On testing exponentiality against IFRA alternatives

Metrika (1989) 36:255-267

On Testing Exponentiality Against IFRA Alternatives

By Emad-E ld in A. A. Aly I

Summary: We present a class of tests for exponentiality against IFRA alternatives. The class of tests of Deshpande (1983) is a subclass of ours. We also treat the same problem when the data is randomly censored from the right. The results of an asymptotic relative efficiency comparison indicate the superiority of our tests.

Key words and Phrases: Brownian bridge, Pitman efficiency, Product-Limit estimator, Weak convergence.

1 Introduction

Let F b e a con t inuous life d i s t r ibu t ion funct ion and_F= 1 - - F b e the co r re spond ing

survival funct ion. F is said to be an increas ing failure rate average ( I F R A ) if and

only if, - log [ '(bx) ~_ - b log f ' (x) for all x > 0 and 0 _~ b ~ 1. I t is well known tha t the

class o f I F R A dis t r ibu t ions is the smal les t class o f life d i s t r ibu t ions which conta ins

the exponent ia l d i s t r ibu t ion and is c losed under fo rma t ion of coherent systems.

D e s h p a n d e (1983) p r o p o s e d a class o f tests for tes t ing exponent ia l i ty aga ins t

I F R A al ternat ives , In this pape r we p ropose a family o f tests for the same p r o b l e m

which conta ins those of Deshpande (1983). We will a lso cons ider the p rob l e m of

test ing with r a n d o m l y r ight censored data .

I Prof. Emad-Eldin Aly, Department of Statistics and Applied Probability, University of Alberta, 434 Central Academic Building, Edmonton, Alberta, Canada T6G 2G 1.

This research was supported by an NSERC Canada operating grant at the University of Alberta.

0026-1335/89/5/255-267 $2.50 © 1989 Physica-Verlag, Heidelberg

256 Emad-Eidin A. A. Aly

2 Motivation and Definitions

Let X1, S2, . . . ,Sn be a random sample from F, F --1 be the corresponding quantile function. Let Fn be the empirical distribution function of the given sample. We wish to test

Ho : F(x) = e -x/h, x > 0, where 2 > 0 is unknown (2.1)

against

H1 :F is an IFRA distribution but not exponential. (2.2)

It is easy to show that F is an IFRA distribution if and only if, any of the following equivalent inequalities hold

i) F(bx) ~_ [Je(x)]b, for all x > 0 and 0 < b < 1, (2.3)

ii) F'(ax)<~[P(x)] a, for all x>O and a > 1, (2.4)

iii) [P(ax)]~<--[P(x)] a~, for all x > 0 , a > 1 and for some c > 0 . (2.5)

We mention here that Desphande (1983) used (2.3) to develop his testing procedure. However, (2.4) could also be used to develop (essentially) the same testing procedure. Our testing approach will be based on (2.5) and consequently the approach of Deshpande will correspond to the special case of c = 1.

Next, we note that (2.5) is equivalent to

[P(aF-l(y))]c~(1 _y)ac for all 0 < y < 1, a > 1 and for some c > 0 . (2.6)

For any fixed a > 1, c > 0 , we define the IFRA plot function ,4(.; F, a, c) by

'4(y; F, a, c)=[P(aF-t(y))] c, O~_ y~_ 1. (2.7)

On Testing Exponentiality Against IFRA Alternatives 257

The empirical IFRA plot function is zl(.; Fn, a, c) and the corresponding empirical IFRA plot process 6n(.; a, c) is defined by

gn(Y; a, c)=nl/2{/ l (y; Fn, a, c ) - z J ( y ; F, a, c)}, 0 6 y ~ _ 1. (2.8)

The deviation from exponentiality in favor of the IFRA class may be measured by the parameter(s)

1

~,(a, c) = yl(F, a, c)=f ~(y; F, a, c)dy. o

(2.9)

Note that, under H0 of (2.1), y ( a , c ) = ( a c + l ) - l and under Hi of (2.2), y(a, c ) < ( a c + 1) -1.

The test statistics corresponding to the above measures are tl (a, c ) : 71(Fn, a, c). The asymptotic theory of these test statistics will follow from the weak convergence of gn('; a, c) of (2.8). In the randomly censored case, we propose the test statistics 7 ( a , c ) = y l ( P , , a , c ) , where Pn is the Product-Limit estimator of F of Kaplan and Meier (1958).

3 Asymptotic Theory in the Noncensored Case

In this section we will study the asymptotic theory of our proposed tests. We will study the weak convergence of 3n('; a, c) of (2.8). First, we give the following definition.

Definition 3.1.' A real-valued function q(.) on [0, 1] is called a Chibisov-O'Reilly function if q(.) is positive on (0, 1), non-decreasing (resp. nonincreasing) in a neighborhood of zero (resp. of one) and

I

f [t( 1 -- t)]- I exp {-- cq2(t)/t( 1 -- t) } dt < 0

for all c > O.

258 Emad-Eldin A. A. Aly

Theorem 3.1: Assume that f - - F is continuous and for some Chibisov-O'Reilly function q( ') we have

sup f(aF-l(Y)) q(y)<oo. o6y_~l f ( F - l ( y ) )

Then, on an appropriate probability space there exists a sequence {B,(.)} of Brownian bridges such that as n - oo,

P sup I d,(y; a, c) + F(y; B,, a, c)[ --- o(1), (3.1)

0.<y~ 1

where

F(y; Bn, a, c ) = cr(y; B,, a, 1)/[ ['(aF--l(y))] l -c. (3.2)

and

r(y; B., a, 1)=B.(F(aF -1 (y))) af(aF-l(Y)) B.(y). f (V - l ( y ) )

The proof of Theorem 3.1 is similar to but much simpler than that of Theorem 3.1 and Corollary 3.1 of Aly and Lu (1988).

Corollary 3.1: Under the conditions of Theorem 3.1, we have

1

nl/2{yl(Fn, a, c) - y1(F, a, c)} -~ T(a, c) = f F(y; B, a, c)dy, o

(3.3)

where B(') is a Brownian bridge.

Theorem3.2: Assume that F(-) is continuous and a > 1, c > 0 are such that the variance of T(a, c) of (3.3) is finite. Then (3.3) holds true.

The proof of Theorem 3.2 is similar to that of Theorem 2.1 of Chen et al. (1983).

Next, we consider the special case when H0 of (2.1) holds true.


Corollary3.2: Assume that H0 of (2.1) holds true. Then, for any a > 1, c > 0 such that c > ( 2 - a)/2a, we have

nl/2{?l(Fn, a, c)--(ac+ 1) -1} ~-~ Z(a, c), (3.4)

where

1 1 1

Z(a, c) = _ c f xC+V - 2 B ( I - - x)dx + ac f x °~- IB(1 - - x)dx. a o o

(3 .5)

It is straightforward to show that Z(a, c) is a mean zero normal random variable with variance

° _ 2 : } tr2(a' c ) = (ac+ 1) 2 ( 2 a c + 1) 2r (2ac+2--a) (aec+ac+ 1)) " (3.6)

4 On Testing Exponentiality Against IFRA Alternatives in the Noncensored Case

In this section we consider the problem of testing H0 of (2.1) against Hi of (2.2) in the noncensored case. We will consider the use of the class of test statistics

tl(a, c) = yl(Fn, a, c). Multiply each observation by a and let Ri be the rank of aXi:n in the combined X

and aXsample, where XI :~, X2:~ . . . . . X~:n are the order statistics of the Xsample. It is easy to show that

t l (a,c):n - l -c ~ {n+i-Ri} c. (4.1) i=1

By (3.4) we know that when c>(2--a)/2a, nl/2{&(a, c)--(ac+ 1) -l} converges in distribution to a mean zero normal random variable with variance o2(a, c) of (3.6). We reject H0 of (2.1) in favor of Hi of (2.2) for small values of tl(a, c).

We mention here that the unbiasedness of &(a, c) will follow by the same argument of Deshpande (1983). In addition, we can also use the argument of section 4 of Deshpande (1983) to show that tl (a, c) is consistent against continuous


IFRA distributions and tl (k, c), k___ 2 is an integer, is consistent against continuous new better than used (NBU) alternatives.

Next, we consider the question of selecting a pair(s) of values of a and c. We have calculated the efficacy of t(a, c) over a large number of pairs (a, c). These results suggest using the pair (a = 3, c = 0.5) whenever the alternative is suspected to belong to the larger NBU class. The pair (a = 3.3, c=0.5) may be recommended whenever the alternative belongs to the restricted IFRA class.

Now, we study the asymptotic relative efficiency (ARE) of tl(3, 0.5) and q(3.3, 0.5) with respect to the following test statistics;

(i) J (Hollander and Proschan 1972),

(ii) ~r-/l/2 (Koul 1978),

(iii) W (cumulative total time of Bickel and Doksum 1969),

and

(iv) Jb, b = 0.5, 0.9 (Deshpande 1983).

We will consider the following three families of distributions;

(i) The Weibull family

[ ' l (x)=exp(-x°) , O~ 1, x>=O,

(ii) The Makeham family

F2(x) :exp{-x - -O(x+e - x - 1)}, 0_~0, x~_0,

and

(iii) The Linear Failure Rate family

F3(x)=exp{-x--Ox2/2}, 0 _ 0 , x ~ 0 .

We mention here that J05 of Deshpande (1983) is essentially tl(2, 1) and his J0.9 is essentially t l (1.1, 1). In Table 1 we give Pitman ARE's of t l (3, 0.5) and t l (3.3, 0.5) with respect to the above-mentioned tests.

It is clear from Table 1 that both t1(3,0.5) and t1(3.3,0.5) are more efficient than J, W, Jl/2 and J0.9 for the three distributions considered. In addition, q(3, 0.5) is as efficient as ~1/2 for the Weibull and Makeham distributions and is more efficient than ~ul/2 for the Linear Failure Rate distribution. It is worth noting here


Table 1. ARE of t1(3, 0.5) and t2(3.3,0.5) with respect to J, ~vl/2, W, Jl/2 and J0.9

Distribu- el:i(tl(3,0.5),. ) eFi(tff3.3,0.5), .) tion

J t//l/2 W Jl/2 J0.9 J ~1/2 W JJ/2 J0.9

F! 1.0924 1.0369 1.0169 1.0852 1.092 1.0975 1.0418 1.0216 1.0903 1.0971 F2 1.2254 0.997 1.0194 1.2953 1.2373 1.217 0.9904 1.0124 1.2864 1.2288 F3 2.3166 1.2368 1.0401 2.4883 2.3196 2.3195 1.2384 1.0414 2.4914 2.3225

that t1(3, 0.5) as a test statistic against NBU alternatives is much easier to calculate than ~1/2.

5 On Testing Exponentiality Against IFRA Alternatives in the Censored Case

Let Xh X2 . . . . , X, be a random sample from Fand Yh Y2, ..., Y~ be a random sample from H which is independent of the X sfimple. In the random censorship model from the right, the X's may be censored by the Y's, so that we only observe the pairs {(Zi, ~i)}~= 1, where Zi = rain (Xi, Yi) and ~i = l(Xi ~_ Yj), the indictor function of the event {X,-~ Yi}, i = 1, 2 . . . . . n. Let Fn be the Product-Limit estimator o f F o f Kaplan and Meier (1958) which is given by

1 -Fn(x)--- 1-I { ( n - i ) / ( n - i + 1)}a(O, {i:Z(o~xl

(5.1)

where (Z(i), 1 - ~(0), i = 1, 2 . . . . , n are the pairs (Zi, 1 - ~i), i = 1,2 . . . . . n in lexicogra- phic order with the exception that Z(n) is always treated as being noncensored so that 8fn)= 1.

Following Chen et al. (1983), throughout this section, we will assume the following two conditions;

(A 1) The distribution functions F and H are continuous and have supports equal to [0,~).


(A2) For some 0 < e < 1,

sup {F(x)} 1 - e{/~(x)} -1 < o o . 0__.x<OO

Next, we introduce the zero mean Gaussian process {G(t), 0 ~ t ~ l } whose covariance function is given by

EG(s) G(t) = (1 -- t)(1 - s)e(sAt), (5.2)

where

t

e(t)= f du du, (5.3) 0 (1-- u)K(F-l(u))

and K(x) = F(x) . / / (x) . The following Theorem is an extension of Theorem 3.2 to the randomly

censored case. Its proof is similar to that of Theorem 2.1 of Chen et al. (1983).

Theorem 5.1." Let ?l( ' , a, c) be as in (2.9), F( . ; . , a, c) be as in (3.2) and G( ' ) be as in (5.2). Define

1

1"(a, c) = f F(y; G, a, c)dy. o

Assume that conditions (A1) and (A2) hold true and that a > 1 and c > 0 are such that T(a, c) has a finite variance. Then

nl/2{7(a, c)--yl(F, a, c)} ~ T(a, c), (5.4)

where 7(a, c ) = yl(Fn, a, c).

Corollary 5.1: Assume that the conditions of Theorem 5.1 and H0 of(2.1) hold true. Then

nl/Z{7(a, c)--(ac+ 1) -1} ~-* 2(a, c), (5.5)


where

1 1 1

Z(a, c)----- c f (1 -- x) c+a - 2 G(x)dx + a c f (1 -- x) "c- 1 a(x)dx. a o o

(5.6)

By straightforward integral evaluations we can show that Z(a, c) is a mean zero normal random variable with variance

ot~

o'2(a, c, 2,/~) = f ~(x; a, c)/I~Otx)dx, (5.7) 0

where

~(X; a, c )= - - c2 {

(ac + 1) 2 e-2(ac+ l)x/a + a2e-2(ac+ l)x

2 a ( a + 1) (a t+ 1) e_(~2c+~c+,,+l)x/~/ (a2c+ac+a+ 1) J

(5.8)

and K(') is as in (5.3). Following Chen et al. (1983), we estimate a2(a, c, 2, K') by ~n = ~n( a, c, 2,/~n),

n n

where 2n = 2~ Z i / 2 ~ 8i and K'n(t) = 1 ~ I(Zi>t) . It is easy to show that i=1 i=1 n i=1

n - 1

~n = Y. n ( n - i + l ) - l (n-O-Xd(ui ) -nd(u , )+d(O) , (5.9) i=1

w h e r e u i = Z(o /2 n a n d

C2 ! a e-2(ac+ 1)t/a .~ a2 e-2(ac+ l)t d(t) = (ac + 1) 2 ( 2(ac + 1) 2(ac + 1)

2a2(a + l)(ac + 1) } ( a2 c + ac + a + 1) 2 e -ta2c + ac + ~ + l )t/a .


It is also easy to show that

7(a, c) = ~ [1 -- F,,(aZ(o)] c. [Fn(Z(0 ) - Fn(Z(i- i))]- (5.10) i=1

Let us assume that a~(a, c, 2,/£) is finite in an interval around 2. By the same argument of Chen et al. (1983), we can show that ~n of (5.9) is a consistent estimator of cr2(a, c, 2, K). We can also show that the testing procedure of approximate size a which rejects H0 of (2.1) in favor of H1 of (2.2) when

n v2{7(a, c ) - (ac + 1- l} /a l . < za

is consistent against all continuous IFRA alternatives under the same conditions of Chen et al. (1983), where za is the a-th quantile of a N(0, 1) r.v.

Now, we consider the question of selecting a pair(s) of values of a and c. We will assume that 2 = 1 in (2.1) and R ( x ) = ~ ux, x > 0 and 0 ~ s t < l . We have then calculated the efficiency loss due to censoring and also the efficacy of 7(a, c) over a large number of pairs (a, c) and for St = 0, 0.1 . . . . . 0.5. These results suggest the four candidates 7(2, 1), 7(3,0.5), ~'(1.1, 1) and 7(3.3,0.5). The first two of them are recommended when the alternative is suspected to belong to the NBU class and the latter two are recommended when the alternative is suspected to belong to the IFRA class. We mention here that 7(2, 1) (resp. 7(1.1, 1)) is more efficient than 7(3, 0.5) (resp. 7(3.3, 0.5)) when the censoring is heavy.

In Table 2 we give efficiency loss due to censoring of both ~'(2, 1) and 7(3, 0.5). We also include in Table 2 a reference point to the amount of censoring which is given by P(X1 < YI) ~ (1 + St)- 1.

The ARE of both 7(2, 1) and t'(1.1, 1) relative to jc of Chen et al. (1983), 7(3, 0.5) and ~'(3.3, 0.5) are given in Table 3 and Table 4, respectively, when H(x)---- ~ux, x > 0 and St ---- 0, 0.1 . . . . . 0.5. In both Tables we will use the same distributions F1, F2, and F3 of Table 1.

It is possible to improve the ARE by using the test statistics

Tl(a)=aT(3,0.5)+(1--a)'~(2, 1), O ~ a ~ l , (5.11)

and

T2(a)=aT(3.3,0.5)+(1-a)7(l .1, 1), 0 _ ~ a _ l . (5.12)


Table 2. ARE of 7(2, 1) relative to t](2, 1) and 7(3,0.5) relative to t1(3,0.5) when/-t(x) = e T M

~: 0.1 0.2 0.3 0.4 0.5

e(7(2, 1), tl(2, 1)) 0.946 0.889 0.830 0.769 0.707 e(7(3,0.5), t](3, 0.5)) 0.850 0.694 0.534 0.375 0.222 P(XI < Yt) 0.909 0.8333 0.769 0.714 0.667

Table 3. ARE of 7(2, 1) relative to J~, 7(3,0.5) and 7(3.3,0.5) when I-:l(x)=e -,~x

Distri- ARE of 7(2, 1) bution relative to

0.5 0.4 0.3 0.2 0.1 0

Fl jc 1.046 1.037 1.032 1.021 1.012 7(3, 0.5) 2.930 1.893 1.433 1.181 1.025 7(3.3,0.5) 3.875 2.095 1.492 1.196 1.026

F2 jc 0.991 0.983 0.978 0.968 0.960 7(3,0.5) 2.459 1.581 1.205 0.990 0.859 7(3.3,0.5) 3.285 1.75 1.262 1.014 0.870

F3 J~ 0.967 0.960 0.956 0.943 0.934 7(3,0.5) 1.276 0.826 0.626 0.514 0.447 7(3.3,0.5) 1.696 0.916 0.654 0.522 0.449

1.009 0.921 0.917

0.957 0.772 0.780

0.933 0.402 0.401

Table 4. ARE of 7( 1.1, 1) relative to jc, 7(3, 0.5) and t'(3.3, 0.5) when / t (x ) = e T M

Distri- ARE of bution 7(1.1, 1)

relative to 0.5 0.4 0.3 0.2 0.1 0

/71 jc 1.001 1.001 1.004 1 1 7(3,0.5) 2.805 1.825 1.394 1.157 1.011 7(3.3,0.5) 3.710 2.022 1.451 1.172 1.012

F2 jc 1 1 1.004 0.998 0.998 7(3,0.5) 2.477 1.612 1.230 1.021 0.894 7(3.3,0.5) 3.314 1.804 1.286 1.047 0.905

F3 JC 1 1 1.004 0.997 0.997 7(3,0.5) 1.320 0.860 0.657 0.545 0.476 7(3.3,0.5) 1.753 0.955 0.685 0.562 0.479

1.003 0.916 0.911

1.002 0.808 0.813

1.001 0.431 0.430


In order to illustrate the advantage and the use of Tn(a) and T2(a), we will briefly consider the family of tests Tl(t~). Similar results also hold for T2(a), but the details are left to the reader. It is easy to show that the variance of Tl(a) is given by

o~2(a , 2 , / ( ) = a2tz2(3, 0.5, 2,/~) + (1 - a)2t72(2, 1, 2,/~)

+ 2a(1 - a)cr12(2,/~),

where

o o

O'12(,~,/~) = f ITJI(X) dx o g ( 2 x ) '

(5.13)

where

7 23 11

~'l(X)= e 3 - - - e l 5 6 - ~ e -1- e 2~ .

We estimate a22(a, 2 , / ( ) by ~n = o~2n(g, 2~,/~n) which is given by

o'~2,, = a2~n(3, 0.5, ~.,,,/'(n) + (I -- a)2~n(2, 1,2,,,/~n)

+ 2a(] -- a)al2n(~.n,/'~n),

where

~n( ' , ", 2n, g'~) is as in (5.9),

n - I

#J2n(2,,/~n) = ~ n(n--i+ 1)-](n - i)-Jdn(ui)+ndn(un)+dv(O), i=1

7 23 11

1 - T t 4 - ~ - t _~_oe_4t+_~_5e-"~-t ' dl( t )=-~e -- 1 l----~e -- and

u,= Z~ol~..

In Table 5 we give the ARE of Ti (0. l) relative to jc and ~'(2, l) when/ / (x) -- e -~'x.


Table 5. ARE of Tl(0.1) relative to jc and 7(2, 1) when/t(x) = e -gx, x > 0

Distri- ARE of TI(0.1) bution relative to

0.5 0.4 0.3 0.2 0.1 0

Fj jc 1.03 1.058 1.07 1.067 1.06 1.056 7(2, 1) 0.985 1.02 1.037 1.045 1.047 1.047

F2 jc 1.005 1.032 1.045 1.042 1.036 1.032 7(2,1) 1.014 1.05 1.068 1.076 1.079 1.078

F3 JC 1.115 1.146 1.161 1.154 1.146 1.144 7(2,1) 1.153 1.194 1.214 1.224 1.227 1.226

References

Aly E-E, Lu M (1988) A unified asymptotic theory for testing exponentiality against NBU, IFR or IFRA alternatives. Statist and Dec 6:261-274

Bickel P J, Doksum KA (1969) Test for monotone failure rate based on normalized spacings. Ann Math Statist 40:1216-1235

Chen Y, Hollander M, Langberg NA (1963) Testing whether new is better than used with randomly censored data. Ann Statist 11:267-274

Deshpande JV (1983) A class of tests for exponentiality against increasing failure rate average alternatives. Biometrika 70:514-518

Hollander M, Proschan F (1972) Testing whether new is better than used. Ann Math Statist 43:1136-

1146 Kaplan EL, Meier P (1958) Nonparametric estimation from incomplete observations. J Amer Statist

Assoc 53:457-481 Koul HL (1978) A class of tests for testing "new is better than used". Canad J Statist 6:249-271

Received 22.8. 1988

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On testing exponentiality against IFRA alternatives