Page 74 - Contributed Paper Session (CPS) - Volume 1

CPS1085 Manoj C.

Inference on P(X < Y) for bivariate normal
distribution based on censored data
Manoj Chacko
Department of Statistics, University of Kerala, Trivandrum, India

Abstract
In this paper, we consider the problem of estimation of = ( < ), when
(, ) follows bivariate normal distribution and measurement on one variable
is dificult. The maximum likelihood estimates (MLEs) and Bayes estimates (BEs)
of are obtained based on censored data, in which censoring is done based
on the easily measurable variate. BEs are obtained based on both symmetric
and asymmetric loss functions. The percentile bootstrap and HPD confidence
intervals for 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 water quality data.

Keywords
Maximum likelihood estimation; Bayesian estimation; importance sampling
method; order statistics.

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 lifetime
of all subjects in the study. There are different types of censoring. The most
common censoring schemes are type-I and type-II censoring schemes. In this
paper, we consider a type-II censoring scheme in which the experiment
continues until a pre-specified number of failures, (≤ ) occur. The
remaining (n-r) items are regarded as censored data. Let (, ) be an
absolutely continuous random vector with cummulative distribution function
(cdf) (, ) and joint probability density function (pdf) (, ). Let
( , ), = 1, 2, . . . , be a random sample of size n drawn from the

distribution of (, ). If the sample is ordered by the i s then the -variate
′

associated with the rth order statistic : is called the concomitant of the rth
order statistic and is denoted by [:] (see, David, 1973). Suppose that we only
observe (≤ ) smallest X-sample and their associated values of the Y -
sample, then ( (:) , [:] ), = 1, 2,· · · , is called a type-II right-censored
sample. The joint pdf of
(X (r) , Y ) = ((X (1:n) , Y [1:n] ), (X (2:n) , Y [2:n] )), . . . , (X (r:n) , Y [r:n] )) is given by
[r]
63 | I S I W S C 2 0 1 9