Bahadur and Hodges-Lehmann approximate efficiencies of tests based on spacings

IWA ELSEVIER

CS &

Statistics & Probability Letters 23 (1995)211-220

Bahadur and Hodges-Lehmann approximate efficiencies of tests based on spacings Jarostaw Bartoszewicz Mathematical Institute, University of Wroclaw, PI. Grunwaldzki 2/4, 50-384 Wroclaw, Poland

Received May 1993; revised February 1994

Abstract

The paper is concerned with the Bahadur and Hodges-Lehmann approximate efficiencies of spacings tests for goodness-of-fit problems. The explicit forms of the approximate slopes are derived for the sums of powers and logarithms of spacings, when the alternative is a distribution with a step density function. Applications to the dispersivity testing problem and numerical values of the efficiencies for various types alternatives are given. Keywords: Goodness-of-fit; Logarithms of spacings; Powers of spacings; Slope; Step density function; Dispersive ordering; TTT transformation

1. Introduction

There are two basic approaches for the goodness-of-fit problem; tests based on the observed frequencies and those based on spacings. Kale (1969) has noticed that while the tests based on frequencies and empirical distribution function are useful to detect differences between the distribution functions, the tests based on spacings are useful to detect differences between the corresponding densities. Kale (1969) has also proposed several criteria for the derivation of tests for goodness-of-fit based on spacings. Bartoszewicz (1986) (see also Bartoszewicz and Bednarski, 1990; Bartoszewicz, 1992) has remarked that spacings tests may be used for testing the dispersive ordering. Asymptotic distributions of the test statistics under the null hypothesis (uniform) and some class of alternatives have been studied by many authors, e.g. Weiss (1957), Gebert and Kale (1969) and CzekMa (1993). Also many authors have studied the Pitman asymptotic relative efficiency (ARE) of spacings tests, e.g. Sethuraman and Rao (1970), Del Pino (1979), Kuo and Rao (1981) and Jammalamadaka and Tiwari (1987). The Bahadur exact ARE of spacings tests has been studied for special cases by Xian Zhou and Jammalamadaka (1989). Rao (1972) has studied the Bahadur approximate efficiency of a spacings test for uniformity on the circle. This paper is concerned with the Bahadur and HodgesLehmann approximate ARE of spacings tests for goodness-of-fit when alternatives have step density functions. Let X1 . . . . X 2 . . . . . . . Xn: n be order statistics of sample from a distribution function F. Define Xo:.=sup{x:F(x)=O} and X . + l ; . = i n f { x : F ( x ) = 1}, if they are finite. The random variables, Vii:. = Xi:. - Xi-1:., i = 1, 2, ..., n + 1, are called spacings from the distribution F. 0167-7152/95/$9.50 © 1995 ElsevierScienceB.V. All rights reserved SSDI 0167-7152(94)001 15-O

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Consider the distribution F on [0, C], C > 0, with a density

f(x) = ~ ajlgj_l.~j)(x ),

(1)

j=l

where aj > 0, j = 1, 2 ..... m, 0 : ~O < ~1 < " ' " < ~ r a : C and m/> 1 are fixed numbers such that ~ = 1 ai(~J - ~i- 1) = 1. For testing the goodness-of-fit hypothesis H o : F is uniform on [0, 1] (F -- U(0, 1)), i.e. m = 1, al = 1, (o = 0, ~1 = C = 1, against the alternative HI :F has the density of the form (1) but not U(0, 1), one can use, e.g. the statistics n+l

n+l

B . , . = )-" V::., r > O , r 4 : l ,

n+l

S . = ~' log V~:.,

i=1

Q.= 2

i=1

V/:.log V~:..

i=1

In Section 2 we recall asymptotic distributions of these statistics under the alternative (1). Section 3 deals with the Bahadur approximate ARE of the tests based on these statistics. The Hodges-Lehmann approximate ARE is considered in Section 4. Applications and numerical examples of etficiencies for some alternatives are given in Section 5. A generalization of the results concerning the Bahadur efficiency is possible (see Remark in Section 3). The restriction of considerations to the density (1) is plausible in the context of applications to the testing the dispersive ordering. It also makes possible a comparison between the Bahadur and Hodges-Lehmann efficiencies and between the tests for the same alternatives.

2. Asymptotic distributions of test statistics The following two lemmas will be useful for calculating the approximate efficiencies in the following sections. These results are straightforward corollaries from theorems of Weiss (1957, Lemma 1) and of Czekata (1963, Lemma 2), which have been stated and proved for the density (1) defined on the interval [0, 1], i.e. C = 1. Lemma 1. If I"1. . . . . . . VI:, + 1 are spacings from the distribution with the density f of the form (1), then lim P .-®

x/

n'-U2B - x/nF(r + 1 ) E [ f - ' ( X ) ] } "" ~< x [F(2r + 1) - 2rF2(r + 1)]E[f-2"(X)] -- {(r - 1)r(r + 1)E[f-'(X)]} 2

where 0 < r, r ~ 1, • is the normal N(0, 1) distribution function and X is a random variable with the density (1). Lemma 2. Under the assumptions of Lemma 1,

p~

J. + n(E{log[nf(X)]} + 7)

(a).~oolim [4~((~/~))= i + ~ ~ X ) ] }

(b) ,~lim P

?((-e/~-

~

~

]

}

<~x

+

=O(x),

xER,

C--~-2;;i - 3) ~< x

= O(x),

x ~ N,

where 7 -= 0.577..- is Euler's constant and X is a random variable with the density (1).

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3. Bahadur approximate ARE

Bahadur (1960) has defined exact and approximate measures of efficiency between two sequences of statistics used to test the same hypothesis. The notion of the slope of the test sequence is fundamental in this approach. Let ~ and fl denote probabilities of Types I and II errors, respectively. If a simple alternative hypothesis is specified and fl is bounded away from 0 or 1, then as the sample increases, the size of or-test typically tends to 0 at an exponential rate. This rate is called the Bahadur exact slope. Formally, the slope of the test T. is defined as follows. Suppose Ho is rejected for large values of T, and Go.(t) = Po { T, ~< t }, where Po is the probability measure under Ho. If under H1, - 2 l o g [ 1 - Go.(/'.)] ~ br, n

then br is the exact slope of T,. The Bahadur ARE of a test 7". relative to another T~ is defined by e,(T,, T~) =

br

provided br and br, are not both zeros. The following results are useful in finding the Bahadur exact slope. Lemma 3 (Bahadur, 1960). I f n-1/2 T P.~C under H1 and

2 lim - - l o g [ 1 n~oo

- Gon(nl/2t)] = 7(t),

t~I

n

for some open interval 1 containing c and for some ~(t) which is continuous on 1, then br = 7(c). Lemma 4. I f the assumptions of Lemma 3 are satisfied and Go, is the normal distribution N(x/~#o,tr2), then (C - - 120) 2

bT-- - -

Proof. The proof is straightforward by applying the well-known relation (see, e.g. Feller, 1961) 1-~(x)~~exp ~/2r~x

-

asx--* oe,

\

to Go,(t) = ~((t - x/~lto)/ao).

[]

The qualification exact in the preceding definition and Lemmas 3 and 4 is to distinguish from an approximate version of the concept, based on the substitution of Go. for G*. in the definition of br, where G*. is the limiting distribution of T. under H0. Bahadur (1960) gives the condition under which the two concepts coincide. Since we do not know the exact distributions of statistics B,., J. and Q. under Ho but only asymptotic ones, we may only calculate the approximate slopes. Our calculations will be provided under the assumptions that the hypothesis Ho is rejected when values of the test statistics are sufficiently large. The results are exactly the same if Ho is rejected for small values of statistics or if tests have two-sided symmetrical regions of rejection. It is also easy to see from Lemma 3, that slopes are the same for the equivalent statistic T* = a, 7", + b. for some a. ~ 0 and b,.

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Theorem 1. Let Ho : F = U(0, 1) and H t : F has the density (1) but not U(0, 1). The Bahadur approximate slopes for the statistics B .... 0 < r ~ l, J. and Q. are, respectively, the following:

r2(r +

1){1 - E [ f - ' ( X ) ] } 2 bsr = r(2r + 1) - (r 2 + 1)r(r + 1)'

(2)

6E2[l°gf(X)] bs

=

bQ =

'IT, 2

-

6

(3) '

3E 2 [(log f ( X ))/f(X)] ~2 _ 9 '

(4)

where X is a random variable with the density (1). nr-1/2Br , n. Thus from Lemma 1 we have that under Ha ~ B*r , n is asymptotically normal N ( x / ~ F ( r + 1)E[f-~(X)], a,21), where P r o o f . Consider the statistic B*r , n =

tr2,,~ = {[F(2r + 1) - 2rFZ(r + 1)]E[f-2"(X)] - {(r - 1)r(r + 1 ) E [ f - ' ( X ) ] } z } and therefore

(5)

n-X/ZB r*, n L F(r + 1)E[f-r(X)]. Under Ho the statistic B*r , n is asymptotically normal N ( x / ~ F ( r + 1), a,o), 2 where a,2o = F(2r + 1) -- (r E + 1)C2(r + 1). N o w from (5) and Lemmas 3 and 4 we obtain (2). Consider now the statistic 1 "+ 1

1

Jn* = / - - • log (n Vi:.) = - 7 J" + ~/n x / n i=1

n +1

,/.

log

n,

which is equivalent to J.. Under the hypothesis H1, by L e m m a 2(a), J* is asymptotically normal N( - x/~{E[logf(X)] + ?}, (~t2/6) - 1 + V a r [ f ( X ) ] ) and hence 1

./~n J* L -- E [ l o g / ( X ) ] -- 7. Under the hypothesis Ho ( f from Lemmas 3 and 4. Consider the statistic

(6)

1) J* is asymptotically normal N( - v/-n?, (~2/6) - 1). Therefore, (3) follows

1 n+l

Q* = x~~ ,=,E (nV,:,)log(nVi:.)= x/~Q. + x / n C logn, which is equivalent to Q.. It follows from L e m m a 2(b), that under H1, Q* is asymptotically normal N(v/-n{ 1 - ? - E[(logf(X))/f(X)]}, a~), where

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and hence 1 ~O*

L 1 --7-E

F!og f(X)-]

(7)

L f(x) _l

Under Ho ( f - 1), Q* is asymptotically normal N(x/~(1 - 7), 3 and 4 we obtain (4). []

(~2/3)

--

3). Applying once again Lemmas

Remark. Theorem 1 may be easily generalized for a wider class of alternatives. Let F be an absolutely continuous distribution with a density f and the support being an interval. Due to the special forms of statistics B.... J, and Q,, using the mean value Lagrange theorem and the construction of uniform spacings given by Malmquist (1950) one can prove (5)-(7) under the assumptions that the expectations E [ f - r ( X ) ] , E [ l o g f ( X ) ] and E ( [ l o g f ( x ) ] / f ( X ) } , respectively, exist (see Pyke, 1965; Proschan and Pyke, 1967 for details).

4. Hodges-Lehmann approximate efliciencies Hodges and Lehmann (1956) have introduced an efficiency measure of test which is pertinent when one is interested in the region of high power. It is the dual of the Bahadur efficiency: the relative importance of Types I and II error probabilities are reversed. If under a fixed alternative the significance level is bounded away from 0 or 1 as the sample size increases, the probability of Type II error often tends to 0 at an exponential rate. The ratio of these rates (slopes) for two test statistics is the Hodges-Lehmann ARE. The calculation of the slopes is similar to the Bahadur case, taking into consideration the reversed role of the error probabilities. The approximate version of the Hodges-Lehmann approach has been investigated by Hettmansperger (1973). The following result, which gives a relation between the Bahadur and Hodges-Lehmann approximate ARE, is useful for calculating the latter.

Lemma 5 (Hettmansperger, 1973). Let T. be a test statistics, which is asymptotically normal distributed: under Ho with the variance a 2 and under H1 with the variance ~ . If br is the Bahadur approximate slope of T., then the Hodges-Lehmann approximate slope of T. is cr = br a2

From Lemmas 1, 2 and 5 and Theorem 1, we obtain immediately the following theorem.

Theorem 2. Let Ho: F = U(0, 1) and Hi: F has the density (1) but not U(0, 1). The Hodges-Lehmann approximate slopes for the statistics B .... 0 < r # l, J. and Q, are, respectively, the following equations: F(r + 1 ) { E [ f - ~ ( X ) ] } 2 cB~ = [F(2r + 1) - 2rF2(r + 1)]E[f-2~(X)] - {(r - 1)F(r + 1 ) E [ f - ~ ( X ) ] } 2'

(8)

E 2 {log I f ( X ) ] } ca = (rt2/6) _ 1 + V a r [ l o g f ( X ) ] '

(9)

E 2{ l o g [ f ( X ) ] / f ( X ) } cQ = ((rt2/3)_ 2)Var[1/f(X)] + C2((~2/3)- 3)' where X is a random variable with the density (1).

(10)

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5. Applications and examples 5.1. Testing dispersivity As was mentioned in Section 1, tests based on spacings may be used for testing the hypothesis Ho: F = G against Hi: F ~< disp G, where G is a known distribution with a continuous density g and F ~< dispG ¢~. F-l(e) -- F-l(6) <<.G-l(e) - G-l(t$)

whenever 0 < 6 < e < 1

(see, Bartoszewicz, 1986, 1992; Bartoszewicz and Bednarski, 1990). Applying the so-called total time on test (TTT) transformation

Hrl(t) =

I

F - ~(t)

,) F- I(0)

o[G-1F(u)] du

(He is distribution function with a density hF) one reduce the above testing problem to the testing the hypothesis Hb: He = U(0, i) against HI: HF <~ disp U(0, I), i.e. h F ~ 1 on the support of HF (see for details Bartoszewicz, 1986). Tests which are appropriate for this problem are based on normalized spacings 1/i-

1~

~'i:n=oG - ~-'--~)Vi:,,

i = 2,...,n

and statistics

log ,n) and i=2

nlog

i=2

,o)

i=2

may be used. Bartoszewicz (1992) has studied asymptotic distributions of these statistics under Ho and also under a class of alternatives in the case, when G satisfies the following conditions: (i)

IoG-l(t + O ) - o G - l ( t ) l < ~ clOl~,t~[O, 1],

for some 6 > 0 and c i> 0 and for all 0 from some neighbourhood of zero; (ii) ~ ( t ) = [oG-l(t)] 1/~ is bounded away from zero on [2, 1 - 2] for some 2 > 0, non-decreasing (nonincreasing) on [0, 2] ([1 - 2, 1]) and

;1

texp[-~q~(t)]dt<°°

for a l l e > 0 ,

i=1,2,

where q , ( t ) = O(t)/x/~t and q2(t)= 0(1 - t)/x/tt. Bartoszewicz (1992) has considered a class of alternatives of the form

F(x) = G(K(x)),

(11)

where K is a piecewise linear increasing function, defined on the interval [A, B] ( - ~ ~< A < B ~< oo), with the derivative

k(x) = ~, ailt¢ .... ¢,)(x)

(12)

i=1

(defined of course everywhere except a finite number of points) m is a fixed integer, ai/> 1, i = 1, 2 ..... m, and aj > 1 for some j, and A = 4o < ¢1 < "'" < ~m = B are fixed. After applying the T T T transformation one

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217

obtain that under the alternative (11) and (12), (13)

hv(t) = ~ ajlgs_,,~}(x), j=l

where J ~j = ~ [ G ( K ( ~ I ) ) - G(K(~i-1))]/ai,

j = 1,2 ..... m,

(14)

i=1

i.e. hv is of the form (1) with C = ~,. < 1. Under the assumptions (i) and (ii) on the distribution G the statistics B,,., J. and (~. have the same asymptotic distributions as the statistics B.... J. and Q., respectively (see Bartoszewicz, 1992, Corollaries 2-4). Thus we can compute easily the Bahadur and Hodges-Lehmann slopes of these statistics for the alternatives given by (11) and (12) using Theorems 1 and 2. 5.2. Numerical examples

For calculating the efficiencies of the considered spacings statistics, notice that if a random variable X has the density (1), then E[f-'(X)]

(15)

= ~ ajX-rzj, j=l

E[logf(X)]

Lf(X)

= ~ ajzj log aj, j=l

_]

j=,

Var[logf(X)] =

(16)

z loga ,

(17)

ajzj log 2 aj j=l

ajzj log a t

,

(18)

j=l

Varl-lff(X)] = ~ zj _ C2 ' j = l aj

(19)

where zj = (j - (~_ l,j = 1, 2 ..... m. These quantities are symmetric functions of the pairs (aj, z j), i.e. they depend only on the lengths zi of subintervals [(j_ 1, (j) and the values at, j = 1, 2 . . . . . m, but not depend on their arrangement within the interval [0, C]. Besides, if f (x) = a t = ak for x e [(j- 1, (j) u [(k- l, (k), j ¢ k, one can join these two subintervals into the one of the length Zg + Zk: the moments (15)-(19) will be equal for two these cases. Thus there are many different alternatives of the form (1) for which slopes (2)-(4) and (8)-(10) and therefore respective efficiencies are the same. Sethuraman and Rao (1970) have studied the Pitman efficiency of spacings statistics when the sequence of alternatives has the form

L.(x)

G,(x) = x + hi~---T ,

x ~ [0, 1],

(20)

where L.(0) = L.(1) = 0, L. is twice continuously differentiable on [0, 1] and L., L'., L~' uniformly converge to functions L, (L(0) = L(1) = 0), L', L", respectively, with the same rate o(n-1/4). They have proved that under this sequence of alternatives the Greenwood statistic, Bz.., has maximal Pitman efficiency in a class of

218

J. Bartoszewicz / Statistics & Probability Letters 23 (1995) 2 1 1 - 2 2 0

the so-called regular spacings statistics. The statistics considered here belong to this class. That is the reason for which we calculate respective efficiencies relatively to B2.,. Let eB(T, B2..) and enL(T, B2,,) denote, respectively, the Bahadur and Hodges-Lehmann approximate ARE of a statistic T relatively to B2,, under a fixed alternative of the form (1). We present below tables giving these efficiencies for various types of alternatives. Table 1 gives values of the efficiencies for some alternatives with the support [0, 1]. It is worth to see, that if an alternative is far from the null hypothesis (A, B, C), tests based on B,,, with large r have great Bahadur efficiency and reversely, small Hodges-Lehmann efficiency. Efficiencies of tests based on J, and Q, are relatively small in these cases, except for enL in B. If an alternative is near to the null hypothesis (D and E), the test based on B2,, (the Greenwood statistic) is the best and the efficiencies of Bahadur, Hodges-Lehmann and also Pitman (see Sethuraman and Rao, 1970, for B,,. and J. under an appropriate sequence of alternatives approaching the null hypothesis) are almost the same. It is a well-known fact remarked by Bahadur (1960), that under normality of limiting distributions of test statistics the limiting Bahadur and Hodges-Lehmann efficiencies coincide with the Pitman one. Table 1 Bahadur and Hodges-Lehmann approximate efficiencies of some tests based on spacings, C = 1 A,m=2

T

B0.L. BI.L. B2., B2.5, n B3. n

B3.5, n B4, . J, Q,

B,m=3

C,m=5

zi

ai

zi

ai

2i

ai

0.5 0.5

1.5 0.5

0.1 0.1 0.8

6 3 0.125

0.2 0.2 0.2 0.2 0.2

0.33 0.67 1 1.34 1.67

eB 0.4496 0.8242 1 1.1729 1.3444 1.5148 1.6830 0.2388 0.6424

eHL 1.7035 1.3757 1 0.6725 0.4272 0.2596 0.1524 1.1523 1.6863

eHL 15.56 2.574 1 0.4156 0.1852 0.0872 0.0426 19.405 6.7573

eB 0.3185 0.7132 1 1.4003 1.9812 2.8466 4.1589 0.1590 0.4960

enL 2.1260 1.5723 1 0.5752 0.3126 0.1648 0.0855 1.3680 2.0891

D,m=2

eB 0.1295 0.4294 1 2.7322 8.4890 29.021 106.10 0.0748 0.2194 E,m=2

F,m=2

2i

ai

Zi

ai

zi

al

T

0.5 0.5

1.01 0.99

0.1 0.9

1.1 0.9889

0.01 0.99

0.01 1.01

eB

enL

eB

enL

eB

eaL

B0.5.. BI.L. B2.. B2.5,. B3, . B3.5. n B4,. J. Q.

0.6722 0.9677 1 0.9726 0.9001 0.7977 0.6795 0.3876 0.8623

0.6727 0.9680 1 0.9722 0.8993 0.7965 0.6780 0.3879 0.8628

0.7304 0.9946 1 0.9466 0.8528 0.7359 0.6106 0.4332 0.9112

0.7353 0.9974 1 0.9434 0.8467 0.7273 0.6004 0.4366 0.9157

7.391" 10- 4 5.186" 10 2 I 28.650 1040.2 43361 1.964" 106 1.453" 10- 4 4.706-10- 3

13.594 3.8031 1 0.3532 0.1486 0.0688 0.0336 2.8962 17.551

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219

Table 2 B a h a d u r a n d H o d g e s - L e h m a n n a p p r o x i m a t e efficiencies of some tests based on spacings, C < 1

T

Bo.5., BI.s., B2,n B2.5., B3, n

B3.5. n B4,, J, Q,

A, C = 0.5, m = 2

B, C = 0.75, m = 2

C, C = 0.99, m = 2

zi 0.25 0.25

ai 1.2 2.8

zi 0.25 0.5

ai 1 1.5

zi 0.09 0.9

al 1.1111 1

ea 8.2640 5.4036 1 0.3209 0.1280 0.0571 0.0272 1.8898 0.6419

eHL 2.5499 3.8505 1 0.4340 0.2274 0.1304 0.0784 0.3424 0.3107

eB 4.6947 4.6298 1 0.3638 0.1604 0.0776 0.0394 0.8259 0.8167

eHL 2.6174 3.9155 1 0.4188 0.2073 0.1099 0.0599 0.3595 0.5684

ea 3.1389 4.0740 1 0.4109 0.2034 0.1098 0.0618 0.4768 0.8593

eHL 3.0640 4.0432 1 0.4138 0.2062 0.1119 0.0634 0.4612 0.8460

Table 3 B a h a d u r a p p r o x i m a t e efficiencies of some tests based on spacings A, C = 0.999, m = 2

B, C = 0.999, m = 2

C, C = 0.999, m = 2

D, C = 0.999 m = 2

Zi 0.001 0.998

Zi 0.001 0.998

Zi 0.001 0.998

Zi 0.001 0.998

Bo.s.n B1.5,n B2,n Bz.5,n B3,a B3.5.n n4. n J,

Q,

al 1 1.001002

al 1.5 1.000501

al 1.75 1.0002505

ai 2 1

eB

ea

ea

eB

2.6934 3.8731 1 0.4320 0.2248 0.1274 0.0753 0.3884 0.8625

3.5335 4.2081 1 0.4005 0.1942 0.1033 0.0576 0.5668 0.8415

4.6560 4.5783 1 0.3706 0.1676 0.0836 0.0440 0.8318 0.8026

6.5625 5.1128 1 0.3333 0.1361 0.0616 0.0295 1.3244 0.7367

The alternative F also seems to be near the null hypothesis, but however, does not approach it in the sense of Sethuraman and Rao (1970). Hence the efficiencies are extremely different. Table 2 provides efficiencies for some dispersive alternatives: C < 1 and al > 1 but enough far off the null hypothesis. Table 3 gives only the Bahadur efficiencies (enL are almost the same with accuracy up to 10-4) for dispersive alternatives which are near the null hypothesis. In both these cases it is seen that the statistics Bo.5,. and B1.5,. have high etiiciencies.

Acknowledgement The author is very grateful to the referee for his valuable comments.

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