Robust Inference of One-Shot Device testing data with Competing Risk under Lindley Lifetime Distribution with an application to SEER Pancreas Cancer Data

The prevalence of one-shot devices can be seen in various fields of human inventions. These devices remain torpid until activated and are immediately destroyed after use. The exact failure time of such devices is not observable in most instances. Therefore, the observation is limited to recording whether the device failed before or after a specified time, leading to dichotomous data. Several studies are devoted to the lifetime study of highly reliable one-shot devices under the accelerated life testing (ALT) setup.

Fan et al. [1] applied one-shot device testing data for the analysis of highly reliable electro-explosive devices using the Bayesian approach with an exponential, normal and beta prior distribution. Balakrishnan and Ling [2] were engaged in the classical inference of the lifetime distribution of one-shot devices with constant stress ALT under exponential lifetime distribution and compared it with the Bayesian framework developed by Fan et al. [1]. Balakrishnan and Ling [3] extended this work [2] for multiple stress models. Under constant stress ALT, Balakrishnan and Ling [4] assumed gamma lifetime distribution and drew classical inference in context one-shot devices data analysis.

There are many possible causes that can result in the failure of any one-shot device. For example, a fuse can be failed due to high temperature, humidity, voltage or overloading. The competing risk analysis confides in identifying the particular cause that caused the ultimate failure of a device. In survival analysis where one-shot device data corresponds to current-status data, there may be multiple causes of death in which there is one cause which occurs earliest and ultimately results in the demise of an individual. Crowder [5] thoroughly studied the statistical inference of classical competing risks. Several studies under competing risk setup were conducted thereafter [6, 7, 8, 9]. In the context of one-shot devices, Balakrishnan et al. [10, 11] drew classical inference with constant stress competing risk set up under exponential and Weibull lifetime distributions, respectively. Balakrishnan et al. [12] also went through the Bayesian approach for competing risk analysis of one-shot devices under exponential lifetime distribution.

Most of the work is focused on exponential, gamma and Weibull as lifetime distributions for one-shot device data in the literature. In this work, we come up with the study of statistical inference of the lifetime distribution of one-shot devices under independent competing risk setup where the lifetime under each risk follows two-parameter Lindley distribution. Lindley distribution was introduced by Lindley [13], [14] to analyse lifetime data, especially to study stress-strength reliability. Ghitany et al. [15] showed with evidence that the Lindley distribution offers better modelling than exponential distribution in terms of mathematical properties like mode, coefficient of variation, skewness, kurtosis and hazard rate shape. Lindley distribution has broad applicability in analysing failure time data. Mazucheli and Achcar [16] used it as an alternative to exponential or Weibull lifetime data under the competing risk setup.

In reality, it is quite likely to get data with outliers. The conventional inferential methods designed for ideal situations fail here, and robust inference methods are needed. In the literature, Basu et al. [17] introduced the density power divergence-based robust method of estimation. In one-shot devices data analysis, Balakrishnan et al. [18, 19, 20] carried the density power divergence based robust inference under gamma, Weibull and exponential lifetime models. It has been observed that very few works on competing risk with robust estimation are available in the literature. Only recently, Balakrishnan et al. [21], Balakrishnan and Castilla [22] developed robust inference under competing risk setup for one-shot devices. Further, it has been observed through extensive literature surveys that robust Bayes estimation has yet to be studied in the context of one-shot devices. The methodology of drawing a robust inference in the Bayesian model was first introduced by Hooker and Vidyashankar [23]. Further, Ghosh and Basu [24] developed robust Bayes estimation where the likelihood function has been substituted by the density power divergence measure in the posterior density function.

The present article focuses on developing a robust estimation method in classical and Bayesian frameworks where the failure of one-shot devices is subject to independent competing risks with an interval monitoring setup under Lindley lifetime distribution. The estimation procedure is based on the density power divergence measure between the true and assumed lifetime distributions. Certain weights are associated with the density power divergence measure and thereafter, a weighted minimum density power divergence estimator (WMDPDE) is obtained. Further, we study the asymptotic behaviour of the robust WMDPD estimator under this setup. With the availability of prior information, Bayesian inference is quite essential. In this work, following the idea by Ghosh and Basu [24], a weighted robust Bayes estimation (WRBE) is proposed where the robustified posterior density was developed on the exponential form of the maximiser equation based on density power divergence measure. The Normal distribution and Dirichlet distribution are taken as priors which are based on the data. Further, the robustness of the derived estimators is studied through the influence function. The numerical study is conducted through simulation experiments and real data analysis to assess the performances of the derived methods.

It is to be noted that in the estimation procedure based on the density power divergence method, the tuning parameter plays a pivotal role in bringing a balance between robustness and efficiency. Warwick and Jones [25] introduced a data-driven algorithm for choosing the optimal tuning parameter, which was further studied and modified by several authors ([26, 27, 28]). In this work, we propose an approach aiming to increase the robustness as well as precision of the estimation. In that direction, a numerical study is also presented.

The data analysis is conducted using information extracted from the database named “Incidence $\cdot$SEER Research Data, 8 Registries, Nov 2021 Sub (1975-2019)” [29] which is maintained by National Cancer Institute (Surveillance, Epidemiology, and End Results Program (SEER)). The data is focused only on individuals diagnosed with pancreatic cancer from 2008-2010. The layout of data is given in Table (7). It was also observed that for some of the patients, the demise of the patient was recorded, but the cause of death was missing. This is frequently followed in real life, where the cause of the failure remains unspecified. Lu and Liang [30] studied that analysis using only known causes of failures would lead to biased results. To avoid this, the missing cause of failure analysis has been conducted separately in this work and the layout of the missing cause of failure data is given in Table (10). This real data set is analysed by exploiting the developed methods and the results are discussed thoroughly.

The rest of the article comprises sections divided as follows. Section 2 is concerned with model building and computation of maximum likelihood estimation. Section 3 focuses on deriving a robust WMDPD estimator and studying its asymptotic property. The robust Bayes method of estimation is introduced in section 4. In section 5, we study the robustness properties of the estimators through the influence function. Section 6 is devoted to the missing cause of failure analysis. Section 7 deals with the numerical study of the theoretical results developed in previous sections through a simulation study followed by the finding of optimal tuning parameter and data analysis. Concluding remarks are given in Section 8.

This section is concerned with the model formulation for the lifetime data analysis of one-shot devices and the development of Maximum Likelihood Estimates (MLE).

One-shot devices are studied under accelerated life testing (ALT), where $G$ devices are distributed into $I$ independent groups. For $i=1,2,\dots,I,$ let the number of devices in $ith$ group be $g_{i}$ where $G=g_{1}+g_{2}+\dots+g_{I}$. These devices are subject to $J$ type of stress factors, quantified by $s_{ij}$ ; $j=1,\dots,J$, where $s_{i0}=1$. The failure mechanism is subject to R independent competing causes. Let, $n_{ilr}$ denote the number of failures in the $ith$ group due to cause $r$ for $r=1,2,\dots,R,$ between the inspection times $(\tau_{i(l-1)},\tau_{il}]$ where $l=1,2,\dots,L$, and $\tau_{i0}=0.$ Therefore, the number of devices that survive after inspection time $\tau_{iL}$ for the $ith$ group is $k_{i}=g_{i}-\sum_{l=1}^{L}\sum_{r=1}^{R}n_{ilr}$. Table (1) shows the layout in tabular form.

In $i$th group for $i=1,2,\dots,I,$ the failure time due to $r$th competing cause for $r=1,\ldots,R,$ is denoted by the random variable $T_{ir},$ which is assumed to follow two-parameter Lindley distribution with shape parameter $\alpha_{ir}$ and scale parameter $\theta_{ir}$. The cumulative distribution function and probability density function of $T_{ir}$ are as follows.

Both shape and scale parameters are related to stress factors in log-linear form as

For the computational conveniences, two competing causes of failure and one stress factor are considered without the additive constant and denote $s_{i1}=s_{i}.$ Hence, $\bm{\Lambda}=\{(a_{1},b_{1},a_{2},b_{2})\}^{{}^{\prime}}$ are the model parameters to be estimated.

The failure probabilities $p_{il1}$, $p_{il2}$ due to cause 1, cause 2, respectively, in the interval $(\tau_{i(l-1)},\tau_{il})$ for $l=1,\ldots,L$ and the survival probability are obtained as follows.

Based on the observed failure count data, the Likelihood function can be obtained as,

Therefore, the log-likelihood function is given as,

Hence MLE of $\bm{\bm{\Lambda}},$ denoted by $\;\hat{\bm{\bm{\Lambda}}}=\{\hat{a}_{1},\hat{b}_{1},\hat{a}_{2},\hat{b}_{2}\}$ would be derived as

provided $\sum_{i=1}^{I}\sum_{l=1}^{L}\sum_{r=1}^{2}n_{ilr}>0.$

In the presence of outliers in the data set, MLE cannot provide a valid estimated value. Therefore, some robust estimation method needs to be developed.

In this section, a robust estimation method based on the density power divergence (DPD) measure will be discussed. The DPD-based estimation was proposed by Basu et al. [17]. Under the assumption that two probability distributions $F$ and $G$ having probability densities $f$ and $g$ respectively, the DPD measure between $g$ and $f$ is given as,

where $\gamma$ is termed as the tuning parameter.

For one-shot device data, the DPD measure is computed between empirical and theoretical probability distributions. The empirical failure probability due to two different causes and empirical survival probability is defined as,

where the theoretical failure probabilities and survival probabilities are given by equation (1). Applying weights proportional to the size of groups, the weighted DPD (WDPD) measure with the weights, $w_{i}=\frac{g_{i}}{G}$ combining all the I groups is obtained as,

When $\gamma$ tends to $0,$ $D^{w}_{\gamma}(\bm{\bm{\Lambda}})$ will converge to weighted Kullback-Leibler (KL) divergence measure $D_{KL}(\bm{\bm{\Lambda}})$ where

DPD measure can also be written in the form of likelihood function as follows:

where $c$ contains the terms independent of parameters.
The weighted minimum density power divergence estimators (WMDPDE) for estimating $\bm{\bm{\Lambda}}$ can be obtained by minimizing the WDPD measure as follows:

Note that, the tuning parameter $\gamma$ plays a role in bringing a balance between robustness and efficiency in this estimation.

As it is seen in the Results (2) and (3), the explicit form of the MLEs and WMDPDEs could not be found. Hence, the Co-ordinate Descent algorithm is used to obtain the estimates. The steps of the algorithm are given in Algorithm (1)

To study the asymptotic behaviour of WMDPDE, the following theorem is presented, which is based on the idea of Calvino et al.[31].

Bayesian inference is of paramount interest when some prior information is available about the model parameters. Conventional Bayes estimation based on likelihood-based posterior is quite popular because of several optimal properties. But the major drawback is that it can not produce a good estimated value when data come with contamination. In the literature, the non-robustness problem is tried to be solved by replacing the likelihood function in the posterior by some robust loss function and the derived posterior is called a pseudo posterior. Readers may see Greco et al. [32], Chérief et al. [33], Jewson et al. [34], Nakagawa and Hashimoto [35] in this reference. Inspired by Ghosh and Basu [24], a robust Bayesian estimation based on the power divergence measure is proposed here for the reliability analysis of a one-shot device with a competing risk interval monitoring set-up. The following developments are done in that direction.
Define,

where WMDPDE with $\gamma>0$ is the maximizer of $B^{w}_{\gamma}(\bm{\Lambda})$. Therefore, the weighted robust posterior density, a pseudo posterior can be defined as follows,

Here, $\pi(\bm{\Lambda})$ is the joint prior density, and $\pi^{w}_{\gamma}(\bm{\Lambda}|data)$ is the proper density for $\gamma\geq 0$. For $\gamma\rightarrow 0$, the robust pseudo posterior will converge to the conventional likelihood-based posterior density. For any loss function $L(.,.),$ the Bayes estimator can be obtained as

For the squared error loss function, the Weighted Robust Bayes Estimator (WRBE) can be obtained as

In previous sections we have derived the estimators of the model parameters using the robust estimation method. In this section robustness of those estimators will be studied through the Influence Function (IF) [38]. If $S(M)$ denotes the functional of any estimator from the true distribution M, then the IF is defined as follows.

There are plenty of incidents where a device fails, but the reason for that failure cannot be identified. In such a situation, the unknown cause of failure is said to be masked or missing. In this section, we develop the previous estimation methods when cause of some of the failures are missing or unidentified.

Let us denote the number of failures of the $i^{th}$ group due to missing cause by $n_{im}$. Then, total number of devices under observation in the $ith$ group is $g_{i}=k_{i}+n_{il1}+n_{il2}+n_{im}$ and $G=g_{1}+g_{2}+\dots+g_{I}$. Therefore, the updated probabilities are described as follows,