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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