A Note on Detecting “More IFR-ness” Property of Life Distributions

doi:10.4236/ojs.2012.24055

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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: [email protected], [email protected]

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





,,,

is decreasing in x, for t 

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

FFxFt





 









P. V. PANDIT, S. INGINASHETTY

444

and

  

ddGxGt







 









Here





and can be considered as the

measure of degree of the IFR-ness. Mugadi and Ahmed

[9] test used this measure as basis in the construction of

test statistic for testing exponentiality against the IFR

alternatives in one sample setting. If F and G belong to

IFR class then













can be taken as a measure for

deciding whether F is “more IFR” than G or not. Under

H0, and it is strictly greater than zero un-

der H1.



,FG 



An unbiased estimator for



,12

which is given

–

UUU



where

 



112 i

ijk2















m mI











and

 



212 i

ijk

n nIy









 2













1if

0otherwise

ab

where



Ia b







The asymptotic normality of the test Um,n is presented

in the following theorem.

Theorem: As n,





,NU FG







,mn



is as-

ymptotically normal with mean zero and variance











where

 

dxFx

9Var 2

Fx F

































and



 

dyGy



9 Var2































Under H0: 222









and is given by



Proof: Proof follows from Hoeffding [10] U-statistics

theory that the limiting distribution of











,NU FG





,mn 





 is asymptotic normal with mean 0

and variance 2



, where N = m + n is the combined

sample size, 22

212









and lim





 and it is

assumed 0 <



< 1.

Here it is to be noted that the asymptotic mean of

,mn is zero, independent of unspecified common

distribution F0. However, the null asymptotic variance









291F





 does depend on F0 through









and must be estimated from the data. To esti-

mate



, one possible way is to obtain consistent esti-

mator for 2



. For that,

Since





111

Var







where









111 2 311

EXXXXx













and









123

123 213

312

122

XXX

IXXXIXX X

IXXX

















we get



111

1ˆ

FxU















where



ˆ,,

iijk

jki

xx x









 

In similar fashion, we define



112

1ˆ

GyU















where



ˆ,,

iijk

jki

yyyy









Then it is easily verified that



1ˆ





 and



1ˆ







Uy.











 are consistent estimators of





and











1, and this consistent estimator 2





and is

obtained by replacing











and 1 by









and







2

1, respectively in the expression of



Hence the estimator for 2



is obtained as

P. V. PANDIT, S. INGINASHETTY 445



99NGn







ˆNNFm



.

By the consistency of



and Slutsky’s theorem



 is asymptotically N(0, 1) under H0.

The approximate



-level test of H0 versus H1 rejects

H0 in favour of H1 if





n z

NUNF mNG











where



is the upper



-percentile point of the normal

distribution. This ensures the consistency of the two

sample IFR test against the class of (F, G) pairs satisfy-

ing .

 

0FG







3. Asymptotic Relative Efficiency

We study the asymptotic efficacy of Um,n, for two pairs of

distribution ,i





. Here, we assume that G is an

exponential distribution with mean one. The different

distributions considered here for ,i



are given below:

1) Linear Failure Rate Distribution



2. exp 2

Fx x





















, 0, 0





2) Makeham Distribution







3. expe 1

Fxx x











, 0, 0x





The Pitman asymptotic efficacy is given by



F G



,. 0d

,, d

mn i

Eff UFGv

















.

The mean of the proposed test is



 

.00

























FxFy

GGxGy



since G is exponential, 4





The asymptotic efficacies of the proposed test for

Linear Failure Rate distribution and Makeham

distributio n are



0.8681



 and



1.4062







respectively.

Next, we compute ARE of the two sample test based

on Um,n proposed here, by considering a sequence of

alternatives







where





is LFR or Makeham

distribution with parameter





, wh ere 1a



 , “a”

being arbitrary positive constant, and G



is LFR or

Makeham with parameter



. It is to be noted that as

, the sequence of alternatives converges to the

null hypothesis. That is G

N





1 when





. The

efficacy of the V test is given by









Hmn

eff UU











,





where mn



is null asymptotic standard deviation

of Um,n, and





,d,

FG FG















.











m,n

The AREs of Um,n relative to V, the test due to Pandit

and Gudaganavar [8] for different values of



are pre-

sented in Table 1.

4. Some Remarks

1) A simple test procedure for testing the null hy-

pothesis that two life time distributions are identical

against the alternative th at one possesses more IFR prop-

erty than another. The test proposed is based on the

measure considered by Mugadi and Ahmad [9] for one

sample problem.

2) The only test available for the above stated problem

in the literature is due to Pandit and Gudaganavar [8].

Hence, only Pitman AREs relative to the test due to Pan-

dit and Gudaganavar [8] are computed for the few alter-

natives by specifying the common null distribution to be

exponential. However, similar testing problems with

more NBU/IFRA/NBU-t0 property are considered by few

authors, namely, Hollander, Park and Proschan [5], Ti-

wari and Zalkikar [6] and Lim, Kim and Park [7].

3) We have also computed Pitman AREs relative to

Pandit and Gudaganavar [8] for those alternatives with

common null distributions being Makeham and Linear

failure rate distributions.

4) It is observed that the new test proposed here is

better than Pandit and Gudaganavar [8] test for the dis-

tributions considered in this paper. However, the per-

formance of Pandit and Gudaganavar [8] test is better for

Weibull distribution.

5. Acknowledgements

Authors thank the Editorial Board and reviewers for their

Table 1. AREs of relative to V.

θ LFR Makeham

2 1.7814 1.5386

3 1.9358 1.4612

4 1.9483 1.4475

5 1.9822 1.4493

P. V. PANDIT, S. INGINASHETTY

446

valuable comments which lead to this revised version of

the paper.

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