Journal of Probability and Statistics

Volume 2017 (2017), Article ID 4860167, 12 pages

https://doi.org/10.1155/2017/4860167

## Maximum Likelihood and Bayes Estimation in Randomly Censored Geometric Distribution

Department of Statistics, Ch. Charan Singh University, Meerut, India

Correspondence should be addressed to Hare Krishna

Received 25 July 2016; Revised 26 December 2016; Accepted 22 January 2017; Published 21 February 2017

Academic Editor: Hyungjun Cho

Copyright © 2017 Hare Krishna and Neha Goel. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

In this article, we study the geometric distribution under randomly censored data. Maximum likelihood estimators and confidence intervals based on Fisher information matrix are derived for the unknown parameters with randomly censored data. Bayes estimators are also developed using beta priors under generalized entropy and LINEX loss functions. Also, Bayesian credible and highest posterior density (HPD) credible intervals are obtained for the parameters. Expected time on test and reliability characteristics are also analyzed in this article. To compare various estimates developed in the article, a Monte Carlo simulation study is carried out. Finally, for illustration purpose, a randomly censored real data set is discussed.

#### 1. Introduction

Lifetime experiments are conducted to collect data on items under study. The data are used for fitting a suitable lifetime model and then inferring about the statistical properties and survival/reliability characteristics of the items. These experiments may be expensive in terms of both cost and time. To save the cost and time, lifetime experiments may be censored intentionally or censoring may occur in an experiment naturally. Many types of censoring schemes have been studied in literature such as type-I, type-II, progressive, hybrid, and random censoring schemes.

Random censoring is a situation when an item under study is lost or removed randomly from the experiment before its failure. In other words, some subjects in the study have not experienced the event of interest at the end of the study. For example, in a clinical trial or a medical study, some patients may still be untreated and leave the course of treatment before its completion. In a social study, some subjects are lost for the follow-up in the middle of the survey. In reliability engineering, an electrical or electronic device such as bulb on test may break before its failure. In such cases, the exact survival time (or time to event of interest) of the subjects is unknown; therefore they are called randomly censored observations.

The random censoring was introduced in literature by Gilbert [1]. Thereafter, Breslow and Crowley [2], Koziol and Green [3], and Csörgó and Horváth [4] also discussed randomly censored data in their work. Kim [5] did chi-square goodness of fit tests for randomly censored data. In the last decade, the recent studies on randomly censored data from exponential distribution include Friesl and Hurt [6] and Saleem and Raza [7]. Rayleigh model with randomly censored data was analyzed by Ghitany [8] and Saleem and Aslam [9]; Burr Type XII was analyzed by Ghitany and Al-Awadhi [10]; generalized exponential and Weibull models were analyzed, respectively, by Danish and Aslam [11, 12]. Krishna et al. [13] studied Maxwell distribution with randomly censored samples.

In all the above cases, survival time or failure time has been assumed to be a continuous variable. However, sometimes it is impossible or inconvenient to measure the life length of a device on a continuous scale.

In real life experiments we come across situations, where failure time data is discrete either through the grouping of continuous data due to imprecise measurement or because time itself is discrete, for example, days, weeks, or months. In such circumstances, one measures the life of a device on a discrete scale. A discrete lifetime model may also consider the number of successful cycles, trials, or operations before failure of a device. In discrete lifetime models, the one parameter geometric distribution has an important position. The geometric distribution can be used as a discrete failure to investigate the ability of electronic tubes to withstand successive voltage overloads and performance of electric switches, which are repeatedly turned on and off.

In reliability theory, geometric distribution has been considered as a lifetime model by Yaqub and Khan [14], Bhattacharya and Kumar [15], Maiti [16], Krishna and Jain [17], Sarhan and Kundu [18], and so forth. In most of above studies, a complete sample or right censoring is considered. No literature is available on random censoring in any discrete distribution.

In view of the above, this paper considers classical and Bayes estimation of the unknown parameters with some reliability characteristics for geometric distribution under randomly censored data. In Section 2, mathematical formulation for randomly censored data with failure and censoring times following geometric distributions is given. Section 3 deals with the maximum likelihood estimation (MLE) for the unknown parameters along with their variances and confidence intervals. Section 4 describes the expected time on test and observed time on test. In Section 5 we obtain the Bayes estimators for the unknown parameters under generalized entropy and LINEX loss functions using beta priors. In Section 6, we consider a Monte Carlo simulation study to explore the properties of various estimates developed in the above sections. Finally, Section 7 deals with a real data example to study the applications of random censoring in geometric distribution. It is essential to mention here that we have used statistical software R [19] for computation purposes throughout the paper.

#### 2. The Model and Its Assumptions

In a life testing experiment or a clinical trial, items or patients are subjected to test. Let be their discrete failure or survival times. Assume that ’s are independently and identically distributed (i.i.d.) random variables (r.v.) with probability mass function (p.m.f.) and cumulative distribution function (c.d.f.) . Further, suppose that these items may be censored before their failure at times . Again, assume ’s to be i.i.d. discrete random variables with p.m.f. and c.d.f. . It is also assumed that ’s and ’s are independent. In a randomly censored experiment, minimum of ’s and ’s, that is, , will actually be observed for . Let be an indicator variable defined by (0) if (). Here, is a Bernoulli random variable with probability [].

In the present article, we consider survival and censoring time variables to follow geometric distributions Geo(*θ*) & Geo(*λ*), respectively, with p.m.f.,The probability of failure of an item on test before its censoring is given by

The above probability of failure is tabulated in Table 1 for various values of *θ* and *λ*. From Table 1 we observe that on increasing the value of lifetime parameter *θ*, the probability of failure decreases. However, as we increase the value of censoring parameter *λ*, the probability of failure increases.