 865.334 12 
W a = 02.5, b = 0.75 0.60 2.404 23 203.391 79 
0.65 4.095 85 231.994 17 
0.70 7.017 11 269.756 36 
0.75 12.236 45 321.834 29 
0.80 22.091 34 398.138 37 
0.85 42.468 02 520.530 22 
0.90 92.028 93 749.194 59 
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TABLE IX.

The GAPW and W distributions’ two risk measures’ simulation outcomes.

DistributionParametersSignificance levelVaRTVaR
GAPW α = 0.75, a = 0.55, b = 0.95 0.60 1.474 90 4.917 50 
0.65 1.746 10 5.453 49 
0.70 2.082 80 6.121 96 
0.75 2.516 31 6.981 12 
0.80 3.136 68 8.113 21 
0.85 3.997 76 9.748 94 
0.90 5.447 88 12.344 54 
W a = 0.55, b = 0.95 0.60 0.951 09 4.337 25 
0.65 1.215 72 4.802 85 
0.70 1.557 27 5.373 56 
0.75 2.010 06 6.093 72 
0.80 2.634 66 7.041 06 
0.85 3.551 73 8.367 79 
0.90 5.052 80 10.435 86 
DistributionParametersSignificance levelVaRTVaR
GAPW α = 0.75, a = 0.55, b = 0.95 0.60 1.474 90 4.917 50 
0.65 1.746 10 5.453 49 
0.70 2.082 80 6.121 96 
0.75 2.516 31 6.981 12 
0.80 3.136 68 8.113 21 
0.85 3.997 76 9.748 94 
0.90 5.447 88 12.344 54 
W a = 0.55, b = 0.95 0.60 0.951 09 4.337 25 
0.65 1.215 72 4.802 85 
0.70 1.557 27 5.373 56 
0.75 2.010 06 6.093 72 
0.80 2.634 66 7.041 06 
0.85 3.551 73 8.367 79 
0.90 5.052 80 10.435 86 
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FIG. 1.
Shapes of the VaR and TVaR using the numerical values in Table VIII.
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Shapes of the VaR and TVaR using the numerical values in Table VIII.

FIG. 1.
Shapes of the VaR and TVaR using the numerical values in Table VIII.
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Shapes of the VaR and TVaR using the numerical values in Table VIII.

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FIG. 2.
Shapes of the VaR and TVaR using the numerical values in Table IX.
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Shapes of the VaR and TVaR using the numerical values in Table IX.

FIG. 2.
Shapes of the VaR and TVaR using the numerical values in Table IX.
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Shapes of the VaR and TVaR using the numerical values in Table IX.

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It is stated to have a larger tail than other distributions that have lower risk metrics. The values in Tables VIII and IX and the plots in Figs. 1 and 2 reveal that the introduced model is a heavier tail than the W distribution, and it may be adopted to model heavy-tailed datasets.

We study two real-world datasets to show the flexibility of the proposed distribution. The first dataset of 27 observations of COVID-19 represents the daily new cases belonging to Angola for 27 days recorded from 8 July to 3 August 2020. The second dataset shows COVID-19 data belonging to Italy of 111 days recorded from 1 April to 20 July 2020. These data are divided by daily new deaths and cases. Both real datasets are available at https://COVID19.who.int/ and studied by Hassan et al.19  Figure 3 discussed auto-correlation unction (ACF) for correlation, covariance, and partial with different lags for each dataset, respectively. By Fig. 3, we note that these datasets have no ACF problem, and we can also use the distribution theory to discuss the best distribution fit of these data.

FIG. 3.
Auto-correlation test with different lag for each dataset.
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Auto-correlation test with different lag for each dataset.

FIG. 3.
Auto-correlation test with different lag for each dataset.
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Auto-correlation test with different lag for each dataset.

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The proposed distribution is compared with some distributions such as Weibull (W), exponentiated Weibull (ExW), Frechet Weibull (FW),3 transmuted Weibull (TW),20 gamma Weibull (GW),21 odd log-logistic Weibull (OLLW),22 transmitted exponentiated Weibull (TExW),23 modified Weibull (MW),24 Beta Sarhan Zaindin modified Weibull (BSZMW),25 and inverse Weibull (IW) distributions.

The competing models are compared using the goodness of fit measures such as Anderson–Darling (AD), Cramér–von Mises (CM), and Kolmogorov–Smirnov (KS) with their p-values (KS p-values) to determine the best fitting model for the considered real datasets.

The Wolfram Mathematica software calculates the goodness of fit measures and MLE. For the two real datasets, 10 and 11 provide GMs, MLEs, and their standard errors (SEs) in parenthesis. These tables show that the proposed distribution fits the COVID-19 datasets better than competing models.

The estimated pdf plots of the GAPW distribution for COVID-19 datasets are introduced in Fig. 4. These plots back up the findings in Tables X and XI, indicating that the suggested distribution fits. COVID-19 datasets well. Figures 5 and 6 provide profile-likelihood plots of the proposed model parameters for the two real datasets. The proposed model’s estimated parameters’ existence and uniqueness are shown graphically in Figs. 7 and 8, respectively, for the two real datasets.

FIG. 4.
Histogram of the COVID-19 datasets with estimated pdf of GAPW distribution.
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Histogram of the COVID-19 datasets with estimated pdf of GAPW distribution.

FIG. 4.
Histogram of the COVID-19 datasets with estimated pdf of GAPW distribution.
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Histogram of the COVID-19 datasets with estimated pdf of GAPW distribution.

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TABLE X.

The first COVID-19 dataset compares the analytical metrics and MLEs of GAPW distribution with those of its rivals.

ModelADCMKSKS p-valueEst. parameters (SEs)
GAPW 0.330 143 0.054 641 3 0.114 243 0.872 612 α̂=0.00147338(0.00844767) 
â=0.159066(0.196476) 
b̂=0.7993(0.269604) 
W 0.625 462 0.102 276 0.139 679 0.667 974 â=1.65592(0.228729) 
b̂=38.6192(4.75327) 
ExW 0.376 386 0.060 973 3 0.117 992 0.846 47 â=0.934603(0.405426) 
b̂=15.8539(13.7389) 
ĉ=3.78934(3.88978) 
FW 1.487 89 0.236 474 0.187 528 0.298 428 α̂=0.200587(92.8766) 
β̂=3.11758(24795.4) 
k̂=6.47119(2996.32) 
λ̂=16.8253(20781) 
TW 0.478 881 0.076 052 5 0.122 45 0.812 942 â=1.81879(0.253895) 
b̂=49.487(7.92168) 
λ̂=0.719054(0.32468) 
GW 0.388 452 0.062 789 8 0.117 794 0.847 901 â=0.687619(2.87124) 
b̂=0.640061(0.635229) 
ĉ=3.41566(4.39368) 
OLLW 10.963 2.407 36 0.534 447 <0.000001 â=0.00306575(0.0004976) 
ĉ=1.39861(0.0195722) 
λ̂=2.97787(0.215434) 
TExW 0.799 245 0.140 809 0.175 674 0.375 265 â=−1.0(0.496104) 
b̂=6.4405(1.29403) 
ĉ=0.296386(0.503636) 
λ̂=0.0432902(0.00834193) 
MW 2.529 57 0.488 272 0.298 431 0.016 307 3 λ̂=0.0291262(0.00560534) 
β̂=1.00903(5.72789) 
k̂=0.0127285(0.00209199) 
BSZMW 0.419 115 0.067 966 4 0.116 323 0.858 362 â=2.88675(1.35478) 
b̂=7.86421(17.8575) 
λ̂=0.00732355(0.00628363) 
β̂=0.00362277(0.00967074) 
k̂=0.874948(0.205954) 
IW 1.487 89 0.236 474 0.187 528 0.298 428 â=20.0573(3.16556) 
b̂=1.29804(0.164943) 
ModelADCMKSKS p-valueEst. parameters (SEs)
GAPW 0.330 143 0.054 641 3 0.114 243 0.872 612 α̂=0.00147338(0.00844767) 
â=0.159066(0.196476) 
b̂=0.7993(0.269604) 
W 0.625 462 0.102 276 0.139 679 0.667 974 â=1.65592(0.228729) 
b̂=38.6192(4.75327) 
ExW 0.376 386 0.060 973 3 0.117 992 0.846 47 â=0.934603(0.405426) 
b̂=15.8539(13.7389) 
ĉ=3.78934(3.88978) 
FW 1.487 89 0.236 474 0.187 528 0.298 428 α̂=0.200587(92.8766) 
β̂=3.11758(24795.4) 
k̂=6.47119(2996.32) 
λ̂=16.8253(20781) 
TW 0.478 881 0.076 052 5 0.122 45 0.812 942 â=1.81879(0.253895) 
b̂=49.487(7.92168) 
λ̂=0.719054(0.32468) 
GW 0.388 452 0.062 789 8 0.117 794 0.847 901 â=0.687619(2.87124) 
b̂=0.640061(0.635229) 
ĉ=3.41566(4.39368) 
OLLW 10.963 2.407 36 0.534 447 <0.000001 â=0.00306575(0.0004976) 
ĉ=1.39861(0.0195722) 
λ̂=2.97787(0.215434) 
TExW 0.799 245 0.140 809 0.175 674 0.375 265 â=−1.0(0.496104) 
b̂=6.4405(1.29403) 
ĉ=0.296386(0.503636) 
λ̂=0.0432902(0.00834193) 
MW 2.529 57 0.488 272 0.298 431 0.016 307 3 λ̂=0.0291262(0.00560534) 
β̂=1.00903(5.72789) 
k̂=0.0127285(0.00209199) 
BSZMW 0.419 115 0.067 966 4 0.116 323 0.858 362 â=2.88675(1.35478) 
b̂=7.86421(17.8575) 
λ̂=0.00732355(0.00628363) 
β̂=0.00362277(0.00967074) 
k̂=0.874948(0.205954) 
IW 1.487 89 0.236 474 0.187 528 0.298 428 â=20.0573(3.16556) 
b̂=1.29804(0.164943) 
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TABLE XI.

The second COVID-19 dataset compares the analytical metrics and MLEs of GAPW distribution with those of its rivals.

ModelADCMKSKS p-valueEst. parameters (SEs)
GAPW 0.451 741 0.064 030 5 0.056 312 2 0.873 047 α̂=0.189759(0.210106) 
â=27.4504(8.97892) 
b̂=1.62853(0.314686) 
W 0.720 499 0.119 255 0.068 349 2 0.677 561 â=2.22234(0.159565) 
b̂=0.187983(0.00844866) 
ExW 0.737 962 0.118 761 0.067 724 3 0.688 626 â=2.02223(0.402285) 
b̂=0.17464(0.027585) 
ĉ=1.19674(0.423325) 
FW 7.901 12 1.434 0.190 731 0.000 621 855 α̂=1.79738(2.01516) 
β̂=0.0737876(1.80794) 
k̂=0.751481(0.842536) 
λ̂=3.40795(6.38141) 
TW 0.550 469 0.083 721 9 0.060 827 5 0.806 024 â=1.88754(0.244163) 
b̂=0.15857(0.0163293) 
λ̂=−0.604748(0.298781) 
GW 0.738 859 0.119 605 0.067 813 7 0.687 045 â=35.373 8(13.7999) 
b̂=1.99971(0.508253) 
ĉ=0.404035(0.921432) 
OLLW 0.667 921 0.093 798 5 0.061 369 2 0.797 314 â=1.32201(0.348178) 
ĉ=1.79944(0.409568) 
λ̂=19.068 9(14.1074) 
TExW 0.580 094 0.090 343 6 0.061 107 7 0.801 533 â=0.859611(0.177011) 
b̂=30.243 3(14.1376) 
ĉ=2.40023(0.309611) 
λ̂=0.0618439(0.333169) 
MW 0.720 499 0.119 255 0.068 349 2 0.677 561 λ̂=8.29668(0.692408) 
β̂=41.0351(12.9686) 
k̂=2.22234(0.258475) 
BSZMW 0.619 759 0.095 567 6 0.062 254 2 0.782 836 â=1.90257(0.903965) 
b̂=1.49591(3.60551) 
λ̂=3.60551(6.38329) 
β̂=34.0341(61.5278) 
k̂=2.1459(0.512621) 
IW 7.901 12 1.434 0.190 731 0.000 621 855 â=0.106197(0.00794849) 
b̂=1.350 7(0.0820069) 
ModelADCMKSKS p-valueEst. parameters (SEs)
GAPW 0.451 741 0.064 030 5 0.056 312 2 0.873 047 α̂=0.189759(0.210106) 
â=27.4504(8.97892) 
b̂=1.62853(0.314686) 
W 0.720 499 0.119 255 0.068 349 2 0.677 561 â=2.22234(0.159565) 
b̂=0.187983(0.00844866) 
ExW 0.737 962 0.118 761 0.067 724 3 0.688 626 â=2.02223(0.402285) 
b̂=0.17464(0.027585) 
ĉ=1.19674(0.423325) 
FW 7.901 12 1.434 0.190 731 0.000 621 855 α̂=1.79738(2.01516) 
β̂=0.0737876(1.80794) 
k̂=0.751481(0.842536) 
λ̂=3.40795(6.38141) 
TW 0.550 469 0.083 721 9 0.060 827 5 0.806 024 â=1.88754(0.244163) 
b̂=0.15857(0.0163293) 
λ̂=−0.604748(0.298781) 
GW 0.738 859 0.119 605 0.067 813 7 0.687 045 â=35.373 8(13.7999) 
b̂=1.99971(0.508253) 
ĉ=0.404035(0.921432) 
OLLW 0.667 921 0.093 798 5 0.061 369 2 0.797 314 â=1.32201(0.348178) 
ĉ=1.79944(0.409568) 
λ̂=19.068 9(14.1074) 
TExW 0.580 094 0.090 343 6 0.061 107 7 0.801 533 â=0.859611(0.177011) 
b̂=30.243 3(14.1376) 
ĉ=2.40023(0.309611) 
λ̂=0.0618439(0.333169) 
MW 0.720 499 0.119 255 0.068 349 2 0.677 561 λ̂=8.29668(0.692408) 
β̂=41.0351(12.9686) 
k̂=2.22234(0.258475) 
BSZMW 0.619 759 0.095 567 6 0.062 254 2 0.782 836 â=1.90257(0.903965) 
b̂=1.49591(3.60551) 
λ̂=3.60551(6.38329) 
β̂=34.0341(61.5278) 
k̂=2.1459(0.512621) 
IW 7.901 12 1.434 0.190 731 0.000 621 855 â=0.106197(0.00794849) 
b̂=1.350 7(0.0820069) 
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FIG. 5.
Behavior of the log-likelihood function with the three estimated parameters for the first real dataset.
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Behavior of the log-likelihood function with the three estimated parameters for the first real dataset.

FIG. 5.
Behavior of the log-likelihood function with the three estimated parameters for the first real dataset.
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Behavior of the log-likelihood function with the three estimated parameters for the first real dataset.

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FIG. 6.
Behavior of the log-likelihood function with the three estimated parameters for the second real dataset.
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Behavior of the log-likelihood function with the three estimated parameters for the second real dataset.

FIG. 6.
Behavior of the log-likelihood function with the three estimated parameters for the second real dataset.
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Behavior of the log-likelihood function with the three estimated parameters for the second real dataset.

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FIG. 7.
For the first actual dataset, the existence and uniqueness of the suggested model parameters were determined.
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For the first actual dataset, the existence and uniqueness of the suggested model parameters were determined.

FIG. 7.
For the first actual dataset, the existence and uniqueness of the suggested model parameters were determined.
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For the first actual dataset, the existence and uniqueness of the suggested model parameters were determined.

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FIG. 8.
For the second actual dataset, the existence and uniqueness of the suggested model parameters were determined.
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For the second actual dataset, the existence and uniqueness of the suggested model parameters were determined.

FIG. 8.
For the second actual dataset, the existence and uniqueness of the suggested model parameters were determined.
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For the second actual dataset, the existence and uniqueness of the suggested model parameters were determined.

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Tables XII and XIII present the values of MLE and GMs of the GAPW distribution for all estimation methods for COVID-19 datasets, respectively. Figure 9 shows estimated PDFs for the proposed model using different estimation methods.

TABLE XII.

Estimates and measures of fit for the specified distribution parameters using actual datasets for the first set.

α̂âb̂ADCMKSKSP
MLE 0.001 473 38 0.159 066 0.799 3 0.330 143 0.054 641 3 0.114 243 0.872 612 
ADE 0.001 542 61 0.142 414 0.832 324 0.320 073 0.050 171 4 0.116 893 0.854 344 
CVME 0.000 808 06 0.140 445 0.846 292 0.336 246 0.048 180 2 0.115 888 0.861 395 
MPSE 0.006 143 48 0.160 45 0.768 16 0.481 157 0.084 230 1 0.123 743 0.802 784 
OLSE 0.052 500 5 0.037 986 6 1.129 9 0.373 552 0.048 012 5 0.108 911 0.905 947 
RTADE 0.000 241 087 0.203 701 0.754 398 0.325 071 0.051 992 2 0.122 522 0.812 376 
WLSE 0.004 820 4 0.110 076 0.886 643 0.322 993 0.050 307 3 0.114 667 0.869 758 
α̂âb̂ADCMKSKSP
MLE 0.001 473 38 0.159 066 0.799 3 0.330 143 0.054 641 3 0.114 243 0.872 612 
ADE 0.001 542 61 0.142 414 0.832 324 0.320 073 0.050 171 4 0.116 893 0.854 344 
CVME 0.000 808 06 0.140 445 0.846 292 0.336 246 0.048 180 2 0.115 888 0.861 395 
MPSE 0.006 143 48 0.160 45 0.768 16 0.481 157 0.084 230 1 0.123 743 0.802 784 
OLSE 0.052 500 5 0.037 986 6 1.129 9 0.373 552 0.048 012 5 0.108 911 0.905 947 
RTADE 0.000 241 087 0.203 701 0.754 398 0.325 071 0.051 992 2 0.122 522 0.812 376 
WLSE 0.004 820 4 0.110 076 0.886 643 0.322 993 0.050 307 3 0.114 667 0.869 758 
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TABLE XIII.

Second real dataset estimations and measurements of fit for the suggested distribution parameters.

α̂âb̂ADCMKSKSP
MLE 0.189 759 27.4504 1.628 53 0.451 741 0.064 030 5 0.056 312 2 0.873 047 
ADE 1.778 99 50.0104 2.607 32 0.671 358 0.075 037 4 0.060 612 6 0.809 447 
CVME 2.758 44 36.4815 2.719 83 0.803 097 0.048 055 9 0.068 957 7 0.666 764 
MPSE 2.758 44 36.4815 2.719 83 0.803 097 0.048 055 9 0.068 957 7 0.666 764 
OLSE 2.742 24 34.4064 2.683 16 0.743 75 0.048 653 6 0.067 353 5 0.695 178 
RTADE 0.041 430 5 25.5074 1.447 49 0.703 773 0.049 742 9 0.069 851 5 0.650 882 
WLSE 2.635 43 30.4431 2.580 8 0.612 215 0.058 627 8 0.061 741 5 0.791 259 
α̂âb̂ADCMKSKSP
MLE 0.189 759 27.4504 1.628 53 0.451 741 0.064 030 5 0.056 312 2 0.873 047 
ADE 1.778 99 50.0104 2.607 32 0.671 358 0.075 037 4 0.060 612 6 0.809 447 
CVME 2.758 44 36.4815 2.719 83 0.803 097 0.048 055 9 0.068 957 7 0.666 764 
MPSE 2.758 44 36.4815 2.719 83 0.803 097 0.048 055 9 0.068 957 7 0.666 764 
OLSE 2.742 24 34.4064 2.683 16 0.743 75 0.048 653 6 0.067 353 5 0.695 178 
RTADE 0.041 430 5 25.5074 1.447 49 0.703 773 0.049 742 9 0.069 851 5 0.650 882 
WLSE 2.635 43 30.4431 2.580 8 0.612 215 0.058 627 8 0.061 741 5 0.791 259 
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FIG. 9.
Fitted PDFs of the proposed model for the considered first and second real datasets, respectively.
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Fitted PDFs of the proposed model for the considered first and second real datasets, respectively.

FIG. 9.
Fitted PDFs of the proposed model for the considered first and second real datasets, respectively.
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Fitted PDFs of the proposed model for the considered first and second real datasets, respectively.

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The three-parameter Gull Alpha Power Weibull model is useful in the statistical sciences, particularly reliability engineering, medicine, and economics. This study utilizes the modified Weibull distribution with the Gull Alpha Power to provide a new adjustment to the Weibull model. The Gull Alpha Power Weibull distribution is proposed as the appropriate model. The maximum likelihood estimators are derived using additional Gull Alpha Power parameter approaches. A Monte Carlo simulation demonstrates the applicability of the Gull Alpha Power Weibull model, which is based on actual COVID-19 pandemic data. The flexibility of the proposed distribution to accommodate two COVID-19 datasets is compared to other distributions.

Comparing four discrimination measures, three goodness-of-fit measures, and the p-value, several analytical methods are studied to establish the excellent fitting of fitted distributions. Using graphical sketching and numerical techniques, we proved that the Gull Alpha Power Weibull model provides a better fit than its competitors. Within the scope of this study, we anticipate that the Gull Alpha Power Weibull model will likely be used to analyze the COVID-19 dataset. By comparing the proposed GAPW distribution with Weibull, exponentiated Weibull, Frechet Weibull, transmuted Weibull, gamma Weibull, odd log-logistic Weibull, transmitted exponentiated Weibull, modified Weibull, Beta Sarhan Zaindin modified Weibull, and inverse Weibull distributions, we note that the GAPW is the best fitting model for these data.

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.

Ahmed M. Gemeay: Software (equal); Validation (equal); Writing – original draft (equal). Yusra A. Tashkandy: Conceptualization (equal); Data curation (equal). M. E. Bakr: Conceptualization (equal); Software (equal); Visualization (equal); Writing – original draft (equal); Writing – review & editing (equal). Anoop Kumar: Investigation (equal); Supervision (equal). Md. Moyazzem Hossain: Supervision (equal); Visualization (equal); Writing – original draft (equal). Ehab M. Almetwally: Investigation (equal); Resources (equal); Writing – review & editing (equal).

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

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