Testing exponentiality against L-distributions

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journal of

statistical planning

Journal of Statistical Planning and and inference

ELSEVIER Inference 64 (1997) 249-255

Testing exponentiality against L-distributions

Gopal Chaudhuri

Stat-Math Unit, Indian Statistical Institute, 203 B.T. Road, Calcutta, India Received 3 April 1996; received in revised form 2 December 1996

Abstract

In this paper we propose a test statistic for testing exponentiality versus L-class of life distributions. This test is based on an estimate of a functional of the cdf which discriminates between the exponential and L-family. @ 1997 Elsevier Science B.V.

A M S classification: 62G10

Keywords: Brownian bridge; Gaussian process; L-class of life distributions; Karhumen Loeve expansion

1. Introduction and summary

We assume that F is an absolutely continuous life distribution with F ( 0 ) = 0, has a finite mean /l and denote the survival function (SF) by ff ( : 1 - F ) . Let D denote the set o f all such life distributions.

Klefsj6 (1983) has defined and extensively studied the L-class o f life distributions.

A distribution function F E D is said to belong to the L-class if for s/> 0,

~0 °° e x p ( - s t ) P ( t ) d t >~ p(1 + slt) -1 . (1.1) The right-hand side o f ( 1.1 ) may be seen to be the Laplace transform o f an exponential distribution with mean #. It is easy to show that the L-class is strictly larger than the harmonically new better than used in expectation ( H N B U E ) class o f life distributions.

Hence, the L-class also contains the smaller NBUE, NBU, IFRA and IFR classes o f life distributions.

An excellent survey o f the tests o f exponentiality versus the various life distributions belonging to IFR, IFRA, NBU, NBUE-classes is given in Hollander and Proschan (1984). The same problem for HNBUE-class was considered by Klefsj6 (1983). For testing exponentiality against I D M R L distribution see Hawkins et al. (1992) and references therein.

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250 G. Chaudhuri/ Journal of Statistical Planniny and Inference 64 (1997) 249-255

Let E denote the class of exponential distributions. Then E C D and equality occurs in (1.1) for all s/> 0 if and only if F E E. Due to this "no-aging" property of F E E, it is of practical interest to know whether a given life distribution F is in E. Alternatively, one may ask if F E L.

Therefore, in this paper, we consider the problem of testing H0: F E E versus Hi: F E L based on a random sample )(1 .... ,Xn, of size n, from F E D (F unknown).

Our test is based on an estimate of a functional which distinguishes F E E from F E L.

This functional for F E D is given by

~p(F) = sup{~b~(F): 0~<s~<F-l(1 - 0 } , (1.2)

where 1 >~ > 0 is a small fixed number, and,

//

O~(F) = e x p ( - s t ) F ( t ) d t - ~(1 + s # ) -1. (1.3)

The functional ~o(F) is clearly 0 for F E E and strictly positive for F EL.

Our test statistic is ~o(F~) where F~(.) is the empirical df. We show that this statistic has a limit distribution which coincides with the distribution of the supremum of certain Gaussian process. Using this limiting distribution, critical values are obtained.

We compute power by Monte-Carlo method.

The paper is organised as follows. In Section 2 the test statistic and its limiting null distribution are given. Section 3 contains the power computations.

2. The test statistic and its limiting null distribution

Let )(n denote the sample mean. Then our test statistic is

Tn = n 1/2yn- l ¢p (Fn). (2.1)

Lemma 2.1. With definition (1.3), we have

sup I I ~ s ( F ~ ) - ~ ' s ( F ) l l = O p ( n - 1 / 2 ) , (2.2) 0~<s~F-I(I--~)

where F~ is the empirical d f and F is assumed to be a life distribution having an absolutely continuous density.

Proof. See the appendix.

Hence, by the above lemma, ~p(F~) is near zero under H0 and large positive under H~, making large values of T~ significant for testing H0 versus H~.

The following computational formula is derived easily, where XO)< . . . < X(n) denote the order statistics (X(o)= 0):

Tn = n l / 2 ) ( n - I m a x 1 - a q -

l<~j<~m i=1 1 ÷(j/m)FL--l(1 - e ) X n

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(7. Chaudhuri/ Journal of Statistical Plannin 9 and Inference 64 (1997) 249-255 251

where

m = [n(1 - e)], and

e x p ( - ( j / m ) F - l (1 -- e )X(i) ) - e x p ( - ( j / m ) F - l (1 - e)A~i+l))

aij = ( j / m ) F - l (1 - e)

The asymptotic null distribution of Tn is given in Theorem 2.1. In this direction, let

P 1

Z ~ ( p ) = J 0 (1 - u ) P U - l W ° ( u ) d u for some # > 0 , (2.4) where W ° is a Brownian bridge. Then {Z~(p), 0~< p~< 1 - e } is a zero-mean Gaussian process with covariance function K ( p , q ) given by

K - ~ ( p , q ) = ( p # + 1)(q# + 1 ) ( p p + q # + 1). (2.5)

Theorem 2.1. Under H0: F E E,

T~ ~ Z~--sup{Z~,(p): 0~<p~<l - e } . Proof. See the appendix.

Approximate asymptotic critical values of the test statistic T~ based on Z~ can be obtained via the following theorem.

Theorem 2.2. For the Gaussian process {Z~(p): 0~<p~< 1 - ~} defined in (2.4), P ( sup Z ~ ( p ) > c <,2P > K 1 / 2 ( l _ e , l _ e ) , (2.6)

\0~p~< 1-~

where X is N(0, 1 ),

K - l ( 1 - ~ , l - ~ ) = [ p ( 1 - e ) + l ] 2 [ ( 1 - e ) # + l ] and 0 < ~ < 1 . Proof. See the appendix.

Table 1 contains selected quantiles of the distribution of Z~, computed from the bound in (2.6).

A comparison of the bound in (2.6) with the exact value is given in Table 3.

3. Power computation

Table 2 contains Monte-Carlo power computations based on 1O00 replications of samples of size n - - 5 0 from G(t), where

G ( t ) - - 1 - t e x p ( 1 - ~ ) , # 0~<t~<#, (3.1)

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252 G Chaudhuril Journal of Statistical Plannino and Inference 64 (1997) 249-255 Table 1

Approximate quantiles of Z~ (e=0.10 and # = 1)

c~ 0.90 0.95 0.99

Z~ quantile 0.5189 0.6164 0.8114

Table 2

Power of the proposed test

Size 0,01 0.05 0.10

#

1 0.0430 0.0512 0.0758

5 0,2061 0.2105 0.2237

10 0.2397 0.2409 0.2816

20 0.4012 0.4653 0.4980

50 0.5574 0.5601 0.5645

100 0.6100 0.6205 0.6291

Table 3

A comparison of the bound in (2.6) with the exact value

Bound Exact

0.10 0.07

0.05 0.03

0.01 0.008

see C h a u d h u r i and D e s h p a n d e (1996). It is k n o w n that G ( t ) is a l o g c o n c a v e SF, w h i c h is a l o w e r b o u n d o f a SF b e l o n g i n g to the L-class. N o t e that the p o w e r increases as /~

increases, as e x p e c t e d .

Acknowledgements

T h e author is e x t r e m e l y grateful to P r o f e s s o r B . V . R a o for his t h o r o u g h c h e c k i n g o f the m a t h e m a t i c a l details o f this p a p e r and thanks are due to P r o f e s s o r s J.K. G h o s h and P.K. S e n for h e l p f u l discussions.

Appendix

Proof of L e m m a 2.1. W e h a v e ,

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G. Chaudhuril Journal of Statistical Plannino and Inference 64 (1997) 249 255 253

= e x p ( - s t ) ( F n ( t ) - F ( t ) ) dt

= - e x p ( - s t ) ( F ~ ( t ) - F ( t ) ) dt.

Thus,

sup n 1/2 I[~bn(F,) - ~s(f)tl ~<

nl/ZllF.(t)

- f(t)ll dt 0~<s~<F-I(1--e)

fo I du

= ]]W~(u)[lf(f_l(u) ).

(By applying the transformation u = F ( t ) and writing W ~ ( u ) = n l / 2 ( F n ( F - l ( u ) ) - u), d u = f ( F - l ( u ) ) d t . Recall that F is assumed to have an absolutely continuous density f ) .

fo ]]W°(u)ll f ( F _ l ( u ) ) < o o du (see Billingsley, 1968). This implies,

fo ]] Wn(u)]] f ( F _ l ( u ) ) du Op(1).

This completes the proof of the lemma. []

Proof of Theorem 2.1. Consider the stochastic process Z n ( p , F ) = nl/2(ffs(Fn) - ~ks(F))

= - f o ~ exp(-s(p)t)nl/2(F~(t) - F ( t ) ) dt, where s ( p ) = F - l ( p ) , 0~<p-G<l - ~ .

Apply the transformation:

u = F ( t ) and define

Wn = n l / Z ( F n ( F - l ( u ) ) - u), O<~u<~ 1.

Since

d F - l ( u ) = g(1 - u) - l ,

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G. Chaudhuril Journal of Statistical Planning and Inference 64 (1997) 249-255

we have,

#-lZn(p,F)=

- f01 (1 -

u)p#-lWn(u)du.

Note that ~ s ( F ) = 0, for F being exponential with mean #.

Hence,

1 p

nl/2~s(Fn)t~l-l "~ --JO

( l -

u)p#-I W°(u)du,

since W~(u) ~

W°(u)

in D[0, 1], where

W°(u)

is a Brownian bridge. []

Remark. The case when p = 0 can be treated separately.

Proof of Theorem 2.2. Since {Z~(p), 0 ~ < p ~ < l - e } has a continuous sample path with probability one, we have an a.s. Karhumen-Loeve expansion for

Z~(p).

Z~(p) = ~ zj4~j(p),

j=l where

fO l --~

zj = Z~(p)~bj(p) dp

and ~bj is an eigenfunction corresponding to the eigenvalue 2j such that

j['o I K(p, q)4)j(P) dp = 2j4~j(q).

Here zj's are independent N(0,2j), j = 1,..., oc. Consider

l(~m)(p) = ~ zj•j(p).

j=l Then

P(\o~p~<,-~sup

Z:~'n)(p)>c)

<~ 2P(Z~m)(1

- ~) >c), since

e(lm<axnZ(m) ( j ~ ' ~ ) ~c)~2e(z(gm)(p)~c)

(see Ash and Gardner, 1975, p. 180) and as n--~ ~z,

P( max Z(m)(J~-~)>c) \o<<.p<~l-~sup z}m)(p)>c),

which is due to the continuity of the sample path of

{Z~(p),

0 < ~ p ~ < l - e } .

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G. Chaudhuril Journal of Statistical Plannin 9 and Inference 64 (1997) 249-255 255

N o w , s i n c e

Z~")(p) ~ Z~(p), u n i f o r m l y in p as m ~ c ~ ( s e e A d l e r , 1 9 9 0 ) ,

s u p z(m)(p) --~ s u p Z~(p) as m --+ oc.

O<~ p<~ l--e O<~ p<~ l--e

H e n c e ,

P(0~<p~<I-eSUp

Z ~ ( p ) > c ) ~ 2 P ( Z ~ ( 1 - ~ ) > c ) .

T h i s c o m p l e t e s t h e p r o o f . [ ]

References

Adler, R.J., 1990. An Introduction to Continuity, Extrema and Related Topics for General Gaussian Processes, IMS Lecture Note - Monograph Series, 12.

Ash, R., Gardner, M.F., 1975. Topics in Stochastic Processes. Academic Press, New York.

Billingsley, P., 1968. Convergence of Probability Measures. Wiley, New York.

Chaudhuri, G., Deshpande, J.V., 1996. A lower bound on the L-class of life distributions and its applications, Technical Report # 3/96. St, at-Math Division, Indian Statistical Insititute, Calcutta.

Hawkins, D.L., Kochar, S., Loader, C., 1992. Testing exponentiality against IDMRL distribution with unknown change point. Ann. Statist. 20 (1), 280-290.

Hollander, M., Proschan, F., 1984. Nonparametric concepts and methods in Reliability. in: Krishnaiah, P.R., Sen, P.K., (Eds.), Handbook of Statistics, vol. 4. pp. 613-655.

Klefsj6, B., 1983. A useful aging property based on the Laplace transforms. J. Appl. Probab. 20, 615 626.

Klefsj6, B., 1983. Testing exponentiality against HNBUE. Scand. J. Statist. 10, 65-75.