Empirical-distribution-function goodness-of-fit tests for discrete models

The Canadian Journal of Statistics Val. 24, No. 1, 1996, Pages 81-93 La Revue Canadienne de Statistique

81

Empirical-distribution-function goodness-of-fit tests for discrete models Norbert HENZE

University of Karlsruhe

Key words and phrases: Goodness of fit, sequence space, Poisson distribution, geometric

AMS 1991 subject classifcations: Primary 62G10; secondary 62630. distribution, parametric bootstrap.

ABSTRACT

We present a simple framework for studying empirical-distribution-function goodness-of-fit tests for discrete models. A key tool is a weak-convergence result for an estimated discrete empirical process, regarded as a random element in some suitable sequence space. Special emphasis is given to the problem of testing for a Poisson model and for the geometric distribution. Simulations show that parametric bootstrap versions of the tests maintain a nominal level of significance very closely even for small samples where reliance upon asymptotic critical values is doubtful.

RESUME

Nous prCsentons un encadrement facile pour Ctudier des tests d’ajustement d’un modtle discret fondts sur la fonction de rdpartition empirique. Un moyen principal de la dtmonstration est la convergence en loi d’un processus empirique discret avec des paramhtres estimb, considtrt comme une variable altatoire ii valeurs dans un espace de suites convenable. A titre d’exemple, nous considerons le cas d’un modtle Poisson et la distribution gtombtrique. Une ttude de simulation illustre que les tests, executts selon la mdthode de bootstrap paramktrique, conservent bien le seuil, mime en cas d’une Cchantillon petite dans lequel il est douteux de se fier ii des valeurs critiques asymptotiques.

1. INTRODUCTION AND SUMMARY

A crucial aspect of data analysis is the problem of testing the goodness of fit (GOF) of given observations with a probabilistic model.

In this respect, much work has been done in the case where the data are assumed to come from a continuous distribution. One of the classical methods is then to assess the GOF by measuring some kind of distance between the empirical distribution function (EDF) of the data and a suitably estimated distribution function from a hypothesized parametric family such as the class of normal distributions (see, e.g., d’Agostino and Stephens 1986).

The present paper is concerned with EDF tests for discrete data. Such data usually arise either from a distribution which is genuinely discrete or from a continuous distribution when the data have been grouped. EDF tests for discrete data have, apart from a Monte Carlo study given by Campbell and Oprian (1979), apparently only been considered in the case of testing for a completely specified distribution, which, however, is of limited practical value [see, e.g., Conover (1972), Horn (1977), Pettitt and Stephens (1977) and Altavela and Wood (1978)l.

82 HENZE Vol. 24, No. 1

Our interest in considering EDF tests for discrete distributions stems from the fact that, for assessing the goodness of fit to the Poisson model for counting data, several authors have recently advocated the use of the empirical generating function (EGF) instead of the EDF [see, e.g., Baringhaus and Henze (1992a), Nakamura and Ptrez-Abreu (1993) and Rueda et al. (1991)l. These tests may be regarded as counterparts to procedures developed for continuous distributions which are based on the empirical characteristic function [see, e.g., Epps and Pulley (1983), Baringhaus and Henze (1988) or Henze and Zirkler (1990)] and GOF tests based on the empirical Laplace transform [see, e.g., Baringhaus and Henze (1991, 1992b) and Henze (1993)l.

We suspect that EDF-based GOF tests for discrete distributions are not yet popular for the following reasons.

First, the pertinent asymptotic distribution theory for the underlying empirical process with estimated parameters (Burke et al. 1979), which uses the sophisticated machinery of strong approximations, does not cover a locally uniform convergence needed to perform a parametric bootstrap. For a bootstrap version of Durbin’s (1973) invariance principle in the continuous case, see Stute et al. (1993).

Secondly, the celebrated and popular chi-square (x2) test has the nice property that the test statistic is asymptotically distribution-free under the null hypothesis, provided that estimation of parameters is done properly. This entails that users of statistical methods usually perform the x2 test with critical values obtained from the asymptotic distribution. However, sample sizes occurring with real data are often not large enough to justify this procedure. A further severe problem inherent in the use of the x2 test is the fact that cell selection is not a clear-cut task. Since this is usually done after observing the data, manipulations of P-values are possible.

In this paper, we present a simple and natural framework for dealing with EDF-based GOF tests for discrete distributions. For the sake of lucidity, restriction will be to the case that the available data are of the counting type, i.e., they take values in the set No = {0,1,2,. . .} of nonnegative integers.

Denoting by Fn(-) the EDF of a random sample of size n from a discrete (No-valued) distribution, and letting F(-,&,,) be a suitably estimated distribution from a hypothesized parametric family { F(., 6) : 6 E Q}, the main idea is to regard the so-called discrete estimated empirical process Z, = (Z,,k)kzo, where Zn,k = &{ F,(k) - F(k, &,,)}, as a random element of the Banach space co of all real sequences x = (xk)kzo converging to zero, equipped with the norm llxll = max IxkIk>o. This will be done in Section 2. As far as we know, the space CO, which is much easier to handle than the Skorohod space D [ - w , w ] , has not yet been used in the context of GOF testing.

In Section 3 we prove that, for a triangular array X,,,, . . . ,X,, having distribution F ( . , 6 , ) such that 6,, + 6, 6 E 0, and provided that certain regularity conditions hold, the discrete estimated empirical process Z, converges in distribution to a Gaussian sequence in co with a covariance function depending on the “true” parameter 6 E 8. In the i.i.d. case, this result follows from Theorem 3.1 of Burke et al. (1979). As a consequence, well-known EDF statistics such as the Kolmogorov-Smirnov or Cramtr- von Mises have limiting distributions under F( . , 6). Since these limit laws depend on the unknown value of 6, we recommend the use of a parametric bootstrap to perform the tests in practice.

In Section 4 we apply our general results to the problem of testing for the Poisson model and for the geometric distribution. It will be seen that bootstrapping yields excellent adherence to the nominal level of Kolmogorov-Smirnov and Cram&-von Mises tests for a wide range of parameter values, even for small sample sizes.

1996 GOOONESS-OF-FIT TESTS 83

2. THE SETTING

On a common probability space (B,A , P), let X,Xl,Xz,. . . ,X,, . . . be a sequence of independent and identically distributed (iid) random variables taking nonnegative integer values. Write F(t) = P(X 5 t ) for the distribution function of X, and let

Ff3 = { F ( . , 6) : 6 E €3)

be a parametric family of discrete distribution functions such that the set of points of discontinuity of F ( . , 6 ) is a subset of NO = {0,1,2,. ..}. The parameter space 0 is assumed to be an open set in Euclidean space RS. A classical problem of statistical inference is then to test, on the basis of XI,. . . ,X,, the hypothesis

Ho : F(*) E Ff3

against general alternatives. Denoting by F,,(t) = It-' & 1{% 5 t } the EDF of XI,. . . ,X,,, a natural appro!ch would be to estimate the unknown parameter (vector) 6 by a suitable estimator a,, = b,,(Xl, . . . ,X,,) and reject HO for large values of the Kolmogorov-Smirnov statistic

or the CramCr-von Mises statistic

m

c,, = It C{ F,,(k) - F(k, b")}*{ F(k, 6,) - F(k - 1, b,,)}. (2.2) k=O

Note that K,, and C,, are functionals of the estimated (discrete) empirical process

z n = (Zn,k)kzO, (2.3)

where Z,,& = ,h{ Fn(k) - F(k, &,,)}. Since limk, Z,,& = 0 almost surely, a convenient setting for asymptotic theorems

will be the separable Banach space co of all sequences x = (Xk)kzO converging to zero, equipped with the norm 11x11 = S U P k a I X k I . The a-algebra B of Bore1 sets of co is generated by the spheres S(x, E) = { y E co : IIx -yll < E} (x E CO, E > 0), and it is easily seen that B coincides with the projection o-algebra !M , i.e., the smallest a-algebra on co such that the projections co 3 x t-, nkx := Xk (k 2 0) are measurable. Since Z,, is a mapping from B into co for which JCk 02, = Zn,k is a random variable, i.e., Borel- measurable, it follows that 2, is a random element of CO, i.e., (A , !M )-measurable. Since !M = 53, the distribution Q,, = P oZ,,-' is a Borel probability measure on CO. To show convergence in distribution of Q,, to a probability measure Q or of 2, to a random element 2 = (zk)k>O (this will be denoted by Q,, - Q Or Z,, - 2, respectively), we have to prove the weak convergence of finite-dimensional distributions and verify that the sequence (2, : n _> 1) is tight (see, e.g., Billingsley 1968).

Since a simple exercise shows that a subset A of co is relatively compact with respect to the topology induced by the norm 11 . 11 if and only if

!D I,


and lim sup sup lxjl = 0,

k- x€A j z k

the following lemma gives a necessary and sufficient condition for tightness in the measurable sequence space (CO, B ).

LEMMA 2.1. A sequence {Q,, : n 2 1) of probability measures on (CO, B ) is tight if and only if these two conditions hold:

(i) For each positive 6 and 1 2 0, there is a finite constant M such that

(ii) For each positive 6 and q, there exists an integer 1 such that

Proof. We only prove that (i) and (ii) are sufficient conditions for tightness. To this end, fix E > 0. Condition (ii) implies that for each integer j there is an integer 1( j) (depending also on E) such that Qn(A,) 2 1 - E - 2-(j+'), n 2 1, where

From condition (i) we obtain constants Mo, ..., M1(1)-1 (depending on E) such that en(&) 2 1 - ~/{21(1)}, n 2 1, where Bk = { x E co : lxkl 5 Mk} [k = 0,1,. . . , l(1)- 11. Letting

00

A = B o n . . - n B l ( l ) - l n n A j , j = 1

an appeal to (2.4), (2.5) shows that the closure K of A is compact. Since Qn(K) 2 1 - E,

n 2 1, the sequence {Qn : n 2 1) is tight. 0

In the following, Pe generically denotes the distribution induced by F ( - , a), '& means expectation under Pe, and wT is the transpose of a row vector w.

In view of the bootstrap procedure described later, it is necessary to generalize the setup specified at the beginning of this section. To this end, let XnlrXn2,. . .,X, (n 2 1) be a triangular array of rowwise iid random variables (defined on a common probability space that may depend on n) having the distribution function F ( - , en), where

Retaining the notation 6" = 6,,(Xn1,Xn2,. . . ,X,,) for an estimator of 6 based on Xnl, Xn2,. . . , X,, we need the following regularity condition on the sequence (6,,)n21 [see conditions 3.1(i), (ii) and (iii) in Burke et al. (1979)l.

ASSUMPTION Al . There is a measurable function 1 : No x 0 + R" such that

1 996 GOODNESS-OF-FIT TESTS 85

where E, = open (1) as n + 00 and

Ee[i(X, S)] = 0, 6 E 0

where 0 is the origin in RS. For fixed k 2 0, l(k, .) is a continuous function of 6. Furthermore,

W ) = %[l(X, 431, 6 E 8,

defines a finite nonnegative definite matrix which depends continuously on 6.

Writing VeF(k,6*) for the gradient vector of F ( k , 6 ) evaluated at 6*, and letting 11 - 112 denote the Euclidean norm in R’, the following regularity assumption specifies a smoothness condition on the distribution function F(- , 6).

ASSUMITION A2. For fixed k 2 0, V e F ( k , 6 ) exists and is a continuous function of 6. Moreover,

lim sup IIVeF(k, **)I12 = 0, k- cr€u(e)

where U (6) is a sufficiently small neighbourhood of 6.

It is interesting to compare the condition (2.7) with condition 3.2(iv) of Burke et al. (1979). They assume that v e F ( x ; 6 ) is uniformly continuous in x and 6 E A, where A is the closure of a given neighbourhood of (the true value) 60. With respect to the discrete case, this formulation is a bit misleading. The condition actually needed for their Theorem 3.l(a) is continuity of VeF(x;6) in 6, uniformly in x , i.e.,

and this is a consequence of (2.7).

3. MAIN RESULTS

For the formulation of the main weak-convergence result, let

k

J(k, 6) = I(r, 6){ F(r, 6) - F(r - 1, 6)}, k L 0, 6 E 0, (3.1) r=O

with l(k, 6) figuring in Assumption Al.

THEOREM 3.1. Under Assumptions A1 and A2, there is a Gaussian sequence ‘W = (wk)ka in Co such rhat [ wk] = 0, [ wk wm] = ce(k, m) (k, m 2 o), where

Ce(k, m) = F(min(k, m), 6) - F(k, 6)F(m, 6)

- J(k, 6)VeF(m,

t ~ e ~ ( k , 6)0(6)VeF(m, 6)T.

- J(m, 6)Vef’(k,

iid Under the triangular array X , , , . . . ,X,, N Pen, we have


where 2, = (&,&)kzO is given in (2.3).

Proof. Fix k 2 0 and write Zn,k = Tn,k -b v n , k , (3-2)

where T,,k = ,h{ F,,(k)-F(k,6,,)}, V n , k = &{ F(k,e,,)--F(k,+,,)}. A standard Taylor expansion and Assumptions Al , A2 yield

and thus

where Ynj(k) = 1{&, 5 k } - F(k,6,,) - VeF(k,6,,)l(Xn,,6,,)'. By straightforward algebra we obtain En[ Y,,,(k)] = 0, En[ Y,,j(k)Y,,j(m)] = C,(k, m), where E,, (C,,) is shorthand for (Ce"). Since lim- C,(k, m) = Ce(k, m), the convergence of finite- dimensional distributions follows from the Lindeberg-Feller central limit theorem and the CramCr-Wold device.

It remains to prove that the sequence {Z,, : n 2 1) is tight under P,, (:= Pen). Since weakly convergent sequences are tight, condition (i) of Lemma 2.1 has already been established. To verify condition (ii), fix positive numbers 6 and q. We have to show the existence of integers 1 and no such that

From (3.2) and the triangle inequality it is enough to have

for some integers 1 and no. To verify (3.3), let (without loss of generality) X,,, = inf{x : F(x, 6,) 2 U,,,}, where U,,,, . . . , U,,, are iid uniform (0,l) random variables, and write a,,(u) = nf (n - l l{Unj I u } - u), 0 I u 5 1, for the uniform empirical process based on U n l , . . . , U,,,,. Choose E > 0 and an integer no such that

Furthermore, choose 1 so that F(k,6,,) > 1 - E for each k 2 1 and each n 2 no. Since, for such k and n,

1996 GOODNESS-OF-FIT TESTS 87

(3.3) follows. To verify (3.4), let U (6) = {a E 0 : 116 - 8112 5 T} be a small neighbourhood of 6 for which the statement of Assumption A2 holds. It follows that the probability occurring in (3.4) is bounded by

Here, the first term tends to zero (and thus is smaller than 6/4 for sufficiently large n), since (fi(&n-6n))n21 = Opn(l) by Assumption Al. Regarding the second term in (3.9, note that

for some 6; satisfying 11% -6, , l l2 5 I[%,, -6,,,112. If n is large enough that 11% - 611~ 5 212, it follows from the triangle inequality that, on the set {I@,, - Sn{ l2 5 z/2},

I v n , k l 5 IlvVF(k 63112 ' I l f i @ n -*")I12

where ak = s u p p , ~ (*)llV~F(k, 6*)112. We may therefore bound the second term in (3.5) for sufficiently large n by

which, since liml- SUpk>/ a k = 0 by Assumption A2, tends to zero in view of the tightness of (fi(%n - 6 n ) ) n l l .

REMARK 3.2. Note that, by Chebyshev's inequality, an upper bound for the probability occurring in (3.3) is given by 4q-2 &I{ 1 - F(k, en)}. Thus, imposing the additional condition of a finite mean that depends continuously on 6, Theorem 3.1 may be proved without falling back upon any empirical process theory at all. This condition holds for all standard families of discrete distributions.

From Theorem 3.1 and the continuous-mapping theorem we get the following result.

COROLLARY 3.3. Let h : co -+ R be a continuous functional with respect to 11 - 11. Then, under a triangular array Xnl , . . . ,Xnn - Pen, we have

0

i id

h(2,) 5 h(" ).

COROLLARY 3.4. Under P 6 6 E 0, we have

and M

k=O

Proof. The first assertion follows from Corollary 3.3, since the functional hl(x) = IIxII, x E co, is continuous. To prove the second assertion, let h2(x) = ox^{ F(k, 6 ) - F ( k -

88 HENZE Vol. 24. No. 1

I, 1,6) ) , x E CO. Note that h2 is a continuous functional on CO, and that h,(Z,,) + h,(‘MI ). Moreover,

00

Ih2(Zn) -cnI 5 xz: ,k/f(k, &n) -f(k k=O

00

5 h:(Zn) C ~f(k, +n) -f(k, k=O

where f ( k , 6 ) = F(k, 6) - F(k - 1,6). Since h:(Z,,) = Op,,(l) and CEO If(k,&,,,,) - f ( k , 6 ) ( = op,(l) by Scheff6’s theorem, we are done. 0

REMARK 3.5. To calculate the statistics K,, and C,, in practice, observe that, with M := maxlys,, Xi, we have

and I

Cn = n Fn(k) - ~ ( k , bn))2f(k, +n) + R, k=O

where, for 1 2 M, the remainder R satisfies 0 5 R 5 n{ 1 - 17(l,&,,)}~. We recommend the choice

1 := min{m 2 M : n{l - F(m, &,,)}3 5 which should provide a sufficient numerical accuracy for all practical purposes. Alterna- tively, one may use the modified Cram6r-von Mises statistic

00

c,* = n Fn(k) - ~ ( k , &n)}*{ Fn(k) - Fn(k - I)), (3.6) k=O

which involves at most M + 1 nonnull.summands. It is easily seen that C,* has the same limiting null distribution as C,,.

It is clear that a test based on a test statistic T,, = T,,(X,, . . . ,X,) such as K,,, C,, or C,* should reject the hypothesis ~0 if T,, exceeds a critical value c. Note that c depends on the sample size n and on the desired level of significance a, 0 < a < 1, but also on the unknown true value of 6. To perform the test we suggest a parametric bootstrap, i.e., estimating c from the data. To be precise, let

Hn,e(t) = PdTn 5 t) (3-7)

be the distribution function of the null distribution of T,, when 6 is the true parameter value. Then a natural critical value for T,, would be the (1 - a)-quantile of H,,,gn. Since the latter is difficult to calculate, it will be estimated by the following Monte Carlo procedure.

Given XI,. . . , X,,, first compute 6,, = *,,(XI,. . . &). Then, conditionally on en, let Xi;, Xi;, . . . , X,’, 1 5 j 5 b, be independent and identically distributed random variables with distribution function F(-,&,,) and compute q,, = Tn(Tl,. . . , X,’), 1 5 j 5 6. Note that, to compute q:,,, parameter estimation has to be done separately for each j . Writing

. b


for the EDF of T;,,,,. . . , T:.,,, the ( 1 - a)-quantile ci,b(a) = (Hi,b)-'(I - a) of Hi,b is

where T;:b 5 T&, 5 - . - 5 T::b are the order statistics of T;,n,. . . , T:,,,. The hypothesis 9 f o is rejected at level a if T,, exceeds c,*Ja).

We now prove that, under general conditions, this bootstrap test for 9& has asymptotic level a as n and b tend to infinity. To this end, assume without loss of generality that the underlying probability space is rich enough to carry a sequence U1, U2,. . . of independent uniform [0,1] variables, independent of the Xi . As a consequence, we may take

(3.10)

where H,,@ is given in (3.7). In the following, let llGlloo = ~up-,~~~,lG(t)l for a function G:R--+R.

THEOREM 3.6. Let T,, = h(Z,,), where h is a continuous functional on cg, and let He(t) = P(h('u) ) 5 t), with 'u) = (Wk)ktO figuring in the statement of Theorem 3.1. Assume that He is continuous and strictly increasing on the set {t : 0 < He(t) < 1). Under the standing assumptions, we then have:

(a) IIH;, -Helloo 2 0 as n, b -+ 00, where Hi, is the bootstrap distribution function

(b) ci,b(a) -+ HG1(1 -a) as n, b -+ 00.

(c) lim Pe(T,, > ci,b(a)) = a as n, b + 00.

q:,, = Hn3"(U,) = inf{t : H,,,%"(t) 2 Uj}, 1 5 j 5 b,

defined in (3.8). Ptl

Proof. From Corollary 3.3 we have lim- IIH,,en -Hell, = 0 for any sequence 6,, E 8 converging to 6 as n -+ 00. On combining this with the convergence in probability of a,, to 6 we obtain

pa llH,,,gm - Hellm - 0 as n -+ 00.

From (3.10) we have (3.11)

where the last expression converges to zero almost surely as b + 00. In view of (3.11), the first assertion follows. Since ci,b(a) = (H&)-'(l- a), (b) is an immediate consequence of the continuity and strict monotonicity of He. (c) follows from (b). Note that we have almost sure convergence in (a) and (b) provided that &,, -+ 6 Pe-a.s.

As a consequence, the parametric bootstrap "works" for K,, == llZ,,ll and C,,, C,t, since C,, and C,t may be approximated by h*(Z,,), where h2(x) = E,"=,x,'f(k,6) (see the proof of Corollary 3.4).

REMARK 3.7. Regarding consistency of the bootstrap tests based on K,,, C,, and C,t, let XI ,X2,. . . ,X,,, . . . be a sequence of independent and identically distributed No-valued random variables with distribution function F such that

0

(3.12)


We claim that the consistency properly

lim P(K,, > c;,Ja)) = 1 (3.13) n, b-x

holds if b,, = &,,(XI,. . . , X,,) converges in probability to some 6 E 0. To this end, recall H,,e from (3.7) and H i b from (3.8). Now, Theorem 3.1 yields

lim-llHn,&n -Helloo = 0 in probability and thus (see the proof of Theorem 3.6) llH,*,,b - Hell, -t 0 in probability as n, b -t 00, whence

Writing F,, for the EDF of XI,. . . ,X,,, note on the other hand that

where lim inf- 11 F - F(.,b,,)loo 2 p as . and limn- 11 Fn -Flloo = 0 a s . from the Glivenko-Cantelli theorem. In view of (3.12), this shows that lim- K,, = +oo as., which, combined with (3.14) proves (3.13).

Since, for the Cram&-von Mises statistic C,, (and likewise for Cf), we have

5 2 { F,(k) - F(k, b,,)}’{ F(k, b,,) - F(k - 1, b,,)} n

for each k 2 0, the convergence of b,, to some 6 E 0 and the Glivenko-Cantelli theorem yield C,, -t 00 and thus the consistency of the Cramtr-von Mises test under the condition

A := inf sup{ ~ ( k ) - ~ ( k , b)}’{ ~ ( k , b) - ~ ( k - I, b)} > 0. (3.15)

P

k e k>O

Note that (3.15)-which, since h 5 p2, is slightly stronger than (3.12)-is “tailored to C,,” in the same way as (3.12) is “tailored to K,,”. However, at least under general conditions, (3.12) is not sufficient to ensure the consistency of the CramCr-von Mises test [consider examples where the alternative F puts mass on points k not supported by the family { F ( . , 6 ) : 6 E O } ] .

4. EXAMPLES

As a first example, we consider the problem of testing for Poissonity. In this case, 8 = (6 E R : 6 > 0) and F(k, 6) = e-’ c=, 6 j / j ! . It is readily seen that Assumption A2 is satisfied, and Assumption A1 obviously holds for &,, = n-’ & Xi.

Although the covariance function of the Gaussian sequence ‘J# figuring in Theorem 3.1 may easily be obtained [note that J ( k , 6 ) defined in (3.1) takes the simple form J ( k , 6) = -e-e&t’/k!], there is little point in recording the algebraic details. Bearing in mind that sample sizes occurring in practice are often not very large, it is the very idea of the parametric bootstrap to avoid the use of an asymptotic distribution which, besides, depends on an unknown parameter. The main thing is that the (parametric) bootstrap versions of K,,, C,, and Cf provide asymptotically level-a tests for the Poisson model (see Theorem 3.6).


TABLE 1: Empirical level of the Kolmogorov-Smimov test for the Poisson model based on loo0 Monte Carlo replications; (I = 0.1.

~ ___ ~

Level

n 6=1 5 10 25 50 100

5 0.097 0.104 0.092 0.107 0.101 0.091 10 0.104 0.108 0.086 0.103 0.108 0.104 12 0.094 0.096 0.086 0.082 0.093 0.091 15 0.088 0.107 0.099 0.105 0.097 0.100 20 0.102 0.080 0.099 0.098 0.112 0.107

TABLE 2: Empirical level of the Cram&-von Mises test for the Poisson model based on lo00 Monte Carlo replications; CI = 0.1.

Level

n 6=1 5 10 25 50 100

5 0.105 0.102 0.090 0.110 0.099 0.093 10 0.101 0.100 0.087 0.103 0.110 0.104 12 0.092 0.082 0.088 0.088 0.087 0.082 15 0.094 0.103 0.104 0.105 0.101 0.106 20 0.103 0.091 0.090 0.100 0.106 0.102

Moreover, the discussion of Remark 3.7 shows that these tests are consistent against each fixed alternative distribution having finite positive expectation, and even that restriction may be dropped. To see this, note that, for an alternative distribution having infinite expectation, we have 6, -t 00 as., which entails that K,,, C, and C,* tend to infinity almost surely. On the other hand, the critical value &(a) stays bounded in probability for almost all sample sequences Xl ,X, , . . ., since the jth resampled value of the test statistic is based on the sequence

Here a.(u), 0 5 u 5 1, denotes the nth uniform empirical process as in the proof of Theorem 3.1, and @I,, = n-' x=l Xiy, where, conditionally on &,,, the random variables X i . . ,Xin are independent with the Poisson(&,) distribution. Now, the first term in (4.1) is bounded uniformly in k by supoluh~lan(u)l, and, using a Taylor expansion and the fact that fi(&,, - ern)&? stays bounded in probability almost surely, also the second term in (4.1) is bounded in probability almost surely, uniformly in k [note that V&k, 6) = - e - W / k ! ] .

To gain some insight into the actual level of the bootstrap test based on K,, and C,, a Monte Carlo experiment was performed for various sample sizes n and the nominal level of significance a = 0.1. A total of lo00 Monte Carlo samples were generated for each n, and the bootstrap sample size b, as in Stute et al. (1993) was taken to be 500.

Each entry in Table 1 is the estimated actual level (percentage of rejections of the hypothesis of Poissonity) of the Kolmogorov-Smirnov test based on lo00 Monte Carlo samples for a wide range of values of 6 and the nominal level a = 0.1. The corresponding results for the CramCr-von Mises test are given in Table 2.

= { F(- , 6) : 0 < 6 < l}, F(k, 6) = 1 - (1 - 6)'+', k 2 0, of geometric distributions. In this case (where Assumption A2

As a second example, we consider the class


TABLE 3: Empirical level of the Kolmogorov-Smirnov test for the geometric model based on lo00 Monte Carlo replications; a = 0.1.

Level

n 6 = 0.7 0.6 0.5 0.4 0.3 0.2

10 0.079 0.089 0.102 0.116 0.115 0.110 15 0.088 0.095 0.082 0.116 0.104 0.112 20 0.096 0.116 0.085 0.117 0.112 0.106 30 0.091 0.089 0.097 0.106 0.106 0.098 50 0.104 0.104 0.095 0.114 0.108 0.106

is easily verified) we may choose the ML estimator satisfies Assumption A1 with the function

= (1 t n-' '& q)-', which

1 - (k t 1)s

W1-6) . l(k, 6) =

Table 3 gives the estimated actual level of the Kolmogorov-Smirnov test for the hypothesis that the underlying distribution is geometric for some parameter 6. The nominal level a is 0.1, and each entry is based on 1000 Monte Carlo replications. As for the Poisson case, the bootstrap sample size b is 500.

The results show that the nominal level is maintained very closely for a large range of parameter values. The characteristics of the CramCr-von Mises test for the geometric distribution are completely similar.

ACKNOWLEDGEMENTS

The author would like to thank Bernhard Klar for assistance in computation and an Associate Editor and the referees for their careful reading of the manuscript and many valuable suggestions.

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Received 26 March 1995 Revised 16 August I995 Accepted 13 September I995

Universitat Karkruhe Irutitut f i r Mathematische Stochastik Englerstrasse 2, 0-76128 Karkruhe

Germany

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Empirical-distribution-function goodness-of-fit tests for discrete models