Abstract
Using a progressive Type-I censoring technique, this article will explore how to estimate unknown parameters of the alpha power exponential distribution (APED) (Type-I PCS). The squared error loss function and the LINEX loss function are used to get the maximum likelihood estimate as well as the Bayesian estimation of the unknown parameters, respectively. It was our intention to use the Markov chain Monte Carlo method in conjunction with the Bayes estimation strategy. We are able to calculate the approximately accurate confidence intervals for the parameters whose values are unknown. In addition to this, we discussed the estimation challenges of reliability and the hazard rate function of the APED while using Type-I PCS, as well as the approximate confidence intervals that relate to these problems. In the last step, the theoretical findings that were acquired are evaluated and compared with the distributions of all of its rivals by making use of two actual datasets that represent the disciplines of engineering and medicine.
1. Introduction
Numerous applications in engineering, health, finance, science, and environmental research have shown that these distributions are poor at characterizing big data sets, prompting the incorporation of additional shape factors in a number of distributions. Therefore, continuous extension of these distributions is required to achieve significant progress in these areas. Mahdavi and Kundu [1] established a unique method for creating distributions by adding an extra shape parameter to well-known baseline distributions. By adopting the exponential baseline distribution, they build a unique extension of the exponential distribution termed the alpha power exponential distribution.
The followings are the probability density function (PDF) and cumulative distribution function (CDF) formulas for the APED’s probability density function:
The following is a representation of the corresponding reliability and hazard rate functions:where , , and . Note that in the rest of this paper, it is assumed that .
Mahdavi and Kundu [1] studied the statistical properties of the APED and used the method of maximum likelihood (ML) to estimate the unknown parameters under the complete sample. Nassar et al. [2] used methods of moments, percentile, maximum product of spacing method, weighted least squares, L-moments, Anderson-Darling, ordinary least square, and Cramer-von-Mises to estimate the parameters of the APED under the complete sample.
Salah [3] studied parameters estimation of APED under progressive Type-II censored scheme (PCS-Type II) using the MLE. Salah et al. [4] considered parameter estimation of APED under Type-II hybrid censored scheme (HCS-Type II) using the MLE and expectation maximizations algorithm. Also, they evaluated the estimated reliability and hazard functions. In addition, the Fisher information matrix is computed by applying the missing information rule to find the asymptotic confidence interval. Finally, Abo-Kasem and Ibrahim [5] introduced the estimating problems of the unknown parameters of the APED using the Type-II progressive hybrid censoring scheme (PHCS-Type II). The MLE and Bayesian estimations of the unknown parameters based on both squared error loss (SE) and LINEX loss functions are obtained. They propose to apply the Markov Chain Monte Carlo (MCMC) technique to carry out a Bayes estimation procedure. Additionally, they introduced the approximate and credible confidence intervals for the unknown parameters. Also, they introduced the estimating problems of reliability and hazard rate function of the APED under PHCS-Type II and the corresponding approximate confidence intervals.
Censored data occur in practical life-testing trials when tests, including the life periods of test units, must be ended before to gathering full observation. The censoring approach is widespread and inevitable in practice for a variety of reasons, including time constraints and cost-cutting measures. Numerous censoring systems have been addressed in the literature, the most prevalent of which are Type-I censoring scheme (CS-Type I) and Type-II censoring scheme (CS-Type II). Lately, a more generic kind of censorship known as progressive censoring schemes (PCS) has garnered considerable interest in the literature due to its more efficient use of available resources when compared to classic censorship designs. Progressive Type-I censoring schemes is one of these PCS.
During the course of the experiment, this method is implemented by deducting prefixed numbered objects from the total number of things that are still in play at a predetermined interval of censoring. It allows the researcher to have additional flexibility throughout the design phase by allowing the test units to be deleted at nonterminal time periods. This mixes the practical ability to know the end time with the extra freedom offered to the researcher. Assume that n units are subjected to a lifetime experiment.
Additionally, we will take into consideration designate the lives of these units extracted from a population, or this sample represents a sample of items or units or products of a certain experiment. Let r be the amount of items that fail and allow censorship to emerge gradually in m stages at times , , At the ith stage of censoring, at each time point a random sample of objects is drawn from the surviving and removed from further observation; thus, Right now, we will suppose that , and are fixed and preassigned. Now, we will write the likelihood function, and it can be written as the following equation:
Such that is the failure time of the ith unit in the experiment, r is the number of failures, and c is a constant which does not depend on parameters.
Many researchers have recently studied the issue estimate for unknown parameters based on Type-I PCS distributions that use the MLE and Bayesian estimation techniques; consider, for instance, Mahmoud et al. [6], Algarni et al. [7], and Elbatal et al. (2021) [8].
The goal of this work is to examine the Type-I PCS where each lifetime of the units has its own APED model. We drive the estimation of the unknown parameters, reliability, and hazard rate functions of APED under Type-I PCS. In Section 2, the MLEs and the information matrix will be discussed to obtain asymptotic confidence intervals for the parameters and estimate reliability and hazard rate functions. Furthermore, Bayesian estimation using SE and LINEX loss functions will be discussed in Section 3. We used two real data sets in the application in Section 4; also, we used these data to determine the superiority of the APED using a progressive Type-I censoring scheme, we compared its fitting with all of its competitors, and we found that it outperform all of them. Finally, we conclude the paper in Section 5.
2. The Maximum Likelihood Estimation
Suppose those n units whose lifetimes are identically and independently distributed (iid) with APED random variables with the probability density function (1) are placed on a life test.
Using some simplifications, we can substitute equations (1) and (2) to equation (4) to get MLE and its information matrix for the APED’s unknown parameters. The likelihood function in this scenario is provided by
Since when , the exponential distribution (ED) logarithm function is obtained. The natural logarithm of the likelihood function (5) is
The MLEs and of and can be obtained by equating the partial differentiation of equation (6) with respect to and to zero. The partial differentiation of with respect to and is given by
It is to be noted that the likelihood equations, in this case, cannot be solved explicitly, so the MLEs of and can be obtained by using any numerical technique.
To get the fisher information matrix, we must find the second derivatives so by using the above equations, we can easily find those partial derivatives.where
The asymptotic variance-covariance matrix of the estimators can be easily obtained as follows:
Therefore, the confidence interval for and , respectively, based on the MLE is given bywhere is the value from the standard normal table.
3. Bayesian Estimation
The Bayesian technique is utilized in this section to estimate the unknown parameters of the APED utilizing symmetric squared error loss functions and asymmetric LINEX loss functions. We will use independent priors for the two parameters as shown in the following equation:
Combining (13) with equation (5), we can easily formulate the joint posterior distribution as it is represented in the following equation:where
The marginal posterior of may be stated aswhere
In a similar manner, one may get the marginal posterior by integrating the joint posterior with respect to λ. This can be done by using the following formula:where
The Bayes estimator about any function, such as, , is supplied by the SE:
The Bayesian estimation techniques for parameters are described using the squared error loss function.
These estimators can be expressed aswhere
The Bayes estimates for parameters of APED and under LINEX loss function arerespectively, where E(.) denotes the posterior expectation. After simplification, we havewhere
Regrettably, equation (20) cannot be calculated. As a result, we recommend the most often used approximate Bayes estimates and MCMC.
MCMC is a computer-assisted sampling method. It enables the characterization of distribution without knowledge of all of its mathematical features by randomly picking values from the distribution (Ravenzwaaij et al. [9]).
For more information about the Metropolis–Hastings algorithm, see Ravenzwaaij et al. [9].
3.1. Highest Posterior Density (HPD)
For the unknown parameters and APED, we use data from the suggested method and a progressive Type-I censoring strategy to produce the HPD intervals. Now, and are the quantile of and , respectively, such that,where , and for a given and , the estimator of can be evaluated as follows:where is the indicator function. So the estimates can be written as follows:where and represent the values of . Now, for can be approximated by
The HPD credible interval for the distribution’s parameters can be written easily using the following formulas:for , where represents the greatest integer that is either less than or equal to .
4. Real Data Applications
To demonstrate the techniques’ adaptation to a real-world phenomenon, we investigate many applications utilizing two real-world datasets from the engineering and medical professions.
4.1. Electrical Appliances Data
We will evaluate real-world engineering data provided by Lawless in this application (2011). This dataset includes the number of cycles required for 60 electrical equipment to fail in a life test. Table 1 lists and summarizes these failure times.
Before proceeding to deduce from these data, we fit the APED to the entire electrical appliances dataset using five competing lifetime distributions, namely the Weibull distribution (WD), the gamma distribution (GD), the generalised exponential distribution (GED), the Burr Type XII distribution (BXIID), and the Lomax distribution (LD). The corresponding PDFs of the competing models (for ) are written in Table 2.
Kolmogorov–Smirnov (K–S), Anderson–Darling (A–D), and Cramer-von Mises (CvM) goodness of fit test statistics and their corresponding values are used to assess the validity of APED. The negative log-likelihood criteria (NLC) and Akaike’s information criterion (AIC) were also employed to evaluate the fit of these distributions it can be written using the equation , Bayesian information criterion (BIC) can be written using and Hannan-Quinn information criterion (HQIC) can be written using , where , , and are the sample size, and the number of model parameters and estimated log-likelihood function, respectively, used. Obviously, according to these parameters, the optimal distribution has the lowest value of the above-mentioned criteria and the greatest value. The “AdequacyModel” package developed by Marinho et al. [15] is used to estimate the parameters of the investigated distributions and also to assess the goodness-of-fit selection metrics. Nevertheless, the derived MLEs for the model parameters and accompanying selection measures, together with their standard errors, are computed and given in Table 3. It demonstrates that the APED lifespan model has the optimum distribution of all fitted competition models under the electrical appliances dataset, with the lowest goodness of statistic values and the greatest p value.
Moreover, for goodness-of-fit of distributions using the graphical presentation method, we draw quantile-quantile (Q-Q) plots of the competitive models using depicts the points where is the MLE of , which are shown in Figure 1. Table 3 show that the APE model has the lowest goodness-of-statistic values and the greatest values for fitting real lifespan data, which makes it superior than other fitted models in the literature.
Additionally, the Q-Q plots corroborate our results. To get a more fitting illustration, we have included two plots in Figure 2 that were computed using the estimated model parameters for APED, LD, GD, WD, GED, and BXIID; plot (a) depicts the histogram of the electrical appliances data and the fitted PDFs, while plot (b) depicts the fitted and empirical survival functions. As seen in Figure 2, the pictorial representations corroborate our numerical results.
(a)
(b)
To demonstrate the presence and uniqueness of the MLEs estimates, we calculated and exhibited the contour plot of the log-likelihood function with respect to the two-parameter APED using the whole electrical appliances dataset as shown in Figure 3. The log-likelihood function’s highest value is represented by a point on the innermost contour. The coordinates of the x-point provide the MLEs of and which become and . Furthermore, it shows that the MLEs are exist and are also unique.
Now, using a variety of various values for m, four progressively Type-I censored samples are constructed based on the information in Table 1. (number of stages) and (number of surviving units are removed) at the (time censoring) refer to the selected quantiles for . Table 4 contains these produced samples as well as the censoring techniques that correspond to them. It is abundantly evident as Scheme-IV is an example of a Type-I censoring scheme, which may be seen as a particular instance of progressive Type-I censoring by setting for and . It is important to note that Scheme-IV provides full sampling, which can also be thought of as an illustrative case of progressive Type-I censoring by setting the value of for and .
The MLEs and Bayes MCMC estimates of the unknown parameters are calculated by using the datasets that were acquired and described in Table 4. Their standard errors, as well as the survival characteristics and and at different mission times, are calculated and reported in Table 5. The Bayes MCMC estimates are developed using SE and LINEX (for ) functions. Because no previous information is known on the APE parameters, and , we turned to the Bayes estimates, and we will use the aid of the MH algorithm sampler, as described in Section 3, under a noninformative priors, . Nevertheless, for computational ease, we set all hyperparameters to 0.0001. After that, we produce 30,000 MCMC samples and remove the first 5000 repetitions as a burn-in. In order to carry out the MCMC sampler method, the initial values of the unknown parameters were thought to be their MLEs. This assumption was made for the sake of convenience. Moreover, two-sided 95% asymptotic/credible intervals with their lengths of , , , and are calculated and listed in Table 6.
From Tables 5 and 6, it can be seen that the point and interval estimates of the unknown parameters , , , and obtained by maximum likelihood and Bayesian inferential approaches are quite close to each other, as expected. Furthermore, the Bayes estimates relative to symmetric (or asymmetric) loss function have efficient work compared to those obtained from the classical approach in terms of their standard errors and confidence interval lengths, as expected. Therefore, the outcomes of the provided estimates under the comprehensive dataset of electrical appliances provide a satisfactory explanation for our model. In order to evaluate whether or not 25,000 MCMC outputs have converged, we will use the whole dataset of electrical appliances and sketch the trace plots of the conditional posterior distributions of , , , and as shown in Figure 4. In each plot, the sample mean for each MCMC trace plot, 95% two-bounds BCI/HPD credible intervals are displayed with solid (—), dashed (- - -), and dotes () horizontal lines, respectively.
Figure 5 also presents the marginal PDF estimates of , , , and along with their histograms, which are based on 25,000 chain values and use the Gaussian kernel. Similarly, the sample mean of each unknown parameter is represented as a vertical dashed line () in each histogram graphic. Estimates reveal that the produced posteriors of all unknown lifespan parameters of the APE model are very symmetric and correspond well to the theoretical posterior density functions. In addition, essential features such as mean, median, mode, standard deviation (SD), and skewness (Sk.) are estimated and reported in Table 7 for MCMC posterior distributions after bun-in.
4.2. Head-Neck Cancer Data
A real-world cancer dataset, derived from Efron [16], is investigated to demonstrate how the various suggested estimators may be used for real-world medical phenomena. The dataset comprises the survival periods of 45 head and neck cancer patients treated with the combination of radiation and chemotherapy. Table 8 shows the comparable survival periods for these cancer patients. Yadav et al. [17] recently studied these data as well. To make computations easier, we divided the original data by 10.
To verify if these data are modeled by the APE distribution, the K–S distance with associated p value is considered. First, we calculate the MLEs (with their SEs) of the unknown parameter and which are 0.0098 (0.0280) and 0.0129 (0.0086), respectively. Thus, the K–S distance is 0.1093 with value 0.617. This result indicates that the APE distribution is a suitable model to fit head-neck cancer data. Again, to show the existence and uniqueness of the MLEs and , the contour plot of the log-likelihood function with respect to and using the complete head-neck cancer dataset is displayed in Figure 6. The coordinates of x-point provide the MLEs of and which are close to 0.0098 and 0.0129, respectively. Furthermore, one can conclude that the MLEs and are exist and are also unique.
Using the complete survival times of head-neck cancer patients, based on various choices of , , and for , different four progressively Type-I censored samples with corresponding censoring schemes are generated and presented in Table 9. Again, Type-I censored (Scheme-IV) and complete sampling (Scheme-V) schemes can be obtained as a special cases from progressive Type-I censoring if putting and . Hence, the MLEs and Bayes MCMC estimates with their SEs of , , , and at time are computed and listed in Table 10.
Using the MH algorithm sampler when , the Bayes MCMC estimates are developed. Then, we generate 30,000 MCMC samples, and then, the first 5000 iterations have a burn-in period. Also, two-sided 95% asymptotic/credible intervals with their lengths of , , , and are calculated and listed in Table 11. Table 12 contains the MCMC statistics of , , , and . Vital statistics about the parameters are also computed from the 25,000 generated sample values. All assessments are carried out by using R statistical computer language and two useful statistical packages suggested by Elshahhat and Nassar [18], namely the “CODA” package, which is used to carry out the computations of the MCMC suggestion made by Plummer et al. [19], and the “maxLik” package, which is used to carry out the computations of the N-R method of maximization proposed by Henningsen and Toomet [20].
In general, the Bayes MCMC estimates of unknown parameters are shown to be better than those derived using the conventional technique in terms of standard errors and smallest confidence intervals. In addition, the convergence of the MCMC algorithm was examined by tuning the variance with the help of a generated sequence of random deviates. It was found that the generated sample from the considered proposal distribution is able to mix well with their sample mean and 95% confidence BCI/HPD credible intervals, as shown in Figure 7. This was one of the findings that emerged from this examination. Figure 8 displays the marginal posterior density estimates of the model parameters, reliability, and hazard rate functions, as well as their histograms. These estimates were calculated using a Gaussian kernel and were based on samples with a size of 25,000. The calculations make it very clear that all of the marginal distributions are quite close to being symmetrical. In addition, the findings of this dataset lend credence to the conclusion that we reached about the electrical appliances dataset.
5. Conclusion Remarks
The research considers the estimated difficulties for the APE lifespan model’s parameters, reliability, and hazard rate functions under Type-I PCS. The MLEs for the unknown parameters and any function of them are calculated, along with their asymptotic confidence intervals. The posterior density function cannot be calculated in closed forms due to the complexity of the likelihood function’s formula. Thus, under the assumption of independent gamma priors and taking into account the SE and LINEX loss functions, Bayesian estimates and accompanying HPD credible ranges are constructed using the Metropolis–Hastings method. To evaluate the behavior of the various estimates, electrical appliance and head-neck cancer data were studied to demonstrate the suggested approaches’ practical applicability in real-world phenomena and to provide the optimal censoring scheme. Lastly, we hope that the findings and approaches presented here will be valuable to practitioners of dependability and/or may be used in other censoring strategies [21].
Data Availability
All data are included within the paper.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
This study was funded by Taif University Researchers Supporting Project (number TURSP-2020/279), Taif University, Taif, Saudi Arabia.