Open Journal of Statistics, 2012, 2, 443-446
http://dx.doi.org/10.4236/ojs.2012.24055 Published Online October 2012 (http://www.SciRP.org/journal/ojs)
A Note on Detecting “More IFR-ness” Property
of Life Distributions
Parameshwar V. Pandit1, Sujatha Inginashetty2
1Department of Statistics, Bangalore University, Bangalore, India
2Department of Statistics, Gulbarga University, Gulbarga, India
Email: panditpv12@gmail.com, sujathasi_gug@rediffmail.com
Received August 25, 2012; revised September 28, 2012; accepted October 11, 2012
ABSTRACT
In this paper, a problem of testing whether one life distribution possesses “more IFR” property than the other is consid-
ered. A new test procedure is proposed and the distribution of the test statistic is studied. The performance of the pro-
cedure is evaluated in terms of Pitman asymptotic relative efficiency. The consistency property of the test procedure is
established. It is observed that the new procedure is better than the existing proced ure in the literature.
Keywords: “More IFR” Property; U-Statistic; Pitman ARE
1. Introduction
A life is represented by a non-negative random variable
X with distribution function F and survival function
1F . Classes of life distributions based on notion
of ageing have been introduced in the literature. One of
the earliest and most important classes is the class of
“Increasing Failure Rate” (IFR). We define IFR class
below.
Definiti on 1.1. A distribution F is said to be increasing
failure rate (IFR), if
xt
x
,,,
is decreasing in x, for t
0.
Proschan and Pyke [1] proposed a test for testing ex-
ponentiality against IFR alternatives followed by Barlow
and Proschan [2], Bickel and Doksum [3], Bickel [4] and
many among others.
In practice, one might be interested in comparing two
life distributions with respect to their possessing positive
ageing property, particularly, IFR. Hollandar, Park and
Proschan [5] developed a test procedure for testing the
null hypothesis that two life distributions F and G are
equal versus the alternative hypothesis that F is more
NBU than G. Tiwari and Za lkikar [6] propo sed a test for
testing the null hypothesis that two life distributions F
and G are identical versus the alternative hypothesis that
F is “More increasing failure rate average” than G. Re-
cently, Lim, Kim and Park [7] developed a class of test
procedures for testing the null hypothesis that two life
distributions F and G are equal against the alternative
that F is “more NBU at specified age” than G. However,
the only test available for testing the null hypothesis that
two life distributions F and G are identical against the
alternative that F is more IFR than G is due to Pandit and
Gudaganavar [8].
In this paper, we develop a simple test procedure for
testing the null hypothesis that two life distributions F
and G are equal against the alternative that F is more IFR
than G. The paper is organized as below: a test statistic is
proposed for the problem of testing whether F is more
IFR than G and its asymptotic distribution is established
in Section 2. Section 3 contains the asymptotic relative
efficiencies of the test proposed with the test due to Pan-
dit and Gudaganavar [8] and some remarks and conclu-
sions are presented in Section 4.
2. The Proposed Two Sample “More IFR”
Test
Let 12 m
XX,,,YY Y and 12 n denote two ran-
dom samples from continuous life distributions F and G
respectively. We want to develop test statistic for testing
the null hypothesis.
H0:F = G (the common distribution is not specified);
Versus H1:F is “more IFR” than G based on the two
independent random samples.
Consider the parameter
,
GFG
,
where
2dd
2
xt
FFxFt
C
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