1.1. Generalized Inverted Exponential Distribution
The generalized inverted exponential distribution (GIED) is a modification of the inverse exponential distribution (IED). In this way, it can fit the lifetime data better. The GIED was introduced by [
1]. The distribution has a non-constant hazard rate function, which is unimodal and positively skewed. Due to these properties, the distribution can model different shapes of failure rates of aging criteria. Reference [
2] proposed the method of the maximum product of spacings for the point estimation of the parameter of the GIED. In [
3], acceptance sampling plans were developed based on truncated lifetimes when the life of an item follows a generalized inverted exponential distribution. Reference [
4] performed a Monte Carlo simulation for the GIED to analyze the performance of the estimations. Reference [
5] studied the point estimation of the parameters of the GIED when the test units are progressively type-II censored. Reference [
6] generated samples from the GIED and computed the Bayes estimates. Reference [
7] investigated the MLEs of the GIED when the test units are progressively type-II censored. Reference [
8] proposed a two-stage group acceptance sampling plan for the GIED under a truncated life experiment.
Provided that
X is a variable and it is subject to the GIED, the following is the form of the corresponding probability density function (pdf), the cumulative distribution function, as well as hazard function. Here,
is the shape parameter and
is the scale parameter. Besides, they are both positive.
The plots of the pdf and hazard function are presented in
Figure 1 and
Figure 2.
1.2. The Joint Progressive Type-II Censoring Scheme
It is difficult to obtain the lifetime data of all the products given the cost and time in many practical situations. Hence, testing experiment censoring is of great importance. A great deal of work has been performed on a variety of censoring schemes. The experimental units cannot be withdrawn during the experiment under the type-I censoring scheme and type-II censoring scheme. Reference [
9] described the progressive type-II censoring scheme, in which some units are allowed to be withdrawn during the test. Afterward, we describe the progressive type-II censoring briefly. Suppose
n units are placed in a lifetime test.
k is the effective sample size. It also represents the number of observed failures that satisfies
.
, ⋯,
stand for the number of units to be withdrawn for each failure time. Furthermore, they are non-negative integers and satisfy
. At the first failure time,
units are withdrawn from the remaining
surviving units randomly. When the second failure happens, we randomly withdraw
units from the remaining
surviving units. Analogously, when the
k-th failure happens, the remaining
surviving units are withdrawn randomly and the test ceases. Reference [
10] provided an amount of work about the progressive censoring schemes. Reference [
11] dealt with the Bayesian inference on step stress partially accelerated life tests using type-II progressive censored data in the presence of competing failure causes.
Much research about progressive censoring schemes for one sample has been performed by many scholars. However, there is little research on two samples. Reference [
12] first introduced the joint progressive censoring schemes for two samples. It is particularly beneficial to compare the life distribution of different products produced by two different assembly production lines on diverse equipment under the same environmental conditions. The joint progressively type-II censoring (JPC) scheme can be briefly described as follows. Suppose the samples are from two different lines, Line 1 and Line 2. The size of the samples of products in Line 1 is
m and in Line 2 is
n. Two samples are combined in the joint progressive censoring scheme, and they are placed on a lifetime test. Suppose
is the size of combined samples.
, ⋯,
stand for the number of units to be withdrawn in each failure time. Additionally, they are non-negative integers and satisfy
. At the first point of failure
,
units are removed from the combined samples at random.
units consist of
units from Line 1 and
units from Line 2. On the second failure
,
units are withdrawn from the remaining
samples at random.
units consist of
units from Line 1 and
units from Line 2. Analogously, at the
k-th failure time point
, the remaining
surviving units are withdrawn randomly and the test ceases. Let
, ⋯,
be random variables. If the
i-th failure is from Line 1, let
. Otherwise, let
. Suppose that the censored sample is
. Here, we introduce
, which means the number of failures from Line 1. Similarly,
, which stands for the number of failures from Line 2.
Figure 3 shows the scheme.
Reference [
12] applied Bayesian estimation techniques to two exponential distributions for the JPC scheme. Reference [
13] considered the JPC scheme for more than two exponential populations and studied the statistical inference. Reference [
14] investigated the conditional maximum likelihood estimations and the interval estimations of the Weibull distribution for the JPC scheme. Reference [
15] discussed the point estimation and obtained the confidence intervals of two Weibull populations for the JPC scheme. Reference [
16] obtained the Bayes estimation when data were sampled in the JPC scheme from a general class of distributions. Reference [
17] studied the expected number of failures in the lifetime tests under the JPC scheme for various distributions. Besides, a new type-II progressive censoring scheme for two groups was introduced by [
18]. Reference [
19] extended the JPC scheme for multiple exponential populations and studied the statistical inference. Reference [
20] studied the likelihood and Bayesian inference when data were sampled in the JPC scheme from the GIED.
A few scholars have studied the statistical inference of the generalized inverted exponential distribution when the test units are progressively type-II censored. However, no one has studied the statistical inference of the generalized inverted exponential distribution under joint progressively type-II censoring. The research on this aspect is of great significance.
In this article, we provide statistical inference and study the JPC scheme for two groups that follow the GIED with the same scale parameter. The expectation maximization algorithm is adopted to calculate the maximum likelihood estimates of the parameters. In light of the missing value principle, we derive the observed information matrix. We obtain the interval estimations by the bootstrap-p method based on the Fisher information matrix. We assume a Gamma prior for the shape and scale parameters. The Bayes estimates and credible intervals for the informative prior and the non-informative prior under the linex loss function and squared error loss function are calculated by adopting the importance sampling technique. The performances of various methods are compared through the Monte Carlo simulation. Besides, we conduct real data analysis. Moreover, in many practical cases, the experimenters may know that the lifetime of various populations is orderly. We investigate the problem that the parameters have order restrictions. We discuss the maximum likelihood estimation and Bayesian inference of the parameters.
The rest of the article is arranged as follows. In
Section 2, we obtain the likelihood function. In
Section 3, we apply the EM algorithm to calculate the MLEs. In
Section 4, we compute the observed information matrix based on the missing value principle. Next, we adopt the bootstrap method to obtain the confidence intervals. The Bayesian inference is presented in
Section 5. In
Section 6, a Monte Carlo simulation and real data analysis are shown. In
Section 7, we derive the maximum likelihood estimation and Bayesian inference of the parameters that have order restrictions.