Predicting observables from Weibull model based on general progressive censored data with asymmetric loss

Abstract

The objective of this paper is to develop a methodology to construct and compute in a Bayesian setting, point and interval predictions based on general progressive Type-II censored data from Weibull model. Prediction bounds for the future observations (2-sample prediction) based on this type of censored will be derived. Bayesian predictions are obtained based on a continuous–discrete joint prior for the unknown two parameters. We have examined point predictions under symmetric and asymmetric loss functions. As application, the total duration time in a life test and the failure time of a $k$-out-of-$m$ system may be predicted. An illustrative example consisting of various types of real data from an accelerated test on insulating fluid reported by Nelson (1982) [17] is presented. Finally, some numerical results using simulation study concerning different sample sizes, and different progressive censoring schemes were reported. A study of 10 000 randomly generated future samples from the same distribution shows that the actual prediction levels are satisfactory.

Introduction

The Weibull distribution has been extensively used to model lifetimes and material strengths.  Johnson et al. [10] presented a detailed account of this distribution and its properties. The prediction problem comes up naturally in several real life situations, and the prediction results are of special interest to engineers who are concerned with setting limits on the performance of a small number of units of a product. For example, in studies concerning the length of life of certain types of manufactured items, the prediction of the life for a future item is often wished based on available life testing information. Specifically, manufacturers would like to have the bounds for the life of their products so that their warranty limits could be more reasonably set, and consumers purchasing manufactured products would like to know the bounds for the life of the unit to be purchased. In addition, prediction has its uses in a variety of disciplines such as medicine (medical prognosis, antibiotic assays and preoperative medical diagnosis), quality control and business. There is a considerable amount of literature on statistical prediction. Sherif and Tan [24] discussed the problem of constructed the predictive distribution based on progressive Type-II censored Weibull data. Patel [21] gave an extensive review of prediction intervals, but nothing in the Bayesian framework. Hahn and Meeker [9] gave a general survey of statistical interval and just a brief description on Bayesian prediction intervals. Nelson [18] gave a simple procedure for computing prediction limit for the number of failures that will be observed in a future inspection, based on the number of failures in a previous inspection when the units have a Weibull failure-time distribution with a given shape parameter. Based on censoring samples, Lu et al. [15], provide some pivotal quantities for constructing prediction intervals for future observations from the 2-parameter Weibull distribution. Recently, several authors considered the problem of prediction based on various types of censored data, see among others [1], [16], [22].

In many life-testing and reliability studies, it is common that the lifetimes of test units may not be able to record exactly. An experimenter may terminate the life test before all $n$ products fail in order to save time or cost. Therefore, the test is considered to be censored in which data collected are the exact failure times on those failed units and the running times on those non-failed units. The most common censoring schemes are Type-I and Type-II censoring, but the conventional Type-I and Type-II censoring schemes do not have the flexibility of allowing removal of units at points other than the terminal point of the experiment. For example, some products have to be withdrawn for more thorough inspection or saved for use as test specimens in other studies. In addition, the reduction of budget or patient dropouts in a clinical study would also result in progressive removal. Viveros and Balakrishnan [30] proposed a conditional method of inference to derive exact confidence intervals. Recently, different inferential procedures based on progressively censored samples have been discussed by several authors, including [5], [19], [20], [26]. A recent account on progressive censoring schemes can be found in the book by Balakrishnan and Aggarwala [4], or in the excellent review by Balakrishnan [3]. This paper considers a general progressive Type-II censoring scheme, this scheme is conducted as follows: We design an experiment in which $n$ units are placed on a life test, and the test can be terminated at the time of any failure. In addition, one or more surviving units may be removed from the test (censored). At the time of the first failure $X_{1,n,}$, a number $R_1$ of the surviving units are randomly removed from the remaining $n-1$ units. At the second failure $X_{2,n}$, $R_2$ units from the remaining $n-2-R_1$ units are randomly removed. The test continues until the $m^{th}$ failure. At this time, all remaining $R_m=n-m-R_1-R_2\cdots -R_{m-1}$ units are removed. The resulting $m$ ordered values, which are obtained as a consequence of this type of censoring, are appropriately referred to as progressively Type-II censored order statistics. This progressive censoring order statistics, can be generalized by allowing for initial left censoring as well. The times of failure for the first $r$ failures are not observed, and the $(r+1)\text{th},\dots ,m\text{th}$ failures are observed at which times $R_{r+1},\dots ,R_m=n-m-{\sum }_{i=1}^{m-1}{R}_i$ surviving units are censored. The resulting $(m-r)$ ordered failure times are referred to as general progressively Type-II censored scheme.

Note that: In this censoring scheme, $R_i(i=r+1,2,\dots ,m)$, $r$, and $m$ are pre-fixed, and If $r=0$, $R_1=R_2=\cdots =R_{m-1}=0$, so that $R_m=n-m$, the scheme reduces to the conventional Type-II one-stage right censoring scheme. Also, if $r\ne 0$, $R_{r+1}=R_{r+2}=\cdots =R_{m-1}=0$, so that $R_m=n-m-r$, the scheme reduces to the doubly Type-II censoring scheme. Finally, if $r=0,R_1=R_2=\cdots =R_m=0$, so that $m=n$, which corresponds to the complete sample case.

Fernandez in [8] discussed Bayesian and non-Bayesian estimators for the exponential parameters using general progressive Type-II censoring data. Kim [13] and Kim and Han [14] using a general progressive censored schemes and discussed the estimation problem for the Burr-XII and Rayleigh distributions respectively. Abdel-Aty et al. [1] are concerned with generalized order statistics in a single sample from exponential distribution with multiply Type-II censoring. They establish prediction intervals of future generalized order statistics. Recently, Schenk et al. [23] addressed the problem of Bayesian estimation and prediction based on multiply Type-II censored samples of sequential order statistics from exponential distributions. This article focuses, via Bayesian approach, on two-sample predictive inferences for the Weibull model based on a general progressively Type-II censored data.

The rest of the article is organized as follows. The model, prior and posterior distributions and the loss functions are described in Section 2. Section 3 presents the details of our main results along with the derivation of all Bayes predictive functions based on general progressive censored data. In the same section, the predictive functions were used to derive both point prediction and prediction intervals for the future observations from the same distribution. Several types of real data are used to illustrate the proposed methodologies in Section 4. The results of a Monte Carlo simulation study are presented in the same section, in which a study of 10 000 randomly generated future samples from the same distribution shows that the actual prediction levels are satisfactory. Conclusions are given in Section 5.