Inference on $$P(X<Y)$$ P ( X < Y ) for Bivariate Normal Distribution based on Censored Data
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RESEARCH ARTICLE
Inference on PðX < YÞ for Bivariate Normal Distribution based on Censored Data Manoj Chacko1 • Shiny Mathew1 Accepted: 19 September 2020 Ó The Indian Society for Probability and Statistics (ISPS) 2020
Abstract In this paper, we consider the problem of estimation of R ¼ PðX\YÞ, when X and Y are dependent and measurement on one variable is difficult. The maximum likelihood estimate and bayes estimate of R are obtained based on censored data when (X, Y) follows bivariate normal distribution. The confidence intervals for R are also obtained. Monte Carlo simulations are carried out to study the accuracy of the proposed estimators. The inferential procedure developed in this paper is also illustrated using a real data. Keywords Bivariate normal distribution Order statistics Concomitants of order statistics Maximum likelihood estimation Bayesian estimation Censored data
1 Introduction Censored sample arises in a life-testing experiment whenever the experimenter does not observe the failure times of all units placed on a life-test. In medical or industrial studies, researchers have to treat the censored data because they usually do not have sufficient time to observe the lifetimes of all subjects in the study or taking observations destruct the units or expensive to measure. There are different types of censoring. The most common censoring schemes are type-I and type-II censoring schemes. Statistical inference based on Type-II censored sample from different bivariate distributions has been studied in the literature. For example, Balakrishnan and Kim (2005) studied the maximum likelihood estimation of the parameters in a bivariate normal distribution based on type-II right-censored sample by using the expectation–maximization (EM) algorithm. Harrel and Sen (1979) considered inference based on type-II censored sample & Manoj Chacko [email protected] 1
Department of Statistics, University of Kerala, Thiruvananthapuram, India
123
Journal of the Indian Society for Probability and Statistics
from a bivariate normal distribution. He and Nagaraja (2011) developed the method of moment estimators for the correlation parameter when the censored bivariate sample follows arising from a Moran–Downton bivariate exponential distribution. Gribkova and Lopez (2015) considered non-parametric copula estimation under bivariate censoring. If X and Y are jointly distributed random variables, then from the available literature we observe that making inference on R ¼ PðX\YÞ attracted wide interest in several areas of studies such as quality control, engineering statistics, reliability, medicine, psychology, biostatistics, stochastic precedence and probabilistic mechanical design (see, Kotz et al. 2003, for a review). In the context of reliability, the stress-strength model describes the life of a component which has a random strength Y and is subjected to a random stress X. The component fails at the instant that the stress applied to it exceeds the strength and the component will function satisfactorily wh
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