Estimation of Lifetime Performance Index for Generalized Inverse Lindley Distribution Under Adaptive Progressive Type-II Censored Lifetime Test

Process capability analysis is a key component of quality management and a core aspect of statistical process control. This analytical method effectively assesses the potential performance and capability of a process, making it widely applicable in quality improvement initiatives within production processes [1]. The process capability index, as the evaluation result of the process capability analysis, measures a process’ ability to meet specification requirements for product quality characteristics. It provides management with an intuitive basis for judgment, helping to determine whether the process capability meets the standards of technical specifications and customer demands [2]. Therefore, the process capability index holds a crucial position in the science of quality management and is of significant importance in reducing waste, enhancing product quality, and improving management levels. Quality management experts and statisticians have proposed various process capability indices, including $C_p$, $C_{pm}$, $C_{pk}$, and $C_{pmk}$ [3,4]. Each of these indices has its unique calculation method and application scenarios, offering diverse choices for process capability assessments across different industries and domains.

With the continuous advancement of manufacturing technologies, consumers are increasingly concerned about the lifetime of products when making purchasing decisions. To enhance their market competitiveness, manufacturers must prioritize the assessment and improvement of product quality and reliability. Although process capability indices hold a significant position in quality management, they are not suitable for measuring product lifetime. Typically, the quality of a product can be assessed by its lifetime, as a longer lifetime often indicates a higher product quality and reliability. Therefore, there is a need for a “bigger is better” metric to evaluate product lifetime.

Montgomery [5] proposed a process capability index suitable for one-sided specifications, named the lifetime performance index $\delta$, which is used to measure the performance of a product’s lifetime. The mathematical expression of $\delta$ is defined as follows: where $L$ is the given lower specification limit, $\mu$ is the process mean, and $\sigma$ is the process standard deviation. This lifetime performance index (LPI) is particularly applicable to products where a longer lifetime represents better performance. It provides a quantitative tool to assist manufacturers in evaluating and improving the lifetime characteristics of their products. Manufacturers can use the value of this index to determine whether the lifetime of a product meets specific quality requirements and take appropriate measures to enhance product reliability and market competitiveness. Shaabani and Jafari [6] conducted a classical estimation on the LPI, assuming that the product lifetime followed a gamma distribution, based on a complete sample. İklim [7] constructed bootstrap confidence intervals of process capability indices for a generalized inverse Lindley distribution, based on a complete sample. However, due to various constraints such as time costs, material costs, and the tools for data collection, it is often not feasible to obtain the lifetimes of all products in a life test. Therefore, scholars have proposed the censored life test as a more efficient method of data collection, which has been widely applied.

The type-I censored life test, type-II censored life test, and hybrid censored life test have been extensively studied by researchers. These censored tests, however, lack flexibility as they do not allow for the removal of test products during the test, which hinders the implementation of related studies. To address these issues, Cohen and Clifford [8] proposed the progressive censored test, which can reduce the testing time and material costs to some extent. Kilany and Lobna [9] considered the maximum likelihood (ML) estimation and Bayesian estimation of the LPI based on the three-parameter Omega distribution, using a progressive type-II censored sample. Mohammad and Mahdi [10] assumed that the product lifetime followed a Weibull distribution and discussed the Bayesian estimation of the LPI based on progressive censored data. Hanan et al. [11] assumed that the sample followed an Ishita distribution and considered the estimation of the LPI based on a progressive type-II censored scheme.

However, with the continuous advancement of science and technology, product lifetimes are becoming longer. Even when using a progressive censored life test, there may still be cases where the testing duration becomes excessively long or no failures are observed during the test. To address these issues, Kundu and Joarder [12] proposed the progressive type-II hybrid censored life test, which combines the progressive type-II censored test with the hybrid censored test. This test ensures that the experiment ends within a predetermined time frame. However, this test may not guarantee that a sufficient number of failures are collected, and there may even be no failures observed. As a result, the statistical inference results and efficiency may be unsatisfactory. Building upon this, Ng et al. [13] proposed the adaptive progressive type-II censored life test. In this test, the censored scheme can be adjusted in a timely manner based on the actual situation to ensure a sufficient number of failure data are collected. This approach can flexibly handle potential changes during the test, improving the statistical inference results and efficiency.

In a censored life test, $n$ products are put into the test. Only $m$ failed products need to be observed. The censored scheme $R=(R_1,R_2,\dots ,R_m)$ satisfies $R_1+R_2+\dots +R_m=n-m$. When the first failed product occurs, the failure time is recorded as $X_{1:m:n}$, and then $R_1$ non-failed products are arbitrarily removed. When the second failed product occurs, the failure time is recorded as $X_{2:m:n}$, and then $R_2$ non-failed products are arbitrarily removed. The process continues until the m-th failure occurs. In the meantime, the remaining $R_m$ non-failed products are removed, bringing the test to a close. The recorded failure times $X_{1:m:n},X_{2:m:n},\dots ,X_{m:m:n}$ are termed progressive type-II censored samples, and the arbitrarily removed non-failed products $R_1,R_2,\dots ,R_m$ are termed the progressive type-II censored scheme.

The adaptive progressive type-II censored test can be seen as a combination of the type-I censored test and the progressive type-II censored test. Before the test begins, a time $T$ is set to represent the ideal duration of the test. In fact, the actual duration of the test is allowed to exceed $T$. If the number of failures observed before the time $T$ reaches $m$, the test ends before $T$. However, if the test time exceeds $T$ and $m$ failures have not yet been observed, the test should end as soon as possible. The experimenters should make some adjustments to ensure the rapid occurrence of $m$ failures. Therefore, it is necessary to retain as many non-failed products as possible throughout the test period. The two cases of the adaptive progressive type-II censored test are as follows:

Case 1. If $m$ failure times have been recorded before the time $T$, then the censored scheme $R=(R_1,R_2,\dots ,R_m)$ remains consistent with the predetermined one.

Case 2. If only $k (k<m)$ failure times have been recorded before the time $T$, that is, In order to retain as many non-failed products as possible in the experiment, the experimenter stops removing non-failed products once the (k + 1)-th failure time is recorded. All remaining non-failed products are removed only after the m-th failure time has been recorded. Therefore, $R_1,R_2,\dots ,R_k$ remain consistent with the predetermined censored scheme. But there are $R_{k+1}=R_{k+2}=\dots =R_{m-1}=0$ and $R_m=n-m-R_1-R_2-\dots -R_k$. In other words, the censored scheme is adjusted to $R=(R_1,\dots ,R_k,0,\dots ,0,R_m)$.

The specific experimental process can be seen in Figure 1.

The process capability index allows for the continuous monitoring of process quality, ensuring that the produced products meet specification requirements and providing a basis for reducing product failure costs. Typically, the process capability index is calculated based on the lifetime of products; thus, a longer lifetime is generally associated with a higher product quality. Considering this, this paper chooses to use the LPI to evaluate product quality. The lifetime of many products may not follow a normal distribution and may instead exhibit characteristics such as an exponential distribution, Weibull distribution, or Burr distribution. Consequently, extensive research has been conducted on the inference of the LPI. Table 1 provides a review of more references on the LPI.

The generalized versions of statistical models offer greater flexibility for modeling and analyzing real-world data. Among these, the generalized inverse Lindley distribution (GILD) is a novel statistical model suitable for analyzing inverted bathtub survival information. More details about the GILD can be found in Section 2. To our knowledge, the inference of the LPI has not been addressed in the case where the shape and scale parameters of the GILD are unknown. Therefore, this paper assumes that the lifetimes of the products follow the GILD and considers the estimation of the LPI under the adaptive progressive type-II censored sample.

The remaining sections are organized as follows: In Section 2, a brief overview of the GILD is provided, and the expressions of the LPI for the GILD are derived. In Section 3, the maximum likelihood (ML) estimator of the LPI is obtained, and the asymptotic confidence interval (ACI) is constructed. The bootstrap confidence interval (BCI) is discussed in Section 4. In Section 5, using the symmetric entropy loss function (SELF), LINEX loss function (LLF), and general entropy loss function (GELF), the Bayesian estimators of the LPI are derived based on the gamma priors, and the highest posterior density (HPD) credible intervals are constructed. In Section 6, the performance of these estimators and confidence intervals are compared through Monte Carlo simulations. In Section 7, two sets of real data are used to illustrate the feasibility of these estimation methods. Finally, Section 8 presents the conclusions of this paper.

In real life, there is a type of data whose hazard function often exhibits a distinctive shape known as the “upside-down bathtub” shape. This type of data includes, but is not limited to, the failure rates of electronic products or mechanical devices, the mortality rates of biological individuals, the occurrence rates of natural disasters, and product demand rates. Specifically, this “upside-down bathtub”-shaped hazard function provides a more accurate depiction of the entire process from the occurrence and development to the eventual decline of certain phenomena. A profound understanding of this pattern holds significant importance for the prediction and management of risks in relevant domains. Addressing this characteristic, Sharma et al. [20] proposed a statistical model called the generalized inverse Lindley distribution. This distribution is an extension of the inverse Lindley distribution and is particularly suitable for modeling survival information with an “upside-down bathtub” shape. By introducing a new parameter, the GILD can more accurately capture and describe the true characteristics of the data, thus providing a more powerful tool for risk prediction and management.

Some research on the GILD has been conducted. Basu et al. [21] considered the classical estimation and Bayesian estimation for the GILD under the progressive type-I hybrid censoring with binomial removal. Vikas [22] assumed that $X$ and $Y$ represented the survival times of two groups of cancer patients under different treatment plans, respectively, and that they were independent and followed the GILD. Under this premise, the Bayesian estimator of $P(X>Y)$ was proposed. Devendra et al. [23] used generalized order statistics to obtain two types of estimators for the parameters of the GILD. Fatma et al. [24] considered the generalized inverse Lindley stress–strength reliability model under different sampling designs and obtained the maximum likelihood estimator of stress–strength reliability. More related literature can be found in the references [25,26,27,28,29].

Let $X$ be a random variable that follows the GILD with the shape parameter $\lambda >0$ and scale parameter $\beta >0$, denoted by $GILD(\lambda ,\beta )$. The probability density function (PDF), cumulative distribution function (CDF), and hazard function (HF) are given as (Sharma et al. [20]) and

Figure 2 presents the diagrams of the PDF and HF for the GILD with different combinations of the parameters $\lambda$ and $\beta$, respectively.

In the following discussion, we always suppose that $\lambda >2$. Let the product lifetime $X$ follow the GILD with the PDF (2) and CDF (3); the LPI $\delta$ can be derived as where

If the product lifetime $X$ exceeds the lower specification limit $L$ (i.e., $X>L$), then the product is marked as a conforming product. Otherwise, the product is marked as an unconforming product. The probability $P(X>L)$ is referred to as the conforming rate of the product, denoted as $P_{cr}$. Therefore, $P_{cr}$ can be defined as

From Equation (8), it is evident that, for the given values of $\lambda$ and $\beta$, the conforming rate $P_{cr}$ and $\delta$ exhibit a strict positive correlation relationship. To show this relationship, Table 2 lists the values of the LPI $\delta$ and the corresponding conforming rate $P_{cr}$ under $\lambda =3.1956$ and $\beta =7.3937$.

Let $X=(X_{1:m:n},X_{2:m:n},\dots ,X_{m:m:n})$ be an adaptive progressive type-II censored sample from $GILD(\lambda ,\beta )$; the relative censored scheme is $R=(R_1,\dots ,R_k,0,\dots ,0,R_m)$, where $R_m=n-m-R_1-\dots -R_k$. Denote $x=(x_{1:m:n},x_{2:m:n},\dots ,x_{m:m:n})$ as the observation of $X$. The likelihood function of $\lambda$ and $\beta$ is where $A={\displaystyle \prod _{i=1}^m(n-i+1-{\displaystyle \sum _{j=1}^{i-1}{R}_j})}$ is a constant and $x_i$ is used instead of $x_{i:m:n}$ for simplicity. Then, the log-likelihood function is given by

The partial derivatives of the log-likelihood function with respect to $\lambda$ and $\beta$ are as follows:

Therefore, the ML estimators of $\lambda$ and $\beta$, say, $\widehat{\lambda }$ and $\widehat{\beta }$, are the solutions of Equation (13).

Due to the invariance of the maximum likelihood estimation, the ML estimator ${\widehat{\delta }}_{ML}$ can be obtained by putting $\widehat{\lambda }$ and $\widehat{\beta }$ into Equation (5), that is,

Equation (13) is nonlinear and does not admit an analytical solution. Therefore, we employ a numerical iterative method to solve it. In this paper, we utilize the Newton–Raphson iteration approach, and the iteration process is provided in Algorithm 1.

Algorithm 1. Newton–Raphson iteration approach using calculate ML estimate of $\delta$

(1) Initial Guess: Start with an initial guess $({\lambda }^{(0)},{\beta }^{(0)})$ for the root of Equation (13).

(2) Calculate the Derivative: Find the first partial derivatives of the Equations (11) and (12).

(3) Iteration: For each iteration $j$, calculate the next approximation $({\lambda }^{(j+1)},{\beta }^{(j+1)})$ by using the following equation: $${({\lambda }^{(j+1)},{\beta }^{(j+1)})}^{\mathrm{T}}={({\lambda }^{(j)},{\beta }^{(j)})}^{\mathrm{T}}-{[I_2({\lambda }^{(j)},{\beta }^{(j)})]}^{-1}{I}_1({\lambda }^{(j)},{\beta }^{(j)}).$$ (15)

(4) Convergence Check: Check if the absolute or relative error between $({\lambda }^{(j+1)},{\beta }^{(j+1)})$ and $({\lambda }^{(j)},{\beta }^{(j)})$ is less than a predetermined tolerance level. If it is, then $({\lambda }^{(j+1)},{\beta }^{(j+1)})$ is considered a root of the Equation (13), and set $\widehat{\lambda }={\lambda }^{(j+1)}$ and $\widehat{\beta }={\beta }^{(j+1)}$.

(5) Repeat: If the convergence criterion is not met, repeat the process from the step (3) with $({\lambda }^{(j+1)},{\beta }^{(j+1)})$ as the new approximation.

(6) Termination: The process is terminated when the accuracy level is achieved or after a maximum number $J$ of iterations is reached.

(7) the ML estimator ${\widehat{\delta }}_{ML}$ can be obtained according to Equation (14).

In the above algorithm, there are and

The preceding discussion has made it evident that Equation (13) is complex, precluding the derivation of an exact expression for the ML estimators of $\lambda$ and $\beta$. Consequently, it is impractical to ascertain precise confidence intervals. Considering the above, we construct an ACI for $\delta$ using the delta method in this paper. An ACI is constructed using the principles of asymptotic theory, which deals with the behavior of estimators and statistical tests as the sample size grows indefinitely. The primary idea is that, as the sample size increases, the distribution of the estimator tends to a normal distribution due to the central limit theorem. This allows for the approximation of the sampling distribution of the estimator with a normal distribution, even if the underlying population distribution is not normal.

The Fisher information matrix is

Let $\Psi (\lambda ,\beta )=({\displaystyle \frac{\partial \delta }{\partial \lambda }},{\displaystyle \frac{\partial \delta }{\partial \beta }})$,

Let $H^{-1}(\lambda ,\beta )$ be the inverse matrix of $H(\lambda ,\beta )$; then, according to the delta method, the estimate of the variance $r$ is obtained as follows:

The $100(1-\gamma )\%$ ACI of the LPI $\delta$ is where $z_{\gamma /2}$ is the upper $\gamma /2$ quantile of the standardized normal distribution.

The bootstrap method is a resampling-based statistical inference approach that is widely used for constructing confidence intervals for unknown parameters. The core aim of this method is to generate some bootstrap samples through repeated sampling from the original sample and then calculate the required statistics for each bootstrap sample to obtain the empirical distribution of the statistic. Compared to the ACI, the BCI has the following advantages:

Therefore, the bootstrap method has received widespread attention and been widely applied in contemporary statistical research, and it has become an important statistical inference tool. Under various distributional assumptions, Ouyang et al. [30] constructed three bootstrap confidence intervals for the process capability index $C_{pc}$ and compared their performance. Saha et al. [31] employed five bootstrap methods to construct the BCIs of the process capability index $C_{pc}$ for an exponentiated exponential distribution. Based on a progressive type-II right censored sample, Tolba et al. [32] constructed the BCI unknown parameter for a one-parameter Akshaya distribution. For more research about the bootstrap method, please refer to [33,34,35,36].

In this section, we use the bootstrap-t method (Hall [37]) to construct the BCI of the LPI $\delta$. The steps are presented in the following Algorithm 2.

Algorithm 2. The calculation process of the bootstrap-t method to construct the BCI of $\delta$

(1) According to Algorithm 1, compute the ML estimates $\widehat{\lambda }$ and $\widehat{\beta }$ under the censored sample $(x_1,x_2,\dots ,x_m)$ and censored scheme $(R_1,R_2,\dots ,R_m)$. Furthermore, the ML estimate of $\delta$, denoted as $\widehat{\delta }$, is obtained.

(2) Generate adaptive progressive type-II censored samples from $GILD(\widehat{\lambda },\widehat{\beta })$ with $(R_1,R_2,\dots ,R_m)$, and denote as $(x_1^{*},x_2^{*},\dots ,x_m^{*})$.

(3) As in step (1), calculate the ML estimates of $\lambda$ and $\beta$ based on $(x_1^{*},x_2^{*},\dots ,x_m^{*})$, say $\widehat{\lambda }*$ and $\widehat{\beta }*$. Then, obtain the Bootstrap sample estimates $\widehat{\delta }*$ by putting $\widehat{\lambda }*$ and $\widehat{\beta }*$ into Equation (5).

(4) Compute the statistic $\tau =\sqrt{J}(\widehat{\delta }*-\widehat{\delta }){[\mathrm{var}(\widehat{\delta }*)]}^{-1/2}$, where $\mathrm{var}(\widehat{\delta }*)$ is estimated by Equation (22).

(5) Repeat steps (2) to (4) for $J$ times and obtain $({\tau }_1,{\tau }_2,\dots ,{\tau }_J)$, where ${\tau }_1<{\tau }_2<\dots <{\tau }_J$.

(6) Let $F_{\tau }(t)=P(\tau \le t)$ be the CDF of $\tau$. For a given $\gamma$, define ${\widehat{\delta }}_{Boot-t}(\gamma )=\widehat{\delta }+J^{-1/2}{F}_{\tau }^{-1}(\gamma )\sqrt{\mathrm{var}(\widehat{\delta })}$. Thus, the BCI of $\delta$ is $$({\widehat{\delta }}_{Boot-t}(\gamma /2),{\widehat{\delta }}_{Boot-t}(1-\gamma /2)).$$ (27)

Bayesian statistics is a branch of statistics that applies the Bayesian theorem to update the probability of a hypothesis as more evidence or information becomes available. It provides a powerful framework for making inferences based on prior knowledge combined with new data. Bayesian statistics allows the incorporation of prior knowledge or expert opinions into the analysis, which can be particularly useful when data are scarce or expensive to obtain.

The gamma prior is a commonly used prior distribution in Bayesian statistics, often employed in establishing Bayesian inference models for parameters, and it is particularly suitable for parameters in the positive domain. The gamma distribution itself is concise in form, allowing for flexible modeling of various prior information, and does not lead to overly complex inference and computational issues. Additionally, the use of a gamma prior can yield a more explicit form of the posterior distribution. Thus, we suppose that $\lambda$ and $\beta$ are independent and follow the gamma prior $\Gamma (a,b)$ and $\Gamma (c,d)$, respectively, that is,

The joint prior distribution of $\lambda$ and $\beta$ is

According to the Bayesian theorem, the joint posterior distribution of $\lambda$ and $\beta$ under the adaptive progressive type-II censored samples is given by where $B={\displaystyle {\int }_0^{+\infty }{\displaystyle {\int }_0^{+\infty }l(\lambda ,\beta |x)\pi (\lambda ,\beta )}}d\lambda d\beta$ is a constant.

In this section, we select the symmetric entropy loss function, LINEX loss function, and general entropy loss function to derive the Bayesian estimators of the LPI $\delta$.

Let $\widehat{\alpha }$ be the estimator of a parameter $\alpha$; then, the three loss functions are defined as follows:

Given the observation $x$, the Bayesian estimators of parameter $\alpha$ under the SELF, LLF, and GELF are expressed by Equation (35), Equation (36), and Equation (37), respectively. In the expressions below, $E(\cdot |x)$ denotes the posterior expectation.

Thus, based on adaptive progressive type-II censored samples, the Bayesian estimator ${\widehat{\delta }}_{SE}$ of $\delta$ under the SELF is presented by

The Bayesian estimator ${\widehat{\delta }}_{LL}$ of $\delta$ under the LLF is

The Bayesian estimator ${\widehat{\delta }}_{GE}$ of $\delta$ under the GELF is

However, the inherent complexity of $\delta$ results in these expressions involving numerous complex integrals, making analytical solutions difficult to obtain. Commonly used methods for obtaining approximate solutions include the Lindley approximation and the Tierney–Kadane (TK) approximation. The Lindley approximation is relatively convenient as it does not involve complex integrals or iterative processes; it only requires the calculation of the partial derivatives of the posterior density of the parameters. However, its drawbacks include the inability to construct credible intervals for the parameters, and the final approximation depends on the maximum likelihood estimates. In contrast, the TK approximation is less complex in terms of computation but involves solving nonlinear equations. After considering various factors, this paper employs the Markov chain Monte Carlo (MCMC) method to perform iterative computations.

Gibbs sampling and the Metropolis–Hastings (MH) algorithm are two of the most commonly used methods in the MCMC method. Gibbs sampling is designed to draw samples from a multidimensional probability distribution, while the MH algorithm is used to sample from complex probability distributions. The key to Gibbs sampling is identifying the conditional distribution for each variable. Typically, this requires us to compute the conditional probability distribution of each variable given the values of the other variables. If the conditional distribution cannot be directly computed, the MH algorithm can be utilized for sampling. The MH algorithm works by setting acceptance–rejection criteria for different candidate values, ensuring that the final set of generated samples adheres to the target distribution.

According to Equation (31), the full condition distributions of $\lambda$ and $\beta$ are given below.

From a formal perspective, both the aforementioned full conditional distributions cannot be expressed as any known distribution. Therefore, this paper combines Gibbs sampling with the MH algorithm to draw samples from Equations (41) and (42). Assuming that the proposal distributions for $\lambda$ and $\beta$ are both normal distributions, the calculation steps are as follows:

Step 1. Set the initial values $({\lambda }^{(0)},{\beta }^{(0)})$ and start with $j=1$.

Step 2. Generate the candidate values $\lambda *$ and $\beta *$ from the proposal distributions $N({\lambda }^{(j-1)},\mathrm{var}(\lambda ))$ and $N({\beta }^{(j-1)},\mathrm{var}(\beta ))$, respectively.

Step 3. Generate $u_1$ from the uniform distribution $U(0,1)$, and calculate $v_1$ from the formula below.

Step 4. Determine ${\lambda }^{(j)}$ based on the following acceptance rules.