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POINT AND INTERVAL ESTIMATION OF P(X < Y):

THE NORMAL CASE WITH COMMON COEFFICIENT

OF VARIATION

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RAMESH C. GUPTA, S. RAMAKRISHNAN AND XINGWANG ZHOU

*Department of Mathematics and Statistics, University of Maine,*

Orono, ME 04469-5752, U.S.A.
(Received September 25, 1996; revised February 9, 1998)

**Abstract.** The problem of estimating *R* = *P*(*X* < *Y*)
originated in the context of reliability where *Y* represents
the strength subjected to a stress *X*. In this paper we
consider the problem of estimating *R* when *X* and *Y* have
independent normal distributions with equal coefficient of
variation. The maximum likelihood estimation of *R* when the
coefficient of variation is known and when it is unknown is
studied. The asymptotic variance of the estimators are
obtained and asymptotic confidence intervals are provided. An
example is presented to illustrate the procedure. Finally
some simulation studies are carried out to study the coverage
probability and the lengths of the confidence interval. In
particular, lengths of the confidence intervals are compared
with and without the assumption of common coefficient of
variation. It is observed that the assumption of common
coefficient of variation results in considerably tighter
intervals.

*Key words and phrases*: Normal distributions,
coefficients of variations, delta method, confidence
intervals, coverage probability.

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