Abstract
A new three-parameter extension of the generalized-exponential distribution, which has various hazard rates that can be increasing, decreasing, bathtub, or inverted tub, known as the Marshall-Olkin generalized-exponential (MOGE) distribution has been considered. So, this article addresses the problem of estimating the unknown parameters and survival characteristics of the three-parameter MOGE lifetime distribution when the sample is obtained from progressive type-II censoring via maximum likelihood and Bayesian approaches. Making use of the s-normality of classical estimators, two types of approximate confidence intervals are constructed via the observed Fisher information matrix. Using gamma conjugate priors, the Bayes estimators against the squared-error and linear-exponential loss functions are derived. As expected, the Bayes estimates are not explicitly expressed, thus the Markov chain Monte Carlo techniques are implemented to approximate the Bayes point estimates and to construct the associated highest posterior density credible intervals. The performance of proposed estimators is evaluated via some numerical comparisons and some specific recommendations are also made. We further discuss the issue of determining the optimum progressive censoring plan among different competing censoring plans using three optimality criteria. Finally, two real-life datasets are analyzed to demonstrate how the proposed methods can be used in real-life scenarios.
1. Introduction
Several extensions of the exponential distribution such as Weibull, gamma, generalized-exponential, and Nadarajah-Haghighi distributions among others have been proposed in literature. A general technique that allows to adding a new shape parameter to expand a family of distributions is originally proposed by Marshall and Olkin [1]. Applying the Marshall-Olkin technique to generalized-exponential distribution, Ristić and Kundu [2] introduced the three-parameter Marshall-Olkin generalized-exponential (MOGE) distribution. However, suppose that the lifetime random variable of an individual testing item follows a three-parameter . Hence, its cumulative distribution function (CDF) , probability density function (PDF) , reliability function (RF) , and hazard function (HF) , at distinct time , are given, respectively, byandwhere , , and are the shape parameters and is the scale parameter. Ristić and Kundu [2] derived several properties of the MOGE distribution and estimated its parameters using the likelihood method.
Using some specified values on the range of , , and , different shapes of the density and failure rate functions of the MOGE distribution are displayed in Figure 1. It shows that the density shapes are decreasing or unimodal while the HF shapes can be increasing, decreasing, bathtub, or inverted tub. Since the bathtub HF shape is quite beneficial in several lifetime data, therefore the MOGE distribution is justly flexible and can be used to provide a good description of different types of censored data, for details, it can be seen in Ristić and Kundu [2]. From (1), three lifetime submodels can be obtained as special cases from the MOGE distribution, namely,(i)Marshall–Olkin exponential distribution, by Marshall and Olkin [1]; if setting .(ii)Generalized-exponential distribution, by Gupta and Kundu [3]; if setting .(iii)Exponential distribution, discussed by Johnson and Kotz [4]; if setting .
Statistical life distributions under censoring plans have received great interest owing to their wide application in numerous fields such as engineering, social sciences, marketing, and medicine. One of the most popular censoring mechanisms, which is of great importance in reliability studies, is known as type-II progressive censoring scheme (PCS-T2). This censoring scheme can be described as follows: suppose that identical and independent units are put on a life-testing experiment at time zero and the progressive censoring scheme is prefixed, where is the number of observed failures that is decided by the experimenter in advance. At the time of the first failure occur (say ), of survival items are randomly removed from the remaining live units. Again, at the time of the next failure occur (say ), of survival items are randomly removed from the remaining live units and continue the test. This procedure continues until all remaining live units are removed from the experiment at the time of the th failure observed. It is clear that . For more details, one may refer to an excellent monograph presented by Balakrishnan and Cramer [5].
However, the likelihood function of the PCS-T2 sample for is defined as where and is parameter vector.
In literature, several authors including Amin [6], Kim et al. [7], Huang and Wu [8], Dey et al. [9], Panahi [10], Muhammed and Almetwally [11], Elshahhat and Rastogi [12], Maiti and Kayal [13], Dey et al. [14], Maiti et al. [15], and Elshahhat and Abu El Azm [16] studied a wide variety of inferential problems for different lifetime distributions under type-II progressively censored samples.
In the context of censoring mechanisms, to the best of our knowledge, we have not encountered any work related to the estimation of parameters and/or survival characteristics of the MOGE distribution under incomplete (censored) sampling, which is of considerable interest and practical significance in many practical situations. So, in this study, we shall exclusively focus on both classical and Bayesian approaches of estimation to derive the both point and interval estimators of the MOGE parameters , , and , as well as its time parameters and under PCS-T2. Our objectives in this study are as follows: first, deriving the maximum likelihood estimators (MLEs) for the unknown MOGE parameters or any function of them. Correspondingly, using observed Fisher information matrix, two-sided asymptotic confidence intervals are also established. Second, obtain the Bayes estimators of the unknown quantities , , , , and using independent gamma priors based on the two popular symmetric and asymmetric loss functions. Markov chain Monte Carlo (MCMC) techniques are also used to carry out the Bayes estimates and associated credible interval estimates from the complex posterior functions. Third, determine the optimum progressive censoring plan from a class of all possible removal patterns that contain a magnificent amount of information regarding the unknown model parameter under consideration. Lastly, the performance of the proposed methods is compared through an extensive simulation study in terms of their simulated root mean squared-error, mean relative absolute bias, average confidence lengths, and coverage percentages. To discuss how the proposed methods can be applied in a real phenomenon, two real-life data sets are analyzed. We also suggest the use of “maxLik” and “coda” statistical packages in programming software to evaluate the theoretical findings. Finally, some specific recommendations are made.
The rest of the paper is arranged as follows: classical and Bayes estimations of unknown parameters and the reliability characteristics are presented in Sections 2 and 3, respectively. Optimum progressive censoring plans are discussed in Section 4. Monte Carlo simulation is performed in Section 5. Two applications using real data sets are presented in Section 6. Finally, we conclude the paper in Section 7.
2. Likelihood Inference
In this section, the point and interval estimators of the unknown parameters and the reliability characteristics of the MOGE distribution based on PCS-T2 are derived.
2.1. Point Estimators
Suppose is a PCS-T2 data obtained from . From (1) and (2), with ignoring any additive constant, the likelihood function (5) becomes where is used instead of and.
From (6), the log-likelihood function, , can be written as
Setting the first-partial derivatives of (7) with respect to each unknown parameter to zero, the respective MLEs of , , and are the solutions to the following normal equations asAnd,where for .
Obviously, from (8) and (10), the MLEs , , and of the parameters , , and , respectively, cannot be obtained analytically but can be evaluated using any iterative approximation technique. Once , , and calculated, using the invariance property of MLEs, the corresponding MLEs and of and at any given time can be easily derived by replacing , , and with , , and in (3) and (4), respectively.
2.2. Asymptotic Intervals
To construct the two-sided ACIs for the unknown parameters , , and , or any life function such as and (say ), the asymptotic variances or covariances (by inverting the Fisher information matrix ) of the MLEs , , and must be obtained as
Under some mild regularity conditions of the MLEs, using the concept of large sample theory, we have the asymptotic normality approximation (NA) of is approximately multivariate normal with mean vector and variance-covariance matrix , i.e., , where is the parametric vector of , , and . By differentiating (7) partially with respect to , , and , locally at their MLEs , , and , the Fisher’s elements for are obtained and provided in Appendix A.
Similarly, based on NA of the MLEs and , it is clear that and . Thus, ACIs for and can be constructed using the corresponding normality. To obtain ACIs of and , we need to compute their variances. Following Greene [17], the delta method is used to find and of and , respectively, as:where,
Then, based on NA, the two-sided ACIs for the unknown parameter , , , or (say ) is given by:where is the upper quantile of the standard normal distribution.
Though, the main problem of asymptotic NA confidence interval is that it may give a negative value in lower bound for a lifetime parameter. To handling this drawback, Meeker and Escobar [18] developed the normal approximation for the log-transformed (NL) of MLE. They also showed that ACI has better CP based on NL compared to NA. However, based on NL, the two-sided ACI of is given by
3. Bayesian Inference
In this section, we discuss the Bayes inference of the model parameters , , and , as well as, the reliability parameters and using progressively type-II censoring.
3.1. Prior Information and Loss Functions
Since the gamma density provides various shapes based on parameter values and is flexible in nature, so the utilizing of independent gamma priors are relatively simple which may yield to results with more explicit posterior density expressions Muse et al. [19]; for more details, it can be seen in [14, 20, 21]. Thus, the gamma conjugate priors , , and are used to adapt the support of MOGE parameters , , and , respectively. However, the joint prior PDF of , , and becomes where the hyperparameters are chosen to reflect prior knowledge about the unknown parameters , , and , and they assumed to be known as non-negative.
The choice of (symmetric and/or symmetric) loss function is an important issue in Bayesian analysis. So, to develop the objective estimates, two well-known loss functions namely, squared-error loss (SEL) and linear-exponential loss (LL), are considered. However, any other loss function can be easily incorporated. The SEL function is defined as where being an estimate of . Under (13), the Bayes estimate is the posterior mean of .
On the other hand, the LL function, , is defined as
The sign and magnitude of represent the direction and degree of symmetry, respectively (if , overestimation is more serious than underestimation and means the opposite). Using (14), the Bayes estimator of is given by:provided the above exception exists, and is finite. When close to , the LL function is approximately the SEL function and it can become almost symmetric, as can be seen in Pandey and Rai [22].
3.2. Posterior Analysis
Using (6) and (12), the joint posterior PDF of , , and can be written as where is the normalizing constant of (15).
Due to nonlinear expression in (6), the closed-form of the marginal PDF for each unknown parameter is not possible. Thus, we purpose to apply some simulation algorithms such as the MCMC methods to approximate the Bayesian estimates and to find associated credible intervals. So the MCMC methods have been widely used in Bayesian computational analysis. At first, from (15), we derive the conditional posterior distributions of , , and , respectively, as And,where .
It can be seen, from (16)–(18), that the conditional distributions of , , and cannot be reduced to any standard distribution so that the exact inferences from their marginal densities cannot be easily obtained. Thus, to generate samples from posterior distribution whenever the posterior PDF cannot be reduced to any familiar distribution, the Metropolis–Hastings algorithm can be easily used for this purpose (see Gelman et al. [23] for detail).
3.3. M-H Algorithm
The Metropolis-Hastings (M-H) algorithm is a general member of the MCMC family of simulation techniques. In this subsection, the M-H algorithm with normal proposal distributions is conducted to generate samples of , , and from (16), (17), and (18), respectively. Precisely, Figure 2 indicates that the distribution of MOGE parameters behaves similar to the normal distribution. Next, to obtain the Bayes estimates and the associated credible intervals, do the following generation steps: Step 1: Put . Step 2: Set initial values . Step 3: Generate , , and from (16)–(18) with normal distributions , , and , respectively, as follows:(a)Calculate , and .(b)Generate sample variates , , and from the uniform distribution.(c)If , set , else set .(d)If , set , else set .(e)If , set , else set . Step 4: Obtain and , for a specified time by replacing , , and with their , , and , respectively. Step 5: Put . Step 6: Redo Steps 3–5 times to obtain the MCMC simulated variates of , , , , or (say ) as for . Step 7: Use outputs in Step 6, compute the Bayesian estimates of under SEL and LL functions given in (13) and (14), respectively, as:And,where is the burn-in period.
3.4. Credible Intervals
To construct the Bayes credible interval (BCI) and highest posterior density (HPD) credible interval for the parameter of interest, the MCMC simulated variates of the target parameter are used. Under HPD credible interval, for every point inside the interval, the study distribution must be greater than that for every point outside the interval. Therefore, it has the shortest length among all possible credible intervals. According to the procedure proposed by Chen and Shao [24], the HPD credible interval can be constructed. Now, to establish the BCIs (or HPD credible intervals) of the unknown parameters , , and or the survival characteristics and , for short say , one can be easily perform the following process: Step 1: Order the simulated MCMC variates of for , as . Step 2: Set the level of significance . Step 3: Obtain the BCI of as . Step 4: Obtain the HPD credible interval is given by,where is chosen, such that,Here, denotes the largest integer less than or equal to .
4. Optimum Progressive Censoring Plans
In reliability field, the problem to determine the “optimal” censoring scheme from a set of available censoring schemes is main issue. So, to great the information included in the unknown parameters under consideration, Balakrishnan and Aggarwala [25] first discussed the optimal censoring plan via several setups. Following Ng et al. [26], the values of (total test units) and (effective sample) must be fixed in advance, then one can be easily determine the optimal censoring design ( where ) under type-II progressive censoring by considering the following optimum criteria:(i)Criterion-A: .(ii)Criterion-B: .(iii)Criterion-C: .
Regarding to criteria A and B, our goal is to minimize the trace and determinant of while regarding to the criterion C our goal is to maximize the main diagonal elements of , with respect to MLE of . The optimized PCS-T2 plan that provides more information corresponds to the smallest value of criteria (A and B) with the highest value of C-criterion.
5. Monte Carlo Simulation
To show the performance of the proposed estimators of the unknown parameters , , and as well as the survival characteristics and , a Monte Carlo simulation study is carried out. Using various combinations of (total test units), (effective sample size), and (progressive censoring), a large 1,000 PCS-T2 samples are generated from the MOGE distribution when the true value of parameters is taken as . Thus, for time , the actual value of the reliability and hazard functions is used as 0.91683 and 0.68856, respectively. In this study, some specified values of , , and are also taken in account such as (small), 60 (moderate), and 90 (large). As soon as the number of failed subjects achieves (or exceeds) , where the failure proportion (FP), , is taken as 30, 60, and 90%, the experiment is terminated.
Moreover, for given and , different censoring schemes to remove survival items during the test are also considered as:
To examine the effects of the prior parameters and for on the Bayesian analysis, two informative sets of the hyperparameters are used; called Prior-1 (say P1): and and Prior-II (say P2): and . These values are specified in such a way that the expected value of prior became the expected value of the corresponding parameter. If one setting , the posterior distribution (20) is reduced in proportion to the corresponding likelihood function (6). So, if one does not have prior information on the unknown parameter, we recommend to use the frequentist estimates instead of the Bayes estimates because the latter are more computationally expensive.
To develop the Bayesian calculations, following the M–H algorithm proposed in Section 3.3, we generate MCMC samples and then the first iterations have been discarded as burn-in. Hence, using 10,000 MCMC samples, the average Bayes point estimates and associated BCI/HPD credible intervals of the unknown parameters , , , , and are computed. Here, the Bayesian estimates relative to SEL and LL are developed.
For each considered setting, the average point estimates (APEs) using the proposed frequentist and Bayes approaches of , , , , or (say ) are computed bywhere is the desired estimate of the parametric function , denotes the obtained estimate at the sample of the unknown parameter , is number of generated sequence data, , , , , and .
Comparison between different point estimates of , , , , or (say ) is made based on their root mean squared-error (RMSE) and mean relative absolute bias (MRAB) values using the following formulae, respectively, asAnd,
Further, the performances of different type of the two-sided asymptotic/credible intervals estimates are compared using their average confidence lengths (ACLs) and the average coverage percentages (CPs) as:And,where and denote the lower and upper bounds, respectively, of asymptotic (or credible) interval of the unknown parameter .
All numerical computations were performed using 4.0.4 software with three useful packages, namely: (i) “coda” package proposed by Plummer et al. [27]; (ii) “maxLik” package proposed by Henningsen and Toomet [28]; (iii) “GoFKernel” package by Pavia [29]. Recently, the same packages are also recommended by Elshahhat and Nassar [30].
The simulation results of , , , , and are presented in Tables 1–10 and are also displayed with heatmaps in Figures 3–7. In each heatmap plot, the “x-lab” display the proposed estimation point (or interval) methods while the “y-lab” represents the different choices of , FP%, and censoring schemes. For specification, using P1 for example, we have used the notation “SEL-P1” for the Bayes (SEL) estimates; “LL1-P1” for the Bayes (LL) estimates when ; “LL2-P1” for the Bayes (LL) estimates when ; “LL3-P1” for the Bayes (LL) estimates when ; “BCI-P1” and “HPD-P1” denote to the BCI and HPD credible intervals, respectively. Also, “ACI-NA” and “ACI-NL” denote to ACI based on NA and NL methods, respectively. All simulation results are presented in Appendix B. Now, we can make the following observations:(i)In general, all estimates of the unknown parameters , , and or the time parameters and are very good in term of minimum RMSEs and MRABs.(ii)As (or ) increases, the proposed estimate become more better. This pattern is also observed when decreases.(iii)Due to gamma prior information, the Bayesian estimates perform better than the competing frequentist estimates.(iv)Bayes estimates using LL function of all unknown parameters are or may be overestimates (or underestimates) when (or ).(v)RMSEs and MRABs of the Bayes (LL) estimates are smaller than the Bayes (SEL) estimates, in most cases, when close to a positive value.(vi)On the basis of smallest ACLs and highest CPs, it is noted that the ACI-NA approach is the best compared to those obtained based on the ACI-NL approach, similarly, the interval estimates based on the HPD method are the best compared to those obtained based on the BCI method.(vii)Comparing CSs 1 and 3, it is observed that RMSEs, MRABs, and ACLs of , , and are greater based on CS-3 than CS-1, thus the associated CPs of the same unknown parameters are greater based on CS-1 than CS-3.(viii)It is also a predictable fact because of the expected duration of the test items using CS-1 (where subjects at the time of failure observed are removed) is greater than the CS-3 (where subjects are withdrawn at the time of failure observed).(ix)Estimates of and have the lowest RMSEs, MRABs, and ACLs as well as the highest CPs based on CS-3 than CS-1.(x)Since the corresponding variance of P2 is less than the variance of P1, one can be seen that the Bayes (point/interval) estimates performed satisfactory based on P2 than P1 as expected.(xi)In summary, the Bayesian inference via M-H algorithm of the unknown life parameters , , , , and of the Marshall-Olkin generalized-exponential lifetime model is recommended.
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6. Real-Life Applications
To show the potentiality and flexibility of the proposed methodologies to real phenomenon, in this section we shall provide two real data sets obtained from different fields. First data set (denoted by Data-I) represents the failure times of electronic devices, as can be seen in Wang [31]. Recently, this data set has also been discussed by Elshahhat and Abu El Azm [16]. Other data set (denoted by data-II), reported by Hinkley [32], consists of the amount of precipitation (inches) having successive values in March at Minneapolis/St Paul. The lifetime points (in order) of both data sets I and II are presented in Table 11.
Using Table 11, plots of the empirical/estimated scaled TTT-transform of the MOGE distribution are provided in Figure 8. It shows that the scaled TTT-transform is (a) convex and then concave for data-I while is concave for data-II. It also suggests that increasing and bathtub shaped failure rates are suitable for the fitting MOGE lifetime model.
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Before discussing our proposed estimators, the MOGE distribution is compared with seven popular Marshall-Olkin extended distributions as competitors, namely: Marshall-Olkin extended Weibull (MOEW), Marshall-Olkin logistic exponential (MOLE), Marshall-Olkin alpha power exponential (MOAPE), Marshall-Olkin Nadarajah-Haghighi (MONH), Marshall-Olkin Gompertz (MOG), Marshall-Olkin Lomax (MOL), and Marshall-Olkin exponential (MOE) distributions. The corresponding probability densities of the competing models (for and ) are tabulated in Table 12.
To judge the performance of the considered models, several goodness-of-fit criteria, namely, estimated negative log-likelihood (ENL), Akaike information (AI), consistent Akaike information (CAI), Bayesian information (BI), Hannan-Quinn information (HQI), Anderson-Darling (AD), Cramer von Mises (CvM) and Kolmogorov-Smirnov (KS) distance with its -value are used. For each model, we estimate the parameters with their standard errors (SEs) by utilizing the method of maximum likelihood, see Table 13. Generally, the lowest values of these statistics (except the highest -value) represent the best fit to the given data. The calculated statistics based on data sets I and II are reported in Table 14. It indicates that the MOGE model provides the best fit than other competing models for the given data sets. Plots of the estimated densities and reliability functions of all fitted models are displayed in Figure 9. It shows that the MOGE distribution for both data sets I and II captures the general pattern of the histograms and also supports the numerical findings established here. Also, quantile-quantile plots of the fitted models are also investigated in Figure 10. It yields that the MOGE model fits the given data sets more closely and supports the numerical findings.
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Now, using some specified choices of and , different PCS-T2 samples are generated from data sets I and II, see Table 15. For brevity, the PCS is abbreviated by . Using the M-H algorithm described in Section 3.3, we generate 30,000 MCMC samples and discard the first 5,000 samples as burn-in. Under SEL and LL (with ), the Bayesian MCMC estimates and associated credible intervals of , , , and using noninformative priors are developed. Also, MLEs and two-sided 95% asymptotic (NA and NL) interval estimates of the unknown parameters are computed. The estimates of RF and HF are calculated when the mission time is taken as 100 and 1 for data sets I and II, respectively. The point estimates (with their SEs) and interval estimates with their confidence lengths (CLs) and reported in Tables 16 and 17, respectively. It is noted that the proposed point and interval estimates, obtained based on data sets I and II, of , , , and are very similar to each other, as expected.
To examine the convergence of M-H algorithm, trace plot based on 25,000 MCMC simulated variates of each unknown parameter (for as an example) using data sets I and II, is plotted in Figure 11. For each trace plot, the sample mean and two bounds of 95% HPD credible intervals are mentioned by solid (—) and dashed (- - -) lines, respectively. It indicates that the M-H algorithm converges very well. Using the Gaussian kernel density, the corresponding histogram plot of each unknown parameter (for as an example) is also plotted and represented in Figure 12. In each histogram, the sample mean is depicted as vertical dash-dotted line (:). It represents that the simulated posteriors of , , and are fairly symmetric while the simulated posteriors of and are close to negative and positive skewed, respectively.
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Using the optimum criteria given in Section 4, using the observed Fisher information (11) obtained from both data sets I and II, the concept of optimal censoring is illustrated, see Table 18. It shows that the scheme is the optimal censoring (for data set I) while is the optimal censoring (for data set II) than other competing scheme plans. However, the parameter and optimal estimates examined here support the same findings reported in the simulation study.
7. Conclusion Remarks
This article investigates the problem of estimating the unknown parameters and reliability characteristics of the MOGE distribution when the data are collected under type-II progressive censoring. Besides the likelihood inference, the Bayes MCMC estimates of the unknown parameters, reliability, and hazard functions are derived. As expected the joint posterior density cannot be obtained explicitly. Therefore, using gamma informative priors under different loss functions, the Bayes point and interval estimates of the unknown parametric functions are approximated utilizing the Metropolis–Hasting algorithm. From the simulation outputs, the Bayes MCMC estimation method for any unknown function of the MOGE parameters is recommended. To show the practical utility of the proposed methods in real-life phenomenon and to determine the optimum censoring plan, two data sets from engineering and physics are analyzed for this purpose. Additionally, based on the given data sets, we showed that the proposed model provides an adequate fit with respect to its competing Marshall–Olkin extended models. We hope that the results and methodologies discussed in this paper will be beneficial to data analyst and reliability practitioners. [39, 40].
Appendix
A. Fisher’s Elements
Differentiating (7) partially with respect to , , and , the Fisher elements are expressed as andwhere .
Appendix B. Simulation Outputs
All simulation outputs including APEs, RMSEs, MRABs, ACLs, and CPs of , , , , and based on different choices of progressive censoring schemes (also different complete and effective sample sizes) are reported here.
Data Availability
The data that support the findings of this study are available from the corresponding author upon reasonable request.
Conflicts of Interest
The authors declare that they have no conflicts of interest.