Inference for Inverse Power Lomax Distribution with Progressive First-Failure Censoring

In the life test of a product, due to the restrictions of test time, cost and other conditions, the complete life test is not generally performed. In these cases, experimenters often use censoring schemes to obtain censored lifetime data. There are many types of censoring schemes, and the most popular censoring schemes are Type-I and Type-II censoring. In Type-I censoring, the test ends at a pre-fixed time, while in Type-II censoring, the test ends when the m-th failure occurs (m is fixed in advance). For the above two censoring schemes, the common disadvantage is that no unit in the test can be removed before the test is terminated. Thus, progressive censoring (PC) was proposed, which has better efficiency in lifetime experiments. Under this censoring scheme, one can remove the test units at various stages of the experiment. For more details, refer to Balakrishnan and Aggarwala [1]. An excellent review of progressive censoring schemes can be found in Ref. [2]. Besides the PC, there is another censoring scheme, namely the first failure censoring scheme. Under this censoring scheme, experimenters group the test units into several sets and then perform all the test units simultaneously until the first failure in each set. The first-failure censoring scheme was studied by Johnson [3], Balasooriya et al. [4], Wu et al. [5] and Wu and Yu [6]. However, this censoring scheme does not allow the removal of units from the test at points other than the final termination point. Wu and Kus [7] combined the advantages of the first failure censoring and progressive censoring to propose mixed censoring, that is, a progressive first-failure censoring (PFFC) scheme. They obtained maximum likelihood estimates (MLEs), interval estimation and expected time on test for the parameters of the Weibull distribution based on the PFFC sample.

The PFFC scheme can be described as follows: suppose that $n$ independent groups with k items within each group are put on a life test at time zero, and the progressive censoring Scheme $\underset{˜}{R}=(R_1,R_2,\dots ,R_m)$ is fixed in advance. At the first failure time $X_{1:m:n:k}$, $R_1$ groups and the group in which the first failure is observed are randomly removed from the test. Similarly, at the second failure time $X_{2:m:n:k}$, $R_2$ groups and the group in which the second failure is observed are randomly removed from the remaining $(n-R_1-1)$ groups. This procedure continues until the mth failure time $X_{m:m:n:k}$ is observed in the remaining groups, and then all the remaining $R_m$ groups are removed. It is clear that $n=m+R_1+R_2+\dots +R_m$. The observed failure times, $X_{1:m:n:k}<X_{2:m:n:k}$ $<\dots <X_{m:m:n:k}$, are called the PFFC sample with the progressive censoring scheme $\underset{˜}{R}=(R_1,R_2,\dots ,R_m)$. Here, $(m,n,k)$ must be pre-specified.

The main advantage of the PFFC scheme is that it reduces time where more items are used, but only m out of $n\times k$ items are observed. It is observed that if $R_1=R_2=\dots =R_m=0$, the PFFC reduces to first failure censoring; If $k=1$, the scheme becomes progressively Type II censoring; when k = 1, $R_1=R_2=\dots =R_{m-1}=0$ and $R_m=n-m$, this scheme reduces to Type II censoring scheme. Furthermore, the progressively first-failure censored sample $X_{1:m:n:k}<X_{2:m:n:k}<\dots <X_{m:m:n:k}$ can be viewed as a progressively Type-II censored sample from a population with the distribution function $1-{(1-F(x))}^k$, which enables us to extend all the results on progressive type-II censored order statistics to progressive first-failure (PFF) censored order statistics.

Because of the flexibility of the PFFC scheme, many scholars have discussed and applied it in reliability studies. Ref. [8] studied statistical inferences of the unknown parameters, the reliability and failure functions of the inverted exponentiated Half-Logistic distribution using PFFC samples. Ref. [9] investigated a competing risks data model under PFFC from a Gompertz distribution using Bayesian and non-Bayesian methods. Ref. [10] considered the estimates of the unknown parameters and reliability characteristics of generalized inverted exponential distribution using PFFC samples. Ref. [11] established different reliability sampling plans using two criteria from a Lognormal distribution based on the PFFC. Some recent studies on the PFFC scheme can be found in Refs. [12,13,14,15,16].

The inverse distributions have a wide range of applications in issues related to econometrics, biological sciences, survey sampling, engineering sciences, medical research and life testing problems. In recent years, some scholars have studied the statistical inference of inverse distribution. For example, Dube et al. [10] studied the MLEs and Bayesian estimators of the unknown parameters and reliability characteristics of generalized inverted exponential distribution using progressively first-failure censored samples. Panahi and Moradi [17] discussed the estimation of the inverted exponentiated Rayleigh distribution based on an adaptive Type II progressive hybrid censored sample. Bantan et al. [18] studied the estimation of the Rényi and q-entropies for inverse Lomax distribution under multiple censored data. An efficient estimation strategy was proposed by using the maximum likelihood and plugging methods. But they did not investigate the statistical inference of the three-parameter inverse power Lomax distribution under the progressive first failure sample. Some other related studies on inverse distribution can be found in Nassar and Abo-Kasem [19], Lee and Cho [20], Xu and Cui [21] and Rashad et al. [22].

In 2019, a new three-parameter lifetime distribution named the inverse power Lomax (IPL) distribution was introduced by Hassan and Abd-Allah [23]. The probability density function (PDF) $f(\cdot )$, cumulative distribution function (CDF) $F(\cdot )$ of the IPL distribution are given, respectively, by where $\alpha >0,\beta >0$ are shape parameters, and $\lambda >0$ is scale parameter. The IPL is very flexible in analyzing situations with a realized non-monotonic failure rate. Therefore, the IPL model can be used for several practical data modeling and analysis, see Ref. [23]. In order to facilitate engineering applications, Ref. [23] studied some statistical properties for the IPL distribution. The MLEs of the model parameters are obtained based on conventional Type I and Type II censored samples. However, they did not discuss the PFFC scheme. The PFFC scheme is more widely used in survival analysis and the life test.

Since the IPL distribution contains three unknown parameters, it is more complicated to estimate the unknown parameters under progressive censoring. So, to date, there has been no published work on statistical inference for IPL distribution under the PFFC scheme. The main aim of this paper is to focus on the classical and Bayesian inference for IPL distribution under the PFFC scheme.

The rest of this paper is organized as follows: In Section 2, the MLEs and asymptotic confidence intervals of the unknown parameters are derived. Based on Tierney–Kadane’s approximation method, Bayesian estimates of the parameters under squared error loss and generalized entropy loss function are obtained in Section 3. In addition, the highest posterior density (HPD) credible intervals of the parameters are constructed by using the importance sampling method. In Section 4, Monte Carlo simulations are carried out to investigate the performances of different point estimates and interval estimates. In Section 5, a real data set has been analyzed for illustrative purposes. The conclusions are given in Section 6.

In this section, the MLEs of the parameters for the IPL distribution will be discussed under the PFFC. Let $X_i=X_{i:m;n},i=1,2,\cdots ,m$ be the progressive first-failure censored order statistics from the IPL distribution with the censored scheme $\underset{˜}{R}=(R_1,R_2,\dots ,R_m)$. Then, using Equations (1) and (2), the likelihood function is given by where $C=n(n-R_1-1)\cdots (n-m+1-R_1-\cdots R_{m-1})$, $\overrightarrow{x}=(x_1,x_2,\cdots ,x_m)$ and ${\xi }_i^{}=(1+{\lambda }^{-1}{x}_i^{-\beta })$. The log-likelihood function is given by

Let $l(\alpha ,\beta ,\lambda |\overrightarrow{x})=\ln L(\overrightarrow{x};\alpha ,\beta ,\lambda )$. By taking the first partial derivative of log-likelihood function with regard to $\alpha ,\beta$ and $\lambda$ and equating them to zero, the following results can be obtained.

The MLEs of $\alpha ,\beta$ and $\lambda$ can be obtained by solving the Equations (5)–(7), but these equations do not yield an analytical solution. Therefore, we use the Newton–Raphson iteration method to obtain the MLEs of the parameters. For this purpose, we first compute the quantity of interest $I_{ij},\hspace{0.17em}i,j=1,2,3$. Here,

The Newton–Raphson iteration method can be implemented according to the following steps:

Step 1: Give the initial values of $\theta =(\alpha ,\beta ,\lambda )$, say ${\theta }^{(0)}=({\alpha }^{(0)},{\beta }^{(0)},{\lambda }^{(0)})$.

Step 2: In the M-th iteration, calculate $\left(\frac{\partial l}{\partial \alpha },\frac{\partial l}{\partial \beta },\frac{\partial l}{\partial \lambda }\right)|{}_{\theta ={\theta }^{(M)}}$ and matrix $I({\theta }^{(M)})$. Here, $I({\theta }^{(M)})=\left[\begin{array}{ccc}-I_{11}& -I_{12}& -I_{13}\\ -I_{21}& -I_{22}& -I_{23}\\ -I_{31}& -I_{32}& -I_{33}\end{array}\right]|{}_{\theta ={\theta }^{(M)}}$, ${\theta }^{(M)}=({\alpha }^{(M)},{\beta }^{(M)},{\lambda }^{(M)})$, $I_{ij},\hspace{0.17em}i,j=1,2,3$ are given by Equations (9)–(14).

Step 3: Update ${(\alpha ,\beta ,\lambda )}^T$ by here, ${(\alpha ,\beta ,\lambda )}^T$ is the transpose of vector $(\alpha ,\beta ,\lambda )$, and $I^{-1}({\theta }^{\left(M\right)})$ represents the inverse of matrix $I({\theta }^{\left(M\right)})$.

Step 4: Setting $M=M+1$, the MLEs of $\theta =(\alpha ,\beta ,\lambda )$, say $\widehat{\theta }=(\widehat{\alpha },\widehat{\beta },\widehat{\lambda })$ can be obtained by repeating steps 2–3 until $|{({\alpha }^{(M+1)},{\beta }^{(M+1)},{\lambda }^{(M+1)})}^T-{({\alpha }^{(M)},{\beta }^{(M)},{\lambda }^{(M)})}^T|<\epsilon$, where $\epsilon$ is a threshold value and fixed in advance.

In this section, we discuss the Bayesian estimates and corresponding credible intervals of the unknown parameters for the IPL distribution. In order to select the best decision in the decision theory, an appropriate loss function must be specified. Here, we consider both the symmetric and asymmetric loss functions. A very well-known symmetric loss function is the square error loss (SEL). The most commonly used asymmetric loss function is the generalized entropy loss (GEL) function. The SEL and GEL function are, respectively, defined by

Here, $\widehat{\theta }$ is an estimation of $\theta$, and the constant $q$ denotes how much influence that an error will have. When $q<0$, negative errors affect the consequences more seriously. When $q>0$, positive errors cause more serious consequences than negative ones.

Under the SEL and GEL function, the Bayesian estimator of $\theta$ are, respectively, given by

The Bayesian analysis requires the choice of appropriate priors for the unknown parameters in addition to the experimental data. Arnold and Press [24] correctly pointed out that there is no clear cut way how to choose prior. Now, we assume the following independent gamma priors for the parameters $\alpha ,\beta$ and $\lambda$ as therefore, the joint prior distribution of $\alpha ,\beta$ and $\lambda$ is given by

The assumption of independent gamma priors is reasonable [10]. The class of the gamma prior distributions is quite flexible as it can model a variety of prior information. It should be noted that the non-informative priors on the parameters are the special cases of independent gamma priors and can be achieved by approaching hyper-parameters to zero [10].

Based on Equations (3) and (18), the joint posterior distribution of the parameters $\alpha ,\beta$ and $\lambda$ can be written as:

Let $g=g(\alpha ,\beta ,\lambda )$ be a function of $\alpha ,\beta$ and $\lambda$, then the posterior mean of $g$ is given by

From Equation (20), we observe that the posterior mean of $g(\alpha ,\beta ,\lambda )$ is in the form of ratio of two integrals for which a closed-form solution is not available [10]. Therefore, we use Tierney–Kadane’s approximation method to obtain the approximate solution of Equation (20).