1. Introduction
In practical reliability and engineering fields, the life characteristic of units is usually featured by lifetime distributions, and there are extensive models proposed in practice to fit different data. The most used models include exponential, Weibull, gamma, and log-normal, among others. Recently, a general family of inverted exponentiated exponential distributions (IEED) has been proposed for data analysis and has attracted wide attention. Let
X be a random variable from the IEED, then the cumulative distribution function (CDF) and the probability density function (PDF) of
X are, respectively, given by
and
The reliability function (RF) and hazard rate function (HRF) of
X can be written in consequence as
In
Figure 1,
Figure 2 and
Figure 3, we have displayed the different shapes of the hazard rate functions of inverted exponentiated exponential distribution (IEE), inverted exponentiated Rayleigh distribution (IER), and inverted exponentiated Pareto distribution (IEP), respectively, with
and
. From the figures, we observed that the hazard rate functions of the IEE and IER distributions experience non-monotonic behavior for each set of parameter values. Further, we observed that the hazard function of IEP distribution decreases rapidly with time when both the shape parameters take values less than one. It is also seen that the behavior can be non-monotone in nature. This implies that the family of IE distributions may offer a suitable fit for datasets displaying decreasing or non-monotonic hazard behavior. However, it is important to note that this distribution family is unsuitable for modeling datasets characterized by increasing, constant, or bathtub-shaped hazard rate behavior.
Recently, Ghitany et al. [
1] proposed the family of inverted exponentiated distributions, reported several characteristics of this distribution, and observed that this family is capable of modeling many real-life phenomena. For instance, different inferences are obtained in the literature for this family. For example, interested readers may refer to the works of Kizilaslan [
2] and Fisher [
3]. In a related study, Kumari et al. [
4] examined a single-component stress–strength model for this family of distributions. Similar works with respect to the stress and strength variables could also refer to some of the contributions by Temraz [
5], de la Cruz et al. [
6], Alsadat et al. [
7], and EL-Sagheer et al. [
8]. Jamal et al. [
9] studied the type II general inverse exponential family of distributions. Dutta et al. [
10] obtained useful estimation results for a general family of inverted exponentiated distributions under unified hybrid censoring, with partially observed competing risk data. Hashem et al. [
11] estimated the reliability of the IER distribution by using the Bayesian method under generalized progressive hybrid censored data, among others. Due to its definition, it is noted that some conventional lifetime models belong to this general family. For example, by considering
and
, the inverted exponentiated exponential, the inverted exponentiated Rayleigh, and the inverted exponentiated Pareto distributions are obtained as special cases of this family, as mentioned above, in distributional plots. Interested readers may further refer to Maurya et al. [
12], Gao et al. [
13], and Wang et al. [
14] for some further discussion on the applicability of this family of densities in real-life situations.
In statistical inference, as an alternative to classical likelihood-based estimation, Bayesian inference is gaining increased popularity among researchers owing to its capacity to incorporate prior information in analyses, making it very valuable in reliability, lifetime study, and other associated fields, where one of the major challenges is the limited availability of data. When proper priori information is adopted in the inferential approach, the Bayesian procedure may obtain improved estimates for considered parametric functions. For instance, Sinha [
15,
16,
17] presented a detailed study on this method by deriving inferences for (RF) under normal, inverse Gaussian, and Weibull distributions, respectively. Lye et al. [
18] and Lin et al. [
19] studied Bayesian estimation under complete and masked data by considering different approximation procedures. Pensky and Singh [
20] studied the empirical Bayes estimation of reliability for an exponential family. In the Bayes framework, among others, Dey [
21,
22] obtained useful parametric estimates for the Rayleigh family of distributions. Amirzadi et al. [
23] discussed Bayes estimation based on informative and non-informative priors by considering various loss functions, such as the general entropy loss function (GELF), squared-log error loss function (SLELF), and weighted squared-error loss function (WSELF). In addition, another new loss function (NLF) is also introduced in Amirzadi et al. [
23]’s work to evaluate reliability estimates. Amirzadi et al. [
23] studied several structural properties of inverse generalized Weibull (IGW) distribution. They showed that the corresponding hazard rate is of non-monotonic behavior. However, it is noteworthy that the family of IE distributions, notably the IEP distribution, is capable of fitting data indicating decreasing hazard rate behavior as well. Due to potential theoretical applications of the IEED, this paper considers the problem of parameter estimation as well as the reliability function of this family of distributions under classical and Bayesian perspectives, and then the associated behavior of different results is compared through numerical studies and real-life examples. For concision, the notations used in the paper are listed in
Table 1 for simplification purposes.
This article is organized as follows.
Section 2 deals with the maximum likelihood and uniformly minimum variance unbiased (UMVU) estimators. In
Section 3, the Bayes estimators are derived under different loss functions.
Section 4 discusses the Bayes estimators under a relatively new loss function.
Section 5 presents Monte Carlo simulation experiments to determine the efficiency of the proposed estimation procedures. Finally, in
Section 5, lifetime datasets are analyzed to support the proposed methods, and a conclusion is presented in
Section 6.
4. Simulation Study
In this section, extensive simulation experiments are carried out to assess the performance of the different estimators of parameter and the RF . All estimates are evaluated based on their computed values and MSEs. The true parameter values are assigned as and 2. Further, the given is taken to be in all cases. The sample sizes are chosen as , and 60, reflecting the small, medium, and large sample sizes, respectively. For Bayesian inference, the values of the hyper-parameter c in the Jeffreys prior are 0.4 and 0.8, whereas the hyper-parameters of the informative gamma hyper-parameters are set to be and , respectively. The values of the weight loss function constant k are considered as 0 and 1. The simulation is conducted based on R software, and the results are obtained through 1000 times repetition. In addition, for generating random samples from the considered family of distribution densities, a special IEP distribution is used, and the data generation procedure is provided as follows.
In sequel, we simulate the estimators
,
,
,
,
, and
for different sample sizes. Subsequently, all these estimates are evaluated in terms of their error and estimated values. Note that the MSE is evaluated as
where
r is the number of replications. The criterion examines the estimator patterns for different parameter assignments and sample sizes. The simulation results for the different estimates are summarized in
Table 2,
Table 3,
Table 4,
Table 5 and
Table 6.
It is seen here that the estimated errors of both the ML and Bayesian estimators of the reliability function tend to decrease with an increase in sample sizes. From the results reported in
Table 2,
Table 3,
Table 4 and
Table 5, it is also found that the Bayes estimators are marginally better than the respective MLEs and, in effect, lower the estimated MSEs. Further, these errors for the UMVU estimator are also lower than the ML estimator, as can be seen in
Table 2. In fact, the Bayes estimators perform better than the UMVU estimates as well. One could observe from
Table 3,
Table 4 and
Table 5 that the Bayes estimators of RF under gamma prior show better MSE performance compared to the non-informative prior. In addition, the estimates obtained under the NLF provide better MSE behavior compared to other loss functions, particularly for the moderate sample size
n = 20. For the large sample sizes proposed, the estimates show similar behavior under the given criterion. Further, we also visually compare the patterns of the ML and Bayes estimators in
Figure 4 and
Figure 5, when
,
, and sample size
. We find that the proposed estimates of reliability converge to zero for large values of
t. In these figures, we have also plotted the respective reliability functions as well. In fact, it is also observed that the UMVU estimates are closer to the reliability function when compared to the ML estimates. Equivalently,
Figure 5 suggests that the estimates obtained under the new loss function show the best performance, as its value is closest to the actual reliability. We have not presented graphs for other values of
n, as no noticeable difference in the behavior of the proposed estimates was observed with respect to the various loss functions.
5. Real Data Analysis
Here, we present the analysis of real-life examples in support of the considered estimation problem. We consider the tensile strength data as presented in Bader and Priest [
24]. The data describe the strength in GPa for single-carbon fibers and impregnated 1000-carbon fiber tow. Single fibers are tested under tension at gauge lengths of 20 mm (Dataset 1) and 10 mm (Dataset 2). Here, sample sizes are taken as
n = 69 and 63, respectively. These data are also discussed by Kundu and Raqab [
25] and Surles and Padgett [
26] for applications under different frameworks.
The details of the tensile strength data, namely Dataset 1, are presented as:
1.312, 1.314, 1.479, 1.552, 1.700, 1.803, 1.861, 1.865, 1.944, 1.958, 1.966, 1.997, 2.006, 2.021, 2.027, 2.055, 2.063, 2.098, 2.14, 2.179, 2.224, 2.240, 2.253, 2.270, 2.272, 2.274, 2.301, 2.301, 2.359, 2.382, 2.382, 2.426, 2.434, 2.435, 2.478, 2.490, 2.511, 2.514, 2.535, 2.554, 2.566, 2.57, 2.586, 2.629, 2.633, 2.642, 2.648, 2.684, 2.697, 2.726, 2.770, 2.773, 2.800, 2.809, 2.818, 2.821, 2.848, 2.88, 2.954, 3.012, 3.067, 3.084, 3.090, 3.096, 3.128, 3.233, 3.433, 3.585, 3.585.
Similarly, Dataset 2 is given as follows:
1.901, 2.132, 2.203, 2.228, 2.257, 2.350, 2.361, 2.396, 2.397, 2.445, 2.454, 2.474, 2.518, 2.522, 2.525, 2.532, 2.575, 2.614, 2.616, 2.618, 2.624, 2.659, 2.675, 2.638, 2.74, 2.856, 2.917, 2.928, 2.937, 2.937, 2.977, 2.996, 3.03, 3.125, 3.139, 3.145, 3.22, 3.223, 3.235, 3.243, 3.264, 3.272, 3.294, 3.332, 3.346, 3.377, 3.408, 3.435, 3.493, 3.501, 3.537, 3.554, 3.562, 3.628, 3.852, 3.871, 3.886, 3.971, 4.024, 4.027, 4.225, 4.395, 5.020.
We compared the fit of the proposed special inverted distributions with other competing models, namely gamma, inverse gamma, and inverse Weibull distributions are very commonly used for fitting various lifetime data.
Table 7 and
Table 8 contain maximum likelihood estimates of all the model parameters and goodness-of-fit statistics (K-S statistic,
p-value) for the first and second datasets, respectively. Based on these two tables, we conclude that the family of IEP distribution provides a satisfactory fit in the sense that it yields the smallest K-S estimates with a large
p-value. Thus, this considered family can be used to obtain inferences from these datasets.
Further, the MLE and Bayes estimates of reliability for both datasets are presented in
Table 9 and
Table 10 by considering the proposed loss functions and prior distributions. To derive Bayes estimates for a real data analysis case, we specified non-informative prior distribution. Accordingly, we set
and
as approaching a zero value. Among others, we found that the Bayes estimates derived under the NLF are marginally bigger than the other studied estimates where
t = 2, 3, 4.