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Two-sample goodness-of-fit tests when ties are present
Journal of Statistical North-Holland
Planning
and Inference
Two-sample goodness-of-fit when ties are present Arnold
tests
Janssen
Mathematisches Germany
Received
399
39 (1994) 399-424
Institut,
29 February
Heinrich-H&e-Universitiit
1993; revised manuscript
Diisseldorf,
received
Universitiitsstr.
1, 40225 Diisseldof
8 June 1993
Abstract The paper contains asymptotic results for two-sample Kolmogorov-Smirnov, Cram&van Mises, Anderson-Darling and other related tests. Whenever the portion of ties cannot be neglected, two procedures are proposed to make the tests (asymptotically) distribution free. The tests can either be carried out as permutation tests or as tests with estimated critical values. All tests turn out to be asymptotically equivalent and their asymptotic power function is established under local alternatives. The same results apply to one-sided tests for stochastically larger alternatives where noncontinuous limit distributions of the test statistics may appear. AMS
Subject Classification:
Primary
62GlQ
Secondary
62G20.
Key words: Ties; permutation tests; Kolmogorov-Smirnov Darling test; asymptotic power function.
tests; Cram&-van
Mises tests; Anderson-
1. Introduction Goodness-of-fit tests are asymptotically distribution-free under the null hypothesis as long as the underlying distribution function F is continuous. We refer to the monograph of Shorack and Wellner (1986) for one-sample tests and HAjek and SidAk (1967, Ch. V), Chibisov and Durbin (1973, p. 39) in connection with two-sample problems where detailed results can be found. However, practical sets of data usually contain ties, and noncontinuous distributions have to be taken into account. In principle, the device for the treatment of Kolmogorov-Smirnov tests under ties is contained in the literature as a special case of more abstract empirical process theory. For instance, Bickel(l969, p. 7) proposed to carry out these tests as permutation tests. He showed consistency and asymptotic equivalence of conditional and unconditional
Correspondence to: A. Janssen, Universititsstr.1, 40225 Diisseldorf,
Mathematisches Germany.
Institut,
Heinrich-Heine-‘Jniversitit,
0378-3758/94/$07.00 0 1994 ~ Elsevier Science B.V. All rights reserved. SSDI 0378-3758(93)E0049-M
Diisseldorf,
400
A. Janssen 1 Two-sample
goodness-of--fit tests when ties are present
tests within a more general setting, see also Roman0 a recent discussion. Throughout, for univariate
(1989) and Praestgaard
(1991) for
we will have a closer look at permutation goodness-of-fit tests tied data. We will also treat integral tests (like the Cramer-von
Mises and Anderson-Darling tests) and one-sided tests for stochastically larger alternatives under ties. It turns out that the limit distribution function under the null hypothesis is not necessarily continuous, which requires further effort. (Note that condition A of Roman0 (1989) is violated.) Section 4 deals with asymptotic power functions of two-sided Kolmogorov-Smirnov tests under local alternatives. Extending earlier results for continuous distributions, we obtain asymptotic admissibility and strict asymptotic unbiasedness of these tests. Whenever the portion of ties cannot be neglected, a further procedure with estimated critical values (Theorem 3.3) is proposed in order to make the tests asymptotically distribution free. Below we will make a few comments concerning references for one- and two-sample Kolmogorov-Smirnov tests if ties are present. It is well known that Kolmogorov-Smirnov tests are conservative and remain valid if ties are ignored; see Kolmogorov (1941), Noether (1963) or Guilbaud (1986) for a recent discussion. In connection with one-sample problems we mention Gleser (1985), who calculated the power under discontinuous distributions. Guilbaud (1986) established stochastic inequalities for KolmogorovSmirnov statistics which are reflected in our Lemma 2.1 within the asymptotic set-up. Acceptance regions for the Kaplan-Meier estimator appear in Guilbaud (1988). Two-sample omnibus tests are studied in Behnen and Neuhaus (1989) also when ties are present. They studied rank tests with estimated scores. In conclusion we give the following recommendation for practical purposes: (1) In the case of small or intermediate sample size the tests should be carried out as permutation tests. For computational reasons a Monte Carlo simulation may necessary. (2) Alternatively, the critical values can be estimated (Theorem 3.3), whenever large. However, these procedures require also Monte Carlo simulations via Brownian bridge. These calculations may be easier as in (1) when a lot of ties
be it is the are
present. (3) If only a few ties show up, they may be ignored for large n and the use of asymptotic critical values seems to be justified. The paper is organized as follows. Section 2 explains the results for two-sample problems. The main results are given in Section 3 and the treatment under local alternatives can be found in Section 4. All longer proofs are represented in Section 5. The results about asymptotic distributions of the Brownian bridge under semi-norms are summarized in the appendix. We will use the following notation. Let 2(X1 P) denote the law of a random variable X under P. The uniform distribution on the unit by supp(P). By interval of Iw is denoted by /I,,,, 1). The support of P is abbreviated
A. Janssen / Two-sample goodness-of-fit tests when ties are present
definition
Xj: n stands for the j-th smallest
and 7
indicate
tion, let f’
among
(P-probability,
X1, . . . ,X,. respectively).
Let z In addi-
tests
This section
explains
and motivates
for the following
Example 2.1. Consider specified by i.i.d. random within group 1 and i.i.d. F2 for group 2. Let us hypotheses:
where K, denotes
procedures.
The results
are first
a two-sample testing problem of sample size yl=nl + n2 function F1 variables Xi, . . . , X,, with common distribution random variables X,, + i, . ,X, with distribution function examine two-sample goodness-of-fit tests for testing the
K1 being the omnibus alternatives, unknown, or for testing Ho: F,=F2
the testing
example.
Ho: F1 = F2 against
Throughout,
order statistic
in distribution
= max(A 0).
2. Two-sample
presented
convergence
401
against
the one-sided
K1: F1 # FZ,
(2.1)
where F1 and F2 are assumed K2: F,3F2, alternative
to be completely
F,#F2, that F2 is stochastically
(2.2) larger than Fi.
let (2.3)
(2.4)
denote the empirical distribution functions of group 1,2 and the pooled respectively. Typically, goodness-of-fit tests are based on test statistics I(&(.)-fin(.))
or
Z((c,(.)-fin(.))+)
sample,
(2.6)
on a suitable function space. As for K1 or KZ, where Z( .) denotes a semi-norm motivation, let us first consider the sup-norm IlfI/:=~up,,~ If(t)1 leading to Kolmogorov-Smirnov tests. As extension of classical invariance theorems, we obtain the limit distribution of (2.6) for arbitrary distributions. Let (Bo(t))tEro.ll denote a standard Brownian bridge. Lemma 2.1 is known as a special case of more abstract empirical process theory.
402
A. Janssen / Two-sample goodness-of-fit tests when ties are present
Lemma
2.1. Assume that X1, . . . ,X,
suppose
that n1 ~n2-+co.
are i.i.d. with joint
distribution
function
F and
Then
112
nln2
supI &(t)-&(r)+
(> n
(2.7)
sup I&(F(t))l
tPR
toi?%
and nl
n2
l/2
sup (d,(t)-&(t))+
C-1 n
5
supB,(F(t))+,
tER
(2.8)
fElR
both in distribution.
For completeness
we will give a proof of Lemma 2.1. Let us introduce
the following
abbreviations:
(I{
c ,.= ~ n,.
li2 lh,
n
-
i=l, ..
(2.9)
i=n,+l,...,n,
Un2,
where the coefficients satisfy Cy=, cii= 1 and max,,iGnlc,il+O Under H,, we may assume
whenever
Xi:=F-‘(Ui),
(2.10)
where F- ’ denotes the left-continuous U1, LIZ, . . . are i.i.d. uniformly distributed
inverse (or quantile on (0,l).
Proof. Taking (2.9) and (2.10) into account, u~(0, l), proves the following representation:
0
nln2 n
the equivalence
(~n(t)-fin(t))=i
principle if
l(-m,t](xi)
t iff u< F(t),
cni
for weighted
(0.~1 C”i)
processes
sup If(s)I soF(R)
on D[O, 11. Then standard plete the proof. 0
implies (2.12)
~(~0(4)s.r0,11
cni ) SE[O,
i=l
empirical
(2.11)
cni.
11
in distribution on D[O, 11, see Shorack and Wellner consider the almost surely continuous functions f~
F-‘(u)<
of F and
i=l
t”i)
(
function)
l/2
=i$ll[O,F(r)l The invariance
n, An2+co.
and f~ arguments
(1986, pp. 88 and 93). Next
(2.13)
sup (f(s))’ ssF(W) of Billingsley
(1968) together
with (2.11) com-
403
A. Janssen / Two-sample goodness-of-fit tests when ties are present
From
Lemma
2.1 we derive
Kolmogorov-Smirnov
easily
the
asymptotic
null
distribution
of the
tests: (2.14)
(2.15) where Z(f)= I/f/l. Evidently, (Pi and $. are asymptotic level CI tests if our critical values c and d are the (1 - u)-quantiles of the limit distributions (2.7) and (2.8) as long as F is not a Dirac distribution and additionally in the case K, we have a< l/2; cf. Remark A of the appendix. However, the test statistics are no longer distributionfree since c=c(F([W)) and d =d(F(R)) depend on the range of the unknown distribution function F. Note that qo, and $,, become conservative if c and d are substituted by the (1 - U)quantile of Smirnov’s statistic (continuous case). Observe also that c(F”(R))
d(F”(R))
and
(2.16)
whenever ~(R)cF(R). A finite sample result of this kind was obtained Guilbaud (1986). To overcome problems with unknown critical values, we will construct of-fit tests that are distribution-free. In particular, we propose permutation fix the idea, consider first the Kolmogorov-Smirnov statistic: SUP
i
l(-m,Xj:,](xi)cni
Permutation permutations
>
goodnesstests. To
I
tests are based on uniformly of 1, . . . , n)
distributed
permutations
(on the set Y,, of
GwH~ni(L3))i
~n=(“ni)r
defined
by
(2.17)
.
(+)
IGIl I( i=l
earlier
on a further
probability
OCQ the permutation
statistic
space (6,2,
P) independent
of Xi : Q+R.
For fixed
of (2.17) is given by (2.18)
which is equal in distribution &++sup j
to
i$I lc-m,X,:,(o)J
For fixed w let x H Gr)(x, w) denote the distribution random variables &(o) and y,(w)~[O, l] as solutions
s
(2.19)
~xi:~~~~~c~~~~~~~)~+~~~
I(
(l(~,,na)(x)+~nl(~.)(~)}G,(dx,o)=a.
function of
of (2.19). Next choose
(2.20)
404
A. Janssen / Two-sample goodness-of-fit tests when ties are present
Then the permutation
test for H,, against
1
&:=
I y.
I
((
0
111112 n >
K1 >
112 (G,(t)-H,(t))
is an exact level c1 test for each F and each
n.
(2.2 1)
=I?"
)
<
Similarly,
permutation
obtained for K2; see (3.20). The main results of the present procedures work well for a large class of semi-norms.
Remark
2.1. To
u 1, . . . , U, defined bution of
paper
tests 6” are
show that these
give further motivation, consider antiranks (D,i(U))i<” of implicitly by Ui: n= UDni(“). Then we have equality in the distri-
by (2.10) and (2.11), since (F-’ (Ui:n))i
If (2.17) is
(2.23)
SUP
jan we get further insight in the permutation new permutations gn. For computational Kolmogorov-Smirnov can be recommended.
statistic (2.19). The antiranks
are replaced
by
reasons one may be interested in additional procedures of type for large n. In this connection estimated critical values Note that the right-hand side of (2.7) reads as (2.24)
sup IBo(s)I. seF(W)
It is reasonable to estimate the unknown fixed w let cn(. , co) denote the distribution
range F(R) by P,(R)= function of
{F^,(Xi):
i
For
(2.25) and
let C,:= c,(. , co)- ’ (1 -a)
critical
value C. similar
be its (1 -a)
to (2.21). It is shown
quantile.
Let
that (Pi - (P,,p
(P,, be the
test
with
0 and (P,,- & p
in probability under nondegenerate distributions of Ho. In this case the nominal level of (P,, converges to CIunder Ho as n+m. The critical values E,, can be obtained by a Monte Carlo study. Similar results hold for one-sided tests.
0
405
A. Janssen J Two-sample goodness-of--fit tests when ties are present
3. Main results In this section
the motivation
above is made precise and the results are extended
to a wider class of goodness-of-fit integral
tests relying
tests given by semi-norms.
examples
are
l/2
(~,(t)-fi,(t))2 q(&t))d&(t) or their one-sided
Typical
on
(3.1)
versions
(3.2) where q: (0,l)-+[O, 00) denotes a suitable weight function. The choice q= 1 gives the Cram&von Mises test (cf. Hijek and SidSk (1967, p. 93)), whereas q(t)=(t(l -t))-‘, 0 < y < 1, leads to tests which give more weight to the extremes. For y = 1 one obtains the weights proposed by Anderson and Darling (1952) for one-sample tests. As motivation of these tests, observe that c,(t)-l?,,(t) has conditional variance var(G,(t)-B,(t)1
X1 :n, . . . ,X,,,)=
&
Q)(l
-R(t))
givenX 1 : “, . . . , X,:, (apply (2.23) and Hhjek and Sidak (1967, p. 61)). Thus (3.1) includes a weighted renormalization via that variance. We now give a precise formulation of our problem. Consider more general regression coefficients C,i such that
(3.3) All our goodness-of-fit
x,,,:= i
tests are based on the general l,_
co,X~,.]:n](Xi)
cni,
OdsG
rank process
(3.4)
l
i=l
([xl denotes the integer part of x), and on the pivoted (via Dirac measures E.) by
empirical
measure
&,, defined
(3.5) In particular,
the Kolmogorov-Smirnov sup SSIO, 11
IXnJl=suP{
IXJ:
test is obtained s=uPP(&l))~
by (3.6)
406
Integral
A. Janssen / Two-sample goodness-of+
tests are obtained
tests when ties are present
by (3.7)
which obviously coincide with (3.1) in the two-sample case. For one-sided testing problems X,,, is substituted by X,7,. The examples have a common feature which is covered
by the following
class of semi-norms.
To explain
_I(~,r)): F distribution
Jq)Jl:={6P(F~F-1~3
this, let
function
on R)
denote the pivoted distributions on [0,11. In addition, consider for A c[O, l] the restriction D(A):= {jjA:fgD [0, l]} of the Skorokhod space D [0, l] on A. Assume now that
001, H-@‘[o, II,
ZH: D(supp(W)+CO, is a given family of measurable
semi-norms.
Then we may define a new function
ll-Kh ~01, z(H,f):=I,(f;,,,,(,,),
I:~,o,l,xDC@
which for fixed H defines a semi-norm f+-+ Z(H,f) Kolmogorov-Smirnov semi-norm is obtained by
Iidff,f):= and the integral
(353)
sup
on D[O, 11. For instance,
by
Zcq) (H 3f).=. Based on this function tests
the
(3.10)
If(s)1
tests are obtained
(3.9)
(3.11) I, (3.9), we now introduce
the general classes of goodness-of-fit
(PII= l@,,Oa,(Z(&I> (Xn,s)sE[O,11) along with the one-sided
(3.12)
versions
*II=1 Cd,,rn~(Z(WXL)SE,O,
1,))
(3.13)
provided the statistics under consideration are measurable. (Note that (2.14) and (2.15) are included in these classes). For Zks and ZCq)keep (3.6) and (3.7) in mind. If ties are present, the evaluation of exact critical values c, and d, yields serious problems. Also asymptotic distributions (if available) usually depend on the unknown distribution. As in Section 2, permutation versions of qn and $,, are proposed. Following the approach of (2.19) let us introduce for fixed w the permutation statistic of X,,, (3.4) by n
(T,HX”n,s(0,6)=2 i=l
l,_ m,Xi,,j:n’w’]txi : ntw)) cno,i ($1
(3.14)
407
A. Janssen / Two-sample goodness-of-jit tests when ties are present
for 0~s~ (6,2,
1, where again the permutations
F) independent
((T”i)i lie on a separate
of the observations.
we will now introduce
the permutation
For technical
probability
and computational
space reasons
process L-1
Z n,s
:!%C[O, 11,
o
1,
Z,,,@):=
c
(3.15)
c.,,<(@+Kl(s)>
i=l
with remainders R,(j/n) = 0, j = 0, . . , n, such that Z,,,, is continuous linear on [j/n, (j+ 1)/n]. It is most important that the permutation depends
in s and piecewise statistic (3.14) only
on o via A”(w) and on & via Z,,,(&).
Lemma 3.1. For each family of semi-norms (3.9) we have (3.16) and a similar result holds for _flS and zc,. one has
Proof. Check that for s=j/nEsupp(&(m)) nnL(O,j/nl x”n, j/n=
C
(3.17)
c~~,i = zn, rh,(O, j/n]
i=l
and
A,(O,j/n] =j/n.
suPP(&).
Thus
s H x”,,,
and
SH Z,,,
coincide
on
the
random
set
0
For fixed OEQ let now Gb+) (.,w) ~Hz(~,(0),(ZbTC,)(~))SEIO,
denote
the distribution
functions
of (3.18)
11),
where the index ‘+’ always indicates the one-sided case. If no other comments are made, both cases are treated equal. Clearly, the exact level a permutation tests associated
to (3.12) and (3.13) are given by >
1
I
@“I= Yn ~(fi,,(X,,S)SS[O,1]) 0
=c”, <
(3.19)
zj.2
(3.20)
and 1 $n:=
0.
I(J%, (XJSSIO, 11)
i 0 where the random variables treated similarly via G,’ .
< yn and c”, are determined
by equation
(2.20) and $,, is
408
A. Janssenl
Two-sample goodness-of-fit
tests when ties are present
Our treatment of permutation tests is based on convergence distributions G!,+‘(. ;) under Ho. Consider FeHo and define
of their conditional
HF:=~(F.F-ll~,(O,l)). Also introduce
unconditional
(3.21) distribution
functions
G and G+
G(+)(x):=Q(Z(HF,(Bo(s)(+)),)~x),
(3.22)
given by the distribution Q of B0 on C[O, 11. Recall from the appendix that G is typically absolutely continuous for Zks and 1(q) but G+ might have a jump at zero. Thus a different treatment of G and G+ is required. Let d(F,,F*)=inf{s:
Fi(x-E)-sEFFz(x)dF1(x+s)+s
for all x)
(3.23)
denote the Levy-distance of two distribution functions F1 and Fz. This is a metric for convergence in distribution. Call FEH, nondegenerate if F is not a dirac measure. It should be mentioned that assertion (a) below is already contained in Bickel (1969) for the two sample-problem (2.9). Distribution free Kolmogorov-Smirnov tests for randomly censored data were recently established by Neuhaus (1992, 1993). Their main concern is with the problems that arise from censoring. Theorem 3.1. Consider the Kolmogorov-Smirnov semi-norm I = ZKS(3.10) or the integral norm I = Zcg)(3.11) with continuous weight function q : (0,l)+(O, co) such that q(s)
O
(3.24)
holds for some K > 0 and 0 < y < 1. Let (Xi), be a sequence of i.i.d. random variables with nondegenerate distribution function F. (a) Zf Z= ZKSor Z= Zcg)with bounded weight function (y = 0) then OH
and
SUP
IG,b,N--(x)1
xe[O,30)
o+-+d(G,+ (.,m),G+(.))
(3.25)
(3.26)
converge to zero almost surely. In both cases the d@erence of the quantile processes o~sup]G(,+)(s,~)-l-GG(+)(~)-l/
(3.27)
SEA
converges to zero almost surely uniformly on compact sets Ac(0,1). (b) Next consider unbounded weight functions q (3.24) and their integral norms Z=Ztg). Then the assertions (3.25)-(3.27) remain valid tfalmost sure convergence to zero is substituted by convergence in probability.
A. Janssen J Two-sample
Proof. Section The proof
5.
goodness-of-@
409
tests when ties are present
Cl
of Theorem
3.1(b) is based
on an approximation
bounded weight functions. As we will see in Theorem under general circumstances. Consider the following family of semi-norms. Condition (A): Let Z(H, f) = 1,(f)
procedure
for q by
3.2 that type of argument works assumptions for the underlying
be as in (3.8) and (3.9) nontrivial
semi-norms
such
that with Q as in (3.22)
QIWF, (B&))MO, m))>O holds for nondegenerate
(3.28)
F and assume
Q(~(H,,(IB,(s)I),)E(O,
a))>0
(3.29)
for the one-sided case. Assume also that G and G+ (3.22) are nondegenerate limit laws. Condition (B): In the case of one-sided tests we assume I(H,,.) to be positive increasing, meaning that I(HF,f)dl(H,,g), Condition
(C): (Unconditional
whenever
O
convergence).
(3.30)
For each nondegenerate
be convergent in distribution as rz+c~ (Their limit distribution were already specified in (3.22).) These assumptions require further comments.
functions
FEH,
let
G and G+
Remark 3.1. (a) Under condition (A) the random variable I(HF, (B,(s)),) has a proper distribution G (3.22) on (0, co). If in addition condition (B) holds then G’ is concentrated on [0, co). Confer the appendix. In both cases the quantile functions G(+)- ’ are continuous on (0,l) which implies the equivalence of tests (see Lemma 3.2). (b) One easily checks that unconditional convergence (3.31) is necessary for conditional convergence (3.25) and (3.26), respectively. For ZKSassertion (3.31) was earlier obtained in Lemma 2.1. Theorem 3.2. Consider afamily of semi-norms IH( .) such that the condition (A)-(C) hold for
the two-sided
semi-norms
and one-sided
case,
respectively.
Assume
that there exist further
(Ik,H)ksN (3.8) with
(3.32)
410
A. Janssen / Two-sample goodness-of-fit tests when ties are present
for all
HE&[,,, II. Define lk(H,f):=Zk,n (f) and let G:yd(.,o) (3.18) and GCkJCf)(3.22) denote the corresponding distribution function belonging to Ik( .). Assume that under nondegenerate FEH, o+d(G:;n)(
., o), GCk)(+)(.))
(3.33)
converges to zero in probability as n-co for each keN. Then we have (a) the same assertion (3.33) holdsfor d(G!,+‘(. ,a), G’+‘( .)), (b) the difSerence of the quantile processes uniformly on compact sets A. Proof. Section
5.
(3.27) converges
to zero in probability
0
These results prove asymptotic
equivalence
of ordinary
and permutation
goodness-
of-fit tests under the null hypothesis. To establish a result of this kind consider conditions (A)-(C). Then there exists xb+‘~R! such that G(+) 1cxr~,mj is absolutely (+).-.- 1 - G(+) (x0) > 0, see the appendix. The choice continuous satisfying txO c,+G-‘(l-cc),
GI
and
d,,+(G+)-‘(l-cc),
a<~$,
(3.34)
lead to asymptotic cc-similar tests cp,, and Ic/, (3.12), (3.13). By the appendix we can choose x0 so that CI~= 1 and c&J= l/2 for I~{l,s, Pq’} of Theorem 3.1. Note that G(+), c, and d, are not really available in practice under ties. Lemma 3.2. In addition to these assumptions suppose that under nondegenerate we have convergence of oHd(G;+)(.,o),G(+)(.))
FEH,
(3.35)
to zero in probability. Then (a) for each ~
(Pl#-d~-O P
(3.36)
in probability under F; (b) for each ~
idtL~0 in probability
(3.37)
under F.
Proof. That proof easily follows from the almost sure subsequence convergence principle for convergence in probability. Choose a subsequence nk such that (3.35) converges almost surely along nk. We have almost sure convergence of quantile functions along nk and (P,,~- &,---+ 0 since G is strictly increasing at G-‘(a) for ~
cf. Witting
and Nijlle (1970, i. 58). Now we may choose a further subsequence
A. Janssen J Two-sample
n; such
that
proved.
Cl
(3.36) converges
goodness-of-/it
almost
surely
tests when ties are present
along
n;.
Lemma 3.2 can be used to evaluate the asymptotic under local alternatives, cf. Section 4.
411
Similarly,
(3.37) can
power function
be
of @, and $”
As explained computational process (Z,,,),
in the introduction other approximations as 9, of (Pi are of interest for reasons whenever n is large. Throughout, it is shown that the rank used for the definition of Gr’ (. , co) can be substituted by its limit process (B,(s)), to get asymptotic quantiles. Assume that the Brownian bridge is defined on a further probability space (fi, d, Q”)independent of the observations. For fixed o~s2 introduce G”(.,u) and GT (.,w) by G:+‘(x,w):=
(3.38)
Q”(I(~~,(Bo(S)(+)),,[O, 1,)d.x)
and let I&, and $n denote
the tests (3.12) and (3.13) given by critical
c,,:=(G,,(.,w))-‘(1-a) These tests work with estimated asymptotically a-similar tests.
and critical
Theorem 3.3. Under the assumption
values
&,=(G~(.,o)))~(~--cr).
(3.39)
values. Again we will see that @, and qn are
of Theorem
3.1 we have for IE{I~~, Icq’]
sup I~,(x,o)-G(x)l+O XE[O,3(1)
(3.40)
and (3.41) in probability.
In addition, we obtain (3.42)
in probability tests.
under nondegenerate
Proof. Section As conclusion tests and further
5.
FEH,,
whenever
a< l/2 holds for the one-sided
0 the Kolmogorov-Smirnov, Cramer-von integral tests Cp, and 4, with estimated
Remark 3.2. Consider that
the two-sample
O
n
problem
1.
Mises, Anderson-Darling critical values work well.
with regression
coefficient
(2.9) such
(3.43)
412
A. Janssenl
Two-sample goodness-of--fit
tests when ties are present
holds. Then the results of Einmahl and Mason (1992) can be used to sharpen assertion of Theorem 3.1(b). Actually, one can prove that (3.25)-(3.27) are almost surely convergent for those weight functions (3.24) determined by 0 < y < l/2. Note that (3.43) yields XI= I c$ = O(n), which is one assumption of Einmahl and Mason.
4. KolmogorovSmirnov
tests under local alternatives
Our treatment of the power of conditional goodness-of-fit tests @, and Cp.is closely related to the continuous case already analysed in the literature. For these reasons we restrict ourselves to the Kolmogorov-Smirnov test in order to show which type of modifications are required. The continuous case was successfully studied in Milbrodt and Strasser (1990). Here the reader finds references about the asymptotic power functions of goodness-of-fit tests. Moreover, we refer to the early paper of Chibisov (1965), where the likelihood ratio (4.3) and the power function was treated within a restricted continuous case. The key for our results is the asymptotic equivalence of both Gn and (Pn to unconditional tests which can be treated under local alternatives. Throughout, the modern approach via tangent vectors is used, cf. Pfanzagl and Wefelmeyer (1982) and references therein. The model (4.1) was earlier proposed by Janssen and Milbrodt (1993). Let 9 + Ps denote an &-differentiable curve at 9 = 0, see Strasser (1985, Section 75), whose tangent gulf’:= {gE&(P,): s g dPo =0} 1s g iven by the L,(P,,)-derivative of 9 -+ 2 (dPg/dPO)“*. Starting with regression coefficients (3.3), we introduce the model of the joint distribution of XI, . . . ,X, as
2(X
x,1=
l,...,
6 Pcni=:Pn,F,p s~q’(po) >
(4.1)
i=l
where F denotes the distribution function of PO. The parametrization allowed within the asymptotic setting since
I6
P,,,-
6
i=l
i=l
p:,,
by F and g is
-+O II
in the variational distance whenever Pg and P$ admit the same tangent g. This is due to local asymptotic normality (LAN). Note that L2-differentiability implies the LAN of the experiment &I=(~“,~~, cf. Strasser
{P”,F,& s~L:“‘(Po))),
(1985, Section dP” F d=U)-; log dP n,F,O
(4.2)
79.2), given by the approximation
g s
g2dF+op,,F,o(l),
(4.3)
413
A. Janssen 1 Two-sample goodness-of--fit tests when ties are present
where I,,( .) denotes
the central
L,(g)(x):=
i
sequence x=(x1,
C7(xi)9
cni
,X,)T.
(4.4)
i=l
Let gl,gELr’
(P,). Then Le Cam’s third lemma
s
in distribution
under
g1 0
s 1
1
Jxg1)~
implies
F-l (+A-,(du)+
0
gloF-‘(u)goF-‘(u)du
(4.5)
0
P,, F,4. Note that the right-hand
local alternatives
side of (4.5)
has mean
s s
gPF-‘g°F-‘d&o,l)=
and variance
h°F-')2d~,co,l,=
These arguments Gaussian
motivate
s
s
(4.6)
glg2dP0
(4.7)
g:dP,,.
that the limit experiment
GF of E, (4.2) is given by the
shift
GF:=(CCO, 11, WCC& 111,{QF, g: s~L’,“‘(J’o)l), with QF,g defined log
by
dQ~,~ _ %,o
(4.8)
’
’ g20F-l(u)du,
g#(u)B,(du)-;
s o
s
(4.9)
0
where QF, o = Q is the distribution of the Brownian bridge Bo. This model is a submodel of the case where the uniform distribution AI(o, 1J (= PO) is restricted to the tangent set (g 0 F-‘: gall’}, cf. Strasser (1985, Section 82.23). Moreover, we have convergence of experiments E,+GF in the sense of Strasser (1985, Section 80.6). For these reasons one expects that the asymptotic power of the Kolmogorov-Smirnov test can be calculated within the limit model CF. These arguments can be made rigorous. Theorem 4.1. Let (P”(3.12), & (3.19) and & (3.39) denote the two-sided asymptotic a~(0,1) Kolmogorov-Smirnov tests dejined through (3.10). Then we have
level
a(s) := n+m lim EP”,~,# cp.= n-m lim EP,,,~,~$J,,=n-+m lim EP,,F,g(Pn (4.10) where c, denotes the (1 - cc)-quantile of (2.24).
A. Janssen / Two-sample goodness-o@
414
tests when ties are present
Proof. The present proof follows standard arguments of Hajek and Sidak (1967 Chap. VI, Section 4.4), see also Milbrodt and Strasser (1990). Consider first -F(t), LE R. Then (4.5) is equal to 91=1,-m,,, F (0 Bo(F(r))+ which is actually i
g”F-‘(u)du, s
(4.11)
0
the limit distribution crd
of (4.12)
l(-mrt](Xi)
i=l
device establishes the convergence of their finite under Pn,F,S. The CramCrWold dimensional marginal distributions. In addition tightness of (4.12) remains valid under contiguous alternatives if we consider s~F(R)n(0, 1) and replace t =F’(s). Thus we may apply the functional f-
(4.13)
sup If(s ssF(R)
which completes
the proof.
Remark 4.1. The limit
0
experiment
GF (4.8) can be written
in an equivalent
form
given by ~F=(CO([W),~(CO([W)),{~F,g:gEL(20'(PO)})
on the set of continuous
functions
C,(R) on If%vanishing
at infinity.
The distributions
OF4 are defined by the shift model
&~,,:=~((Bo(F(x))+~~_~,~,
gdF)xeR)
of the Brownian bridge. Obviously, the corresponding log-likelihood as in (4.9). The right-hand side of (4.10) then reads as
Next we briefly sketch some applications continuous case. We restrict our attention regression coefficients (2.9).
ratio is the same
of Theorem 4.1 and some extensions of the to the two-sample testing problem given by
Example 4.1 (Continuation of Example 2.1). (a) The conditional two-sample Kolmogorov-Smirnov tests &, (3.19) and (Pn (3.39) given by IKS are asymptotically admissible in the following sense. Assume that qn denotes a further asymptotic level LXtest for Ho (2.1) and suppose
lim inf&,, n-30
Fr
g
r?,2 B(s)
(4.14)
A. Janssen / Two-sample
goodness-of--fit
tests when ties are present
415
for each gEL$‘) (P,) and each PO. Then we have
in
probability. The proof follows the lines of Strasser (1985, p. 436). Pn,F,g (b) (strict asymptotic unbiasedness of (Pn and I$,,). For each gELi’) (P,)\(O), we have p(g) > CI,where p is as in (4.10).
(c) (consistency). For g#O and t,+co, we have fl(t”g)+l as y1--tcc. (d) As further consequence one immediately generalizes the principal component decomposition for the power of Kolmogorov-Smirnov tests of Milbrodt and Strasser (1990) if ties are present. Note that for each gEL$“(P,) we have ~(tg)=cl++a(g)t2+o(t2)
(4.15)
as t+O,
where a(g)=a(g, c()30 denotes the curvature of the power function along the ray {tg: PER} at t=O. It can be shown that for fixed PO the curvature g-a(g) admits a principal component decomposition in the sense of Milbrodt and Strasser (1990, Theorem 2.8).
5. Main proofs The proof of Theorem 3.1 is based on an conditional invariance principle. denote i.i.d. uniformly distributed random Throughout let Ui, U,, . . . : Q-(0,1) variables with standard empirical process cn(m,t). Let M denote (the probability one set) M=
W: SUP Iti,(O,t)--tl+O,
Ui(W)#Uj(O)
for all i#j
rsm, 11
For REM introduce
the process
SH Y,,, on [0,1] into C[O, l] by
Y,,,(w,c5):= i 1[O,s](Ui:n(u))C,,~i(~)+~Rn(S),
(5.1)
i=l
with remainders making (5.1) continuous and Y,_, is piecewise linear in between. Lemma 5.1. For.$xed
o~A4
YE,s (QA6) 5 weakly in distribution
in s such that R,(O) = R,( 1) = 0, R,( Ui : ,,) = 0
we have
Bo(4
on C[O, l] under uniformly distributed
(5.2) permutations
(a,i(G))i,,.
416
A. Janssenl
Two-sample goodness-of--fit
Proof. Let REM be fixed. According (1968, Section 24), we have
tests when ties are present
to Hajek and Sidak (1967, p. 186) or Billingsley
weakly in C[O, 11, where Z,,, is given in (3.15). Check that Ui : n(w)
yn,s=~“,ii”(o,s,. Let (Pi : [0, l] + [0, l] denote
the one-to-one continuous transformation given by i n, which is linear on [ Ui _ 1 : ,, (CO), Ui : n (co)]. Conserp,(O)=O, $%(I)= 1, cPn(Ui:, (m))= / quently, we have
yn, s=
&I, qpn (s) .
Since sup(l~,(s)---sl:s~[O,
(5.5) l]}-0,
it is easy to see that (5.6)
llfnO(Pn-fll+O> whenever /fn - f 11+O in C [0, 11. If we now combine 0 Billingsley (1968 p. 34) complete the proof.
(5.3))(5.6) standard
arguments
of
Proof of Theorem 3.1 (part (a)). The proof is given in two steps. Step 1: I = ZKs. Consider a fixed element OEM and Xi = F- 1(Ui). The basic identity (3.17) implies sup lZ!I’,’ I= sup lr7:;: I sssupp(ni.) SPS”PP (4)
if we take (2.11) into account.
fH
Now we can apply the continuous
function
sup Im’+‘I SOP(R)
and the invariance principle (5.2) proves the desired results. Note that G is continuous according to the appendix. Step 2: Z=Ztq), q bounded. Lemma 5.2 below shows weak convergence of &,(o)+HF on [0, l] for all o lying in a probability one set N. Let oeN be fixed and
A. Janssenl
consider
Two-sample
fn, fE{gEC[O,l],
goodness-of--fit
g(O)=g(l)=O}.
tests when ties are present
Then
417
it is easy to see that
llfn-flI+O
implies (5.7)
jffqd&+f’qdH,. The invariance
principle
(5.3) together
with Theorem
5.5 of Billingsley
(1968) now
imply (.G~~‘)‘q(s)d&,(~,s)+ which completes
BbfW2q(s)dH&),
s
s
the proof of part (a).
0
Lemma 5.2. Under H, we have weak convergence
of
A,jHF
(5.8)
on [0,1) almost surely, where HF is de$ned
Proof. Let US use again Xi = F 5.1, and set A:=
by (3.21).
’ (Ui). Define N:= Mn@,
where M is as in Lemma
.
o: supI&(F(x)I+O XER
Let S(q) denote the set of continuity points (5.8) will be established for fixed oeN. (1) First let us verify that
of a monotone
function
cp. Throughout,
lim inf &, [0, y] 3 HF [0, y) “-02 holds except for a countable distribution the inverse
subset
(5.9) of (0,l).
Note
implies F^;’ (y)-+F-’ (y) for all ~ES(F-‘), of t H f,(o, t). Moreover,
that convergence
choose z
Since o~h?,
we have by (5.10)
lim inf riz, [0, y] > F(z), n+m where F(z)fF(F-’
(y)-)
as zfF_’
(5.11) (y). Note
that in addition
HFCO,Y)=E,~~~,~,(U:F~F-~(~)
This statement,
together
in
denotes
(5.10)
&J-o~Yl3F^,(R(Y)-).
For ~ES(F-‘)
of t+F,(t)
y~(0, l), where F^;’
F-‘(u)
with (5.1 l), implies
assertion
(5.9).
418
A. Janssen / Two-sample goodness-of@
tests when ties are present
(2) The converse inequality of (5.9) for limsup can Consider random variables I’, on (0,l) with distribution fixed woN (cf. Lemma 5.1). Consequently,
V,$+
be obtained as follows. function TV fi,,(~, t) for
U, and FOF-‘(I’,)*
FOF’(Ui)
converge both in distribution since F 0 F- ’ is IIIco,1j almost everywhere Check that F- ‘(V,) has distribution function t H $,,(a~,t). Thus c!Z’(Fd,‘lA,,,,
~,)=Z(FOF-~(I’~))+H,
weakly on [0, l] as n-03. ^ & = y(F, Choose observe
now that
(5.12)
On the other hand, note that
^ o F, ’ 11,co,1)).
continuity
continuous.
points
(5.13)
y, y+ E~(0, l), s>O,
of x H HFIO, x].
In addition
I F^,(x)-F(x)1 de for all x and II large enough. large n &cO,Yl=~,(,,l,(~:
If we take (5.12) and (5.13) into account,
F^,(F^,‘(u))dy)
dA,(o, l)(u: F(F^,l(u))d~+~)~H~CO,~+~l. Altogether, (5.9) and (5.14) yield convergence of (0,l). 0
q(s) dH&) dQ =
s
VWM4Ms)
The present proof is based on Theorem sequence of seminorms (3.32) as
h&f)
=
(f
f’ mink, 4 dH
(5.14)
of &,[O, y]+HFIO,y]
Proof of Theorem 3.1 (part (b)). First, it can be checked finite almost surely. Observe that by Fubini’s theorem
Is&(s)~
we have for
for a dense set
that JB,,(s)2q(s)dHF(s)
dH&) < 00
3.2. We may choose
.
is
(5.15)
the approximating
11-7
,
(5.16)
with bounded weight function min(q, k) for kEN. Note that assumption (3.33) is valid according to part (a) of the theorem. In order to apply Theorem 3.2, it is now enough to prove unconditional convergence, see condition (C). This is done in Lemma 5.3. The convergence of the quantile functions (3.27) can be obtained as follows. Note that supp(G’+‘) is a closed interval and that the inverse G(+)-’ is continuous on (0,l). Thus convergence in distribution implies here pointwise convergence of their inverse functions. Monotonicity arguments yield uniform convergence on compact sets.
A. Janssen/ Two-sample goodness-of-fit tests when ties are present
Lemma
5.3. For each weight function
s
(X!CA2 q(s) d&,(s) --%
in distribution
under nondegenerate
419
q given by (3.24), we have
s
(&,(s)‘+‘)~ q(s)dH,(s)
distributions
(5.17)
FEH~.
Proof.
Throughout consider continuity points 6 and l-6, 0~ 6 < 1, of the distribution function of HF (3.21). Introduce qs(s):=q(s) 1C6,1_6j (s). According to Billingsley (1968, Theorem 4.2), it remains to check the following three conditions. (1) For 6 > 0 we have convergence in distribution of
s
(X!Z2qs(s)
d% (s) *
(Bb+‘(s))2qd(s) dH,(s). s
(2) We have
s
(@,+’ (s))2qa(s) dHF (s) ---%
in distribution
(Bb”(s))2q(s)
dH,(s)
as 6 JO.
(3) lunli_m supE
(X:::)’
(q(s)-q4s(s))d@,(s)
(S Assertion (1) follows from the conditional convergence theorem for the bounded weight function qs, see Remark 3.1(b). Note that qs is HF almost surely continuous and (5.7) carries over. It should be mentioned that (1) also follows directly along the lines of the proof of Theorem 3.1(a). Moreover, observe that condition (2) is verified provided ;gE U holds.
Obviously,
(Bd+)(~))~(q(s)-qqs(s)dH,(s)
(5.18)
=0 1
it remains
to check
the conditions
(3) and (5.18) for two-sided
tests. Note that E
B:(s)(q(s)-&))dHF(s)=
s
E(B;(s)(q(s)-&))dH,(s)
converges to zero as 6 JO which implies (5.18). Similarly, the third condition can be verified which is done below. As in Section 3, we get that the conditional variance of X,,, given the order statistics is
420
A. Janssen / Two-sample goodness-of--fit tests when ties are present
which is equal to [n/(nthe order statistics, E
U
l)] (~(1 -s))
for s=j/n~supp(&,).
If we now condition
under
we have
X,2,,(q(s)-qqd(~))d~,(~)
=SS~(s(‘-r))(q(S)-q6(S))dl(s)d~(X1:”,
s
where the upper bound
X”,,)
~,((0,1)\(6,1_6))d~(X,:.,...,X,:.), converges
to (5.19)
KHF((O, l)\@, 1 - 6)). This limit follows from Lemma
5.2 and the strong law of large number,
which yields
&l({l})+HF((l)) almost surely for a proper treatment of the upper endpoint. Since (5.9) becomes arbitrary small for 6 10 the conditions (l)-(3) are established and the proof is complete. 0 Proof of Theorem 3.2. The proof is carried apply to GJ (. , w). Notice that for each x
out for G,( . , co). Analogous
arguments
(5.20)
Gk, n(x, w) 3 G,(x, 0). Let B denote the intersection of the sets of continuity points Throughout let XEB be fixed. The assumption implies as k-co
Gck’(x)- G(x)+0
(5.21)
On the other hand, we have Gk, ,,(x, o) - Gck)(x) 7 keN. This result combined
and
observe
(cni)i
as n+ cc for each
proves (5.22)
P
of (5.22) to negative that
0 in probability
with (5.20) and (5.21) immediately
(G,(x,o)-G(x))+-0. The extension
of G and Gk for each k.
(Dni(u))i
(Xi : n)i = (F- ’ (UC:n))i. Returning (2.22), that OH (X,,,(c.$ &,(~))
parts
runs
which
to our definition
and
as follows. turns
out
Choose
Xi=F-‘(Ui)
to be independent
(3.4) and (3.14) we see, similarly
(w, &)H (znJ~,
G), h,(w))
of to
A. Janssen J Two-sample goodness-of-jt tests when ties are present
have the same distribution we have
since fi, only depends
on order statistics.
421
By Lemma
3.1,
s
G,(x,O)dP(o)=P(I(~,,(X,,,),)Qx)~G(x),
which converges
according
to our assumptions.
Thus (5.23)
{G,(vJ-G(x)}dP(w)+O
s
holds for fixed XEB. On the other hand, we obtain
s s
(G,(x, co)- G(x))+ dP(o)+O
by (5.22). Combining
(5.24)
(5.23) and (5.24), we have
IG,(x,co-G(x)IdP(o)+O.
Now we may choose a countable
s
dense subset
{xiEB: igN}
of [0, co). Then
itI IG,(xi,~)-G(xi)l/2’dP(~)~O.
For each subsequence $I
there exists a further
subsequence
nk such that
IG,,(xi,~)-G(xi)//2’~0
holds almost surely. Along that subsequence we get G,,(x, w)- G(x)-+0 almost surely again for all continuity points x of G. This gives us convergence of d(G,(. , w), G( .)) to zero in probability. 0 Proof of Theorem 3.3. First consider
sup {lz::l)SSSUPPbw
sup
the semi-norm
IKs. Then, we have
{IBL+‘(s)l} d ll&(~)-G,. II.
(5.25)
SES”PPl~“)
According to (5.3) and Skorokhod’s embedding theorem (Shorack and Wellner (1986, p. 47), one can find versions of B,( .) and Z,, such that the right-hand side of (5.25) converges almost surely to zero. Together with (3.25) and (3.26) one gets assertions (3.40) and (3.41). The same arguments apply to I (4) for bounded weight functions. The device of the proof of Theorem 3.2 can now be used to establish the result for unbounded weight functions q (3.24). Consider again the approximating sequence Ik,H (5.16) of semi-norms and let G, T+)n denote their conditional distribution functions (3.38). Then we have, similarly to (5.20),
422
A. Janssen 1 Two-sample goodness-of-@
We already
proved
tests when ties are present
(3.40) and (3.41) for k. Hence
Theorem 3.2 can be adapted. Lemma 3.2. 0
The proof of statement
the arguments
of the proof
of
(3.42) is the same as that proof of
Appendix Let I : C [0, l] -[O, co] denote a measurable semi-norm and let Q = _.Y((B,(s),)) be the distribution of the standard Brownian bridge. It is well known that each measurable linear subspace Vc C[O, l] has either Q-measure zero or one, cf. Kallianpur (1970). A survey of results of that type is given in Janssen (1984). Evidently, Ni:= {f: I(f)=O} and N2:= {f: I(f) < co} are O-l sets. Throughout, we are concerned with the distribution of the semi-norm I. Lemma A.l(a) is due to Hoffmann-Jorgensen et al. (1979) who applied convexity arguments of Bore11 (1974). Let I be nontrivial, following lemma.
i.e. Q(N,uN;)=O.
Then, we have the
Lemma A.l. Let H:‘)(t):= Q(I((Bb+’ (s)),) < t) denote the distribution functions of I under B,, and BJ. Choose xbf):= sup{t: Hj+)(t)=O}. Then we have: (a) log H,(t) is concave on (x,,, co) and HI is absolutely continuous on that interval, (b) if1 is positive increasing (3.30) and Q(f: Z(lfl) < 00) = 1 the same result holds for H; on (x0’, co). This H:
result
implies
may have typically
Remarks condition
A2. (a) The
absolute a jump
continuity
distribution
QV((Bo(s),)@O,r))>O
of HI under
mild
conditions
whereas
at zero. HI
is absolutely
continuous
whenever
the (Al)
holds for all t >O. Obviously, (Al) is valid for nontrivial 11./I-continuous semi-norms on C[O, 11. (b) However, continuity of I does not imply H: (0) =0 and absolute continuity of Ht. For instance choose I(f) = If(y) )for some y@O, 1). More generally consider ZE {Zks, Ztq’} (3.10), (3.11) and nondegenerate FEH~. Then there exists y~supp(H~)n(O, 1). Thus
interval with lower endpoint (c) The support of H :+) is an (usually unbounded) xr’. (For reasons of concavity H:” cannot be constant on whole intervals between xy’ and the upper endpoint, since log Hi+‘(t)+0 as t+co). Hence the quantile function Hi+‘- ’ is continuous on (0,l).
A. Jamsen/
Two-sample
goodness-of-fit
tests when ties are present
423
Proof. The present proof relies on an obvious extension of Theorem 1.2 of subspace with Hoffman-Jorgensen et al. (1979). Let C,, c C [0, l] be a measurable Q((B,(s)),EC~)= 1. Consider a convex function q: Co-+[O, co), i.e. q(Af+(l -A)g)< Then t c, log Q(q((B,(s)),) < t) is concave nq(f)+(l-n)q(g), OGA< 1 and JgEC,. and
dP(q((Bo(s))s))
is absolutely
continuous
on
the
interval
(to, co), t,:=sup{t:
QM(&WMdt)=Ol. To prove this, consider
s, te[to,
l”{f: q(f)bt}+(l-A){g:
co). Then convexity q(g)ds}c{f:
of q implies
q(f)dIt+(l-+}.
If we proceed as in Hoffmann-Jorgensen et al. (1979), (l.ll), the statement follows from the results of Bore11 (1974). Obviously, this result gives assertion (a). To prove part (b) define q(f):=I(f +). Now it is easy to check that q is convex iff I is positive increasing, which is done below. Assume (3.30). Then (,?f+(l -i)g)+
take 0 d f< g and consider f= 4 (f-g)
+3 (g +f).
Convexity
of q yields
I(f)GfI((.!-g)+)+t1(g+f)d&U(g)+1(f)), since I((f-g)+)=O. Next we take care that q remains finite on Co= {j r(lfl)< 00). Note that (3.30) implies Z(f’)
Acknowledgement I am grateful to D.M. Mason Lemma 5.3.
who showed me the &method
(l)-(3)
of the proof of
References Anderson, T.W. and D.A. Darling (1952). Asymptotic theory of certain ‘goodness of fit’ criteria based on stochastic processes. Ann. Math. Statist. 23, 193-212. Behnen, K. and G. Neuhaus (1989). Rank Tests with Estimated Scores and their Application. Teubner, Stuttgart. Bickel, P. (1969). A distribution free version of the Smirnov two sample test in the p-variate case. Ann. Math. Statist. 50, l-23. Billingsley, P. (1968). Weak Conuergence of Probability Measures. Wiley, New York. Borell, C. (1974). Convex measures on locally convex spaces. Ark. Math. 12, 239-252. Chibisov, D.M. (1965). An investigation of the asymptotic power of the tests of fit. Theory Prob. Appl. X, 421437. Durbin, J. (1973). Distribution theory for tests based on sample distribution function. Reg. Conf: Ser. Appl. Math., Vol. 9., SIAM, Philadelphia.
424
A. Janssenj
Two-sample goodness-of-@
tests when ties are present
Gleser, L.J. (1985). Exact power of goodness-of-fit tests of Kolmogorov type for discontinuous distributions. J. Amer. Statist. Assoc. 80, 954-958. Guilbaud, 0. (1986). Stochastic inequalities for Kolmogorov and similar statistics, with confidence region applications. Stand. J. Statist. 13, 301-305. Guilbaud, 0. (1988). Exact Kolmogorov-type tests for left truncated and/or right-censored data. J. Amer. Statist. Assoc. 83, 213-221. HBjek, J. and Z. Sidak (1967). Theory of Rank Tests. Academic Press, New York. Hoffmann-Jerrgensen, J., L.A. Shepp and R.M. Dudley (1979). On the lower tail of Gaussian seminorms. Ann. Prob. 7, 319-342. Janssen, A. (1984). A survey about zero-one laws for probability measures on linear spaces and locally compact groups. In: Lecture Notes in Mathematics, Vol. 1064, Springer, Berlin, 551-563. Janssen, A. and H. Milbrodt (1993). R&nyi type goodness of fit tests with adjusted principal direction of alternatives. Stand. J. Statist. (to appear). Kolmogorov, A. (1941). Confidence limits for an unknown distribution function. Ann. Math. Statist. 12, 461-463. Mason, D. and U. Einmahl(l992). Approximations to permutation and exchangeable processes. J, Theoret. Prob. 5, 101&126. Milbrodt, H. and H. Strasser (1990). On the asymptotic power of the two-sided Kolmogorov-Smirnov test. J. Statist. Planning Inference 26, l-23. Neuhaus, G. (1992). Conditional rank tests for the two-sample problem under random censorship: treatment of ties (to appear). Neuhaus, G. (1993). Conditional rank tests for the two-sample problem under random censorship. Ann. Statist. (to appear). Noether, G. (1963). Note on the Kolmogorov statistic in the discrete case. Metrika 7, 115-116. Pfanzagl, J. and W. Wefelmeyer (1982). Contributions to a General Asymptotic Sfatistical Theory. Lecture Notes in Statistics, Vol. 13, Springer, New York. Praestgaard, J.T. (1991). General-weights bootstrap of the empirical process. Ph.D. dissertation, Department of Statistics, University of Washington. Romano, J.P. (1989). Bootstrap and randomization tests for some nonparametric hypotheses. Ann. Statist. 17, 141-159. Shorack, G.R. and J.A. Wellner (1986). Empirical Processes with Applications to Statistics. Wiley, New York. Strasser, H. (1985). Mathematical Theory of Statistics. De Gruyter, Berlin. Witting, H. and G. N(ille (1970). Angewandte Mathematische Statistik. Teubner, Stuttgart.
Planning
and Inference
Two-sample goodness-of-fit when ties are present Arnold
tests
Janssen
Mathematisches Germany
Received
399
39 (1994) 399-424
Institut,
29 February
Heinrich-H&e-Universitiit
1993; revised manuscript
Diisseldorf,
received
Universitiitsstr.
1, 40225 Diisseldof
8 June 1993
Abstract The paper contains asymptotic results for two-sample Kolmogorov-Smirnov, Cram&van Mises, Anderson-Darling and other related tests. Whenever the portion of ties cannot be neglected, two procedures are proposed to make the tests (asymptotically) distribution free. The tests can either be carried out as permutation tests or as tests with estimated critical values. All tests turn out to be asymptotically equivalent and their asymptotic power function is established under local alternatives. The same results apply to one-sided tests for stochastically larger alternatives where noncontinuous limit distributions of the test statistics may appear. AMS
Subject Classification:
Primary
62GlQ
Secondary
62G20.
Key words: Ties; permutation tests; Kolmogorov-Smirnov Darling test; asymptotic power function.
tests; Cram&-van
Mises tests; Anderson-
1. Introduction Goodness-of-fit tests are asymptotically distribution-free under the null hypothesis as long as the underlying distribution function F is continuous. We refer to the monograph of Shorack and Wellner (1986) for one-sample tests and HAjek and SidAk (1967, Ch. V), Chibisov and Durbin (1973, p. 39) in connection with two-sample problems where detailed results can be found. However, practical sets of data usually contain ties, and noncontinuous distributions have to be taken into account. In principle, the device for the treatment of Kolmogorov-Smirnov tests under ties is contained in the literature as a special case of more abstract empirical process theory. For instance, Bickel(l969, p. 7) proposed to carry out these tests as permutation tests. He showed consistency and asymptotic equivalence of conditional and unconditional
Correspondence to: A. Janssen, Universititsstr.1, 40225 Diisseldorf,
Mathematisches Germany.
Institut,
Heinrich-Heine-‘Jniversitit,
0378-3758/94/$07.00 0 1994 ~ Elsevier Science B.V. All rights reserved. SSDI 0378-3758(93)E0049-M
Diisseldorf,
400
A. Janssen 1 Two-sample
goodness-of--fit tests when ties are present
tests within a more general setting, see also Roman0 a recent discussion. Throughout, for univariate
(1989) and Praestgaard
(1991) for
we will have a closer look at permutation goodness-of-fit tests tied data. We will also treat integral tests (like the Cramer-von
Mises and Anderson-Darling tests) and one-sided tests for stochastically larger alternatives under ties. It turns out that the limit distribution function under the null hypothesis is not necessarily continuous, which requires further effort. (Note that condition A of Roman0 (1989) is violated.) Section 4 deals with asymptotic power functions of two-sided Kolmogorov-Smirnov tests under local alternatives. Extending earlier results for continuous distributions, we obtain asymptotic admissibility and strict asymptotic unbiasedness of these tests. Whenever the portion of ties cannot be neglected, a further procedure with estimated critical values (Theorem 3.3) is proposed in order to make the tests asymptotically distribution free. Below we will make a few comments concerning references for one- and two-sample Kolmogorov-Smirnov tests if ties are present. It is well known that Kolmogorov-Smirnov tests are conservative and remain valid if ties are ignored; see Kolmogorov (1941), Noether (1963) or Guilbaud (1986) for a recent discussion. In connection with one-sample problems we mention Gleser (1985), who calculated the power under discontinuous distributions. Guilbaud (1986) established stochastic inequalities for KolmogorovSmirnov statistics which are reflected in our Lemma 2.1 within the asymptotic set-up. Acceptance regions for the Kaplan-Meier estimator appear in Guilbaud (1988). Two-sample omnibus tests are studied in Behnen and Neuhaus (1989) also when ties are present. They studied rank tests with estimated scores. In conclusion we give the following recommendation for practical purposes: (1) In the case of small or intermediate sample size the tests should be carried out as permutation tests. For computational reasons a Monte Carlo simulation may necessary. (2) Alternatively, the critical values can be estimated (Theorem 3.3), whenever large. However, these procedures require also Monte Carlo simulations via Brownian bridge. These calculations may be easier as in (1) when a lot of ties
be it is the are
present. (3) If only a few ties show up, they may be ignored for large n and the use of asymptotic critical values seems to be justified. The paper is organized as follows. Section 2 explains the results for two-sample problems. The main results are given in Section 3 and the treatment under local alternatives can be found in Section 4. All longer proofs are represented in Section 5. The results about asymptotic distributions of the Brownian bridge under semi-norms are summarized in the appendix. We will use the following notation. Let 2(X1 P) denote the law of a random variable X under P. The uniform distribution on the unit by supp(P). By interval of Iw is denoted by /I,,,, 1). The support of P is abbreviated
A. Janssen / Two-sample goodness-of-fit tests when ties are present
definition
Xj: n stands for the j-th smallest
and 7
indicate
tion, let f’
among
(P-probability,
X1, . . . ,X,. respectively).
Let z In addi-
tests
This section
explains
and motivates
for the following
Example 2.1. Consider specified by i.i.d. random within group 1 and i.i.d. F2 for group 2. Let us hypotheses:
where K, denotes
procedures.
The results
are first
a two-sample testing problem of sample size yl=nl + n2 function F1 variables Xi, . . . , X,, with common distribution random variables X,, + i, . ,X, with distribution function examine two-sample goodness-of-fit tests for testing the
K1 being the omnibus alternatives, unknown, or for testing Ho: F,=F2
the testing
example.
Ho: F1 = F2 against
Throughout,
order statistic
in distribution
= max(A 0).
2. Two-sample
presented
convergence
401
against
the one-sided
K1: F1 # FZ,
(2.1)
where F1 and F2 are assumed K2: F,3F2, alternative
to be completely
F,#F2, that F2 is stochastically
(2.2) larger than Fi.
let (2.3)
(2.4)
denote the empirical distribution functions of group 1,2 and the pooled respectively. Typically, goodness-of-fit tests are based on test statistics I(&(.)-fin(.))
or
Z((c,(.)-fin(.))+)
sample,
(2.6)
on a suitable function space. As for K1 or KZ, where Z( .) denotes a semi-norm motivation, let us first consider the sup-norm IlfI/:=~up,,~ If(t)1 leading to Kolmogorov-Smirnov tests. As extension of classical invariance theorems, we obtain the limit distribution of (2.6) for arbitrary distributions. Let (Bo(t))tEro.ll denote a standard Brownian bridge. Lemma 2.1 is known as a special case of more abstract empirical process theory.
402
A. Janssen / Two-sample goodness-of-fit tests when ties are present
Lemma
2.1. Assume that X1, . . . ,X,
suppose
that n1 ~n2-+co.
are i.i.d. with joint
distribution
function
F and
Then
112
nln2
supI &(t)-&(r)+
(> n
(2.7)
sup I&(F(t))l
tPR
toi?%
and nl
n2
l/2
sup (d,(t)-&(t))+
C-1 n
5
supB,(F(t))+,
tER
(2.8)
fElR
both in distribution.
For completeness
we will give a proof of Lemma 2.1. Let us introduce
the following
abbreviations:
(I{
c ,.= ~ n,.
li2 lh,
n
-
i=l, ..
(2.9)
i=n,+l,...,n,
Un2,
where the coefficients satisfy Cy=, cii= 1 and max,,iGnlc,il+O Under H,, we may assume
whenever
Xi:=F-‘(Ui),
(2.10)
where F- ’ denotes the left-continuous U1, LIZ, . . . are i.i.d. uniformly distributed
inverse (or quantile on (0,l).
Proof. Taking (2.9) and (2.10) into account, u~(0, l), proves the following representation:
0
nln2 n
the equivalence
(~n(t)-fin(t))=i
principle if
l(-m,t](xi)
t iff u< F(t),
cni
for weighted
(0.~1 C”i)
processes
sup If(s)I soF(R)
on D[O, 11. Then standard plete the proof. 0
implies (2.12)
~(~0(4)s.r0,11
cni ) SE[O,
i=l
empirical
(2.11)
cni.
11
in distribution on D[O, 11, see Shorack and Wellner consider the almost surely continuous functions f~
F-‘(u)<
of F and
i=l
t”i)
(
function)
l/2
=i$ll[O,F(r)l The invariance
n, An2+co.
and f~ arguments
(1986, pp. 88 and 93). Next
(2.13)
sup (f(s))’ ssF(W) of Billingsley
(1968) together
with (2.11) com-
403
A. Janssen / Two-sample goodness-of-fit tests when ties are present
From
Lemma
2.1 we derive
Kolmogorov-Smirnov
easily
the
asymptotic
null
distribution
of the
tests: (2.14)
(2.15) where Z(f)= I/f/l. Evidently, (Pi and $. are asymptotic level CI tests if our critical values c and d are the (1 - u)-quantiles of the limit distributions (2.7) and (2.8) as long as F is not a Dirac distribution and additionally in the case K, we have a< l/2; cf. Remark A of the appendix. However, the test statistics are no longer distributionfree since c=c(F([W)) and d =d(F(R)) depend on the range of the unknown distribution function F. Note that qo, and $,, become conservative if c and d are substituted by the (1 - U)quantile of Smirnov’s statistic (continuous case). Observe also that c(F”(R))
d(F”(R))
and
(2.16)
whenever ~(R)cF(R). A finite sample result of this kind was obtained Guilbaud (1986). To overcome problems with unknown critical values, we will construct of-fit tests that are distribution-free. In particular, we propose permutation fix the idea, consider first the Kolmogorov-Smirnov statistic: SUP
i
l(-m,Xj:,](xi)cni
Permutation permutations
>
goodnesstests. To
I
tests are based on uniformly of 1, . . . , n)
distributed
permutations
(on the set Y,, of
GwH~ni(L3))i
~n=(“ni)r
defined
by
(2.17)
.
(+)
IGIl I( i=l
earlier
on a further
probability
OCQ the permutation
statistic
space (6,2,
P) independent
of Xi : Q+R.
For fixed
of (2.17) is given by (2.18)
which is equal in distribution &++sup j
to
i$I lc-m,X,:,(o)J
For fixed w let x H Gr)(x, w) denote the distribution random variables &(o) and y,(w)~[O, l] as solutions
s
(2.19)
~xi:~~~~~c~~~~~~~)~+~~~
I(
(l(~,,na)(x)+~nl(~.)(~)}G,(dx,o)=a.
function of
of (2.19). Next choose
(2.20)
404
A. Janssen / Two-sample goodness-of-fit tests when ties are present
Then the permutation
test for H,, against
1
&:=
I y.
I
((
0
111112 n >
K1 >
112 (G,(t)-H,(t))
is an exact level c1 test for each F and each
n.
(2.2 1)
=I?"
)
<
Similarly,
permutation
obtained for K2; see (3.20). The main results of the present procedures work well for a large class of semi-norms.
Remark
2.1. To
u 1, . . . , U, defined bution of
paper
tests 6” are
show that these
give further motivation, consider antiranks (D,i(U))i<” of implicitly by Ui: n= UDni(“). Then we have equality in the distri-
by (2.10) and (2.11), since (F-’ (Ui:n))i
If (2.17) is
(2.23)
SUP
jan we get further insight in the permutation new permutations gn. For computational Kolmogorov-Smirnov can be recommended.
statistic (2.19). The antiranks
are replaced
by
reasons one may be interested in additional procedures of type for large n. In this connection estimated critical values Note that the right-hand side of (2.7) reads as (2.24)
sup IBo(s)I. seF(W)
It is reasonable to estimate the unknown fixed w let cn(. , co) denote the distribution
range F(R) by P,(R)= function of
{F^,(Xi):
i
For
(2.25) and
let C,:= c,(. , co)- ’ (1 -a)
critical
value C. similar
be its (1 -a)
to (2.21). It is shown
quantile.
Let
that (Pi - (P,,p
(P,, be the
test
with
0 and (P,,- & p
in probability under nondegenerate distributions of Ho. In this case the nominal level of (P,, converges to CIunder Ho as n+m. The critical values E,, can be obtained by a Monte Carlo study. Similar results hold for one-sided tests.
0
405
A. Janssen J Two-sample goodness-of--fit tests when ties are present
3. Main results In this section
the motivation
above is made precise and the results are extended
to a wider class of goodness-of-fit integral
tests relying
tests given by semi-norms.
examples
are
l/2
(~,(t)-fi,(t))2 q(&t))d&(t) or their one-sided
Typical
on
(3.1)
versions
(3.2) where q: (0,l)-+[O, 00) denotes a suitable weight function. The choice q= 1 gives the Cram&von Mises test (cf. Hijek and SidSk (1967, p. 93)), whereas q(t)=(t(l -t))-‘, 0 < y < 1, leads to tests which give more weight to the extremes. For y = 1 one obtains the weights proposed by Anderson and Darling (1952) for one-sample tests. As motivation of these tests, observe that c,(t)-l?,,(t) has conditional variance var(G,(t)-B,(t)1
X1 :n, . . . ,X,,,)=
&
Q)(l
-R(t))
givenX 1 : “, . . . , X,:, (apply (2.23) and Hhjek and Sidak (1967, p. 61)). Thus (3.1) includes a weighted renormalization via that variance. We now give a precise formulation of our problem. Consider more general regression coefficients C,i such that
(3.3) All our goodness-of-fit
x,,,:= i
tests are based on the general l,_
co,X~,.]:n](Xi)
cni,
OdsG
rank process
(3.4)
l
i=l
([xl denotes the integer part of x), and on the pivoted (via Dirac measures E.) by
empirical
measure
&,, defined
(3.5) In particular,
the Kolmogorov-Smirnov sup SSIO, 11
IXnJl=suP{
IXJ:
test is obtained s=uPP(&l))~
by (3.6)
406
Integral
A. Janssen / Two-sample goodness-of+
tests are obtained
tests when ties are present
by (3.7)
which obviously coincide with (3.1) in the two-sample case. For one-sided testing problems X,,, is substituted by X,7,. The examples have a common feature which is covered
by the following
class of semi-norms.
To explain
_I(~,r)): F distribution
Jq)Jl:={6P(F~F-1~3
this, let
function
on R)
denote the pivoted distributions on [0,11. In addition, consider for A c[O, l] the restriction D(A):= {jjA:fgD [0, l]} of the Skorokhod space D [0, l] on A. Assume now that
001, H-@‘[o, II,
ZH: D(supp(W)+CO, is a given family of measurable
semi-norms.
Then we may define a new function
ll-Kh ~01, z(H,f):=I,(f;,,,,(,,),
I:~,o,l,xDC@
which for fixed H defines a semi-norm f+-+ Z(H,f) Kolmogorov-Smirnov semi-norm is obtained by
Iidff,f):= and the integral
(353)
sup
on D[O, 11. For instance,
by
Zcq) (H 3f).=. Based on this function tests
the
(3.10)
If(s)1
tests are obtained
(3.9)
(3.11) I, (3.9), we now introduce
the general classes of goodness-of-fit
(PII= l@,,Oa,(Z(&I> (Xn,s)sE[O,11) along with the one-sided
(3.12)
versions
*II=1 Cd,,rn~(Z(WXL)SE,O,
1,))
(3.13)
provided the statistics under consideration are measurable. (Note that (2.14) and (2.15) are included in these classes). For Zks and ZCq)keep (3.6) and (3.7) in mind. If ties are present, the evaluation of exact critical values c, and d, yields serious problems. Also asymptotic distributions (if available) usually depend on the unknown distribution. As in Section 2, permutation versions of qn and $,, are proposed. Following the approach of (2.19) let us introduce for fixed w the permutation statistic of X,,, (3.4) by n
(T,HX”n,s(0,6)=2 i=l
l,_ m,Xi,,j:n’w’]txi : ntw)) cno,i ($1
(3.14)
407
A. Janssen / Two-sample goodness-of-jit tests when ties are present
for 0~s~ (6,2,
1, where again the permutations
F) independent
((T”i)i lie on a separate
of the observations.
we will now introduce
the permutation
For technical
probability
and computational
space reasons
process L-1
Z n,s
:!%C[O, 11,
o
1,
Z,,,@):=
c
(3.15)
c.,,<(@+Kl(s)>
i=l
with remainders R,(j/n) = 0, j = 0, . . , n, such that Z,,,, is continuous linear on [j/n, (j+ 1)/n]. It is most important that the permutation depends
in s and piecewise statistic (3.14) only
on o via A”(w) and on & via Z,,,(&).
Lemma 3.1. For each family of semi-norms (3.9) we have (3.16) and a similar result holds for _flS and zc,. one has
Proof. Check that for s=j/nEsupp(&(m)) nnL(O,j/nl x”n, j/n=
C
(3.17)
c~~,i = zn, rh,(O, j/n]
i=l
and
A,(O,j/n] =j/n.
suPP(&).
Thus
s H x”,,,
and
SH Z,,,
coincide
on
the
random
set
0
For fixed OEQ let now Gb+) (.,w) ~Hz(~,(0),(ZbTC,)(~))SEIO,
denote
the distribution
functions
of (3.18)
11),
where the index ‘+’ always indicates the one-sided case. If no other comments are made, both cases are treated equal. Clearly, the exact level a permutation tests associated
to (3.12) and (3.13) are given by >
1
I
@“I= Yn ~(fi,,(X,,S)SS[O,1]) 0
=c”, <
(3.19)
zj.2
(3.20)
and 1 $n:=
0.
I(J%, (XJSSIO, 11)
i 0 where the random variables treated similarly via G,’ .
< yn and c”, are determined
by equation
(2.20) and $,, is
408
A. Janssenl
Two-sample goodness-of-fit
tests when ties are present
Our treatment of permutation tests is based on convergence distributions G!,+‘(. ;) under Ho. Consider FeHo and define
of their conditional
HF:=~(F.F-ll~,(O,l)). Also introduce
unconditional
(3.21) distribution
functions
G and G+
G(+)(x):=Q(Z(HF,(Bo(s)(+)),)~x),
(3.22)
given by the distribution Q of B0 on C[O, 11. Recall from the appendix that G is typically absolutely continuous for Zks and 1(q) but G+ might have a jump at zero. Thus a different treatment of G and G+ is required. Let d(F,,F*)=inf{s:
Fi(x-E)-sEFFz(x)dF1(x+s)+s
for all x)
(3.23)
denote the Levy-distance of two distribution functions F1 and Fz. This is a metric for convergence in distribution. Call FEH, nondegenerate if F is not a dirac measure. It should be mentioned that assertion (a) below is already contained in Bickel (1969) for the two sample-problem (2.9). Distribution free Kolmogorov-Smirnov tests for randomly censored data were recently established by Neuhaus (1992, 1993). Their main concern is with the problems that arise from censoring. Theorem 3.1. Consider the Kolmogorov-Smirnov semi-norm I = ZKS(3.10) or the integral norm I = Zcg)(3.11) with continuous weight function q : (0,l)+(O, co) such that q(s)
O
(3.24)
holds for some K > 0 and 0 < y < 1. Let (Xi), be a sequence of i.i.d. random variables with nondegenerate distribution function F. (a) Zf Z= ZKSor Z= Zcg)with bounded weight function (y = 0) then OH
and
SUP
IG,b,N--(x)1
xe[O,30)
o+-+d(G,+ (.,m),G+(.))
(3.25)
(3.26)
converge to zero almost surely. In both cases the d@erence of the quantile processes o~sup]G(,+)(s,~)-l-GG(+)(~)-l/
(3.27)
SEA
converges to zero almost surely uniformly on compact sets Ac(0,1). (b) Next consider unbounded weight functions q (3.24) and their integral norms Z=Ztg). Then the assertions (3.25)-(3.27) remain valid tfalmost sure convergence to zero is substituted by convergence in probability.
A. Janssen J Two-sample
Proof. Section The proof
5.
goodness-of-@
409
tests when ties are present
Cl
of Theorem
3.1(b) is based
on an approximation
bounded weight functions. As we will see in Theorem under general circumstances. Consider the following family of semi-norms. Condition (A): Let Z(H, f) = 1,(f)
procedure
for q by
3.2 that type of argument works assumptions for the underlying
be as in (3.8) and (3.9) nontrivial
semi-norms
such
that with Q as in (3.22)
QIWF, (B&))MO, m))>O holds for nondegenerate
(3.28)
F and assume
Q(~(H,,(IB,(s)I),)E(O,
a))>0
(3.29)
for the one-sided case. Assume also that G and G+ (3.22) are nondegenerate limit laws. Condition (B): In the case of one-sided tests we assume I(H,,.) to be positive increasing, meaning that I(HF,f)dl(H,,g), Condition
(C): (Unconditional
whenever
O
convergence).
(3.30)
For each nondegenerate
be convergent in distribution as rz+c~ (Their limit distribution were already specified in (3.22).) These assumptions require further comments.
functions
FEH,
let
G and G+
Remark 3.1. (a) Under condition (A) the random variable I(HF, (B,(s)),) has a proper distribution G (3.22) on (0, co). If in addition condition (B) holds then G’ is concentrated on [0, co). Confer the appendix. In both cases the quantile functions G(+)- ’ are continuous on (0,l) which implies the equivalence of tests (see Lemma 3.2). (b) One easily checks that unconditional convergence (3.31) is necessary for conditional convergence (3.25) and (3.26), respectively. For ZKSassertion (3.31) was earlier obtained in Lemma 2.1. Theorem 3.2. Consider afamily of semi-norms IH( .) such that the condition (A)-(C) hold for
the two-sided
semi-norms
and one-sided
case,
respectively.
Assume
that there exist further
(Ik,H)ksN (3.8) with
(3.32)
410
A. Janssen / Two-sample goodness-of-fit tests when ties are present
for all
HE&[,,, II. Define lk(H,f):=Zk,n (f) and let G:yd(.,o) (3.18) and GCkJCf)(3.22) denote the corresponding distribution function belonging to Ik( .). Assume that under nondegenerate FEH, o+d(G:;n)(
., o), GCk)(+)(.))
(3.33)
converges to zero in probability as n-co for each keN. Then we have (a) the same assertion (3.33) holdsfor d(G!,+‘(. ,a), G’+‘( .)), (b) the difSerence of the quantile processes uniformly on compact sets A. Proof. Section
5.
(3.27) converges
to zero in probability
0
These results prove asymptotic
equivalence
of ordinary
and permutation
goodness-
of-fit tests under the null hypothesis. To establish a result of this kind consider conditions (A)-(C). Then there exists xb+‘~R! such that G(+) 1cxr~,mj is absolutely (+).-.- 1 - G(+) (x0) > 0, see the appendix. The choice continuous satisfying txO c,+G-‘(l-cc),
GI
and
d,,+(G+)-‘(l-cc),
a<~$,
(3.34)
lead to asymptotic cc-similar tests cp,, and Ic/, (3.12), (3.13). By the appendix we can choose x0 so that CI~= 1 and c&J= l/2 for I~{l,s, Pq’} of Theorem 3.1. Note that G(+), c, and d, are not really available in practice under ties. Lemma 3.2. In addition to these assumptions suppose that under nondegenerate we have convergence of oHd(G;+)(.,o),G(+)(.))
FEH,
(3.35)
to zero in probability. Then (a) for each ~
(Pl#-d~-O P
(3.36)
in probability under F; (b) for each ~
idtL~0 in probability
(3.37)
under F.
Proof. That proof easily follows from the almost sure subsequence convergence principle for convergence in probability. Choose a subsequence nk such that (3.35) converges almost surely along nk. We have almost sure convergence of quantile functions along nk and (P,,~- &,---+ 0 since G is strictly increasing at G-‘(a) for ~
cf. Witting
and Nijlle (1970, i. 58). Now we may choose a further subsequence
A. Janssen J Two-sample
n; such
that
proved.
Cl
(3.36) converges
goodness-of-/it
almost
surely
tests when ties are present
along
n;.
Lemma 3.2 can be used to evaluate the asymptotic under local alternatives, cf. Section 4.
411
Similarly,
(3.37) can
power function
be
of @, and $”
As explained computational process (Z,,,),
in the introduction other approximations as 9, of (Pi are of interest for reasons whenever n is large. Throughout, it is shown that the rank used for the definition of Gr’ (. , co) can be substituted by its limit process (B,(s)), to get asymptotic quantiles. Assume that the Brownian bridge is defined on a further probability space (fi, d, Q”)independent of the observations. For fixed o~s2 introduce G”(.,u) and GT (.,w) by G:+‘(x,w):=
(3.38)
Q”(I(~~,(Bo(S)(+)),,[O, 1,)d.x)
and let I&, and $n denote
the tests (3.12) and (3.13) given by critical
c,,:=(G,,(.,w))-‘(1-a) These tests work with estimated asymptotically a-similar tests.
and critical
Theorem 3.3. Under the assumption
values
&,=(G~(.,o)))~(~--cr).
(3.39)
values. Again we will see that @, and qn are
of Theorem
3.1 we have for IE{I~~, Icq’]
sup I~,(x,o)-G(x)l+O XE[O,3(1)
(3.40)
and (3.41) in probability.
In addition, we obtain (3.42)
in probability tests.
under nondegenerate
Proof. Section As conclusion tests and further
5.
FEH,,
whenever
a< l/2 holds for the one-sided
0 the Kolmogorov-Smirnov, Cramer-von integral tests Cp, and 4, with estimated
Remark 3.2. Consider that
the two-sample
O
n
problem
1.
Mises, Anderson-Darling critical values work well.
with regression
coefficient
(2.9) such
(3.43)
412
A. Janssenl
Two-sample goodness-of--fit
tests when ties are present
holds. Then the results of Einmahl and Mason (1992) can be used to sharpen assertion of Theorem 3.1(b). Actually, one can prove that (3.25)-(3.27) are almost surely convergent for those weight functions (3.24) determined by 0 < y < l/2. Note that (3.43) yields XI= I c$ = O(n), which is one assumption of Einmahl and Mason.
4. KolmogorovSmirnov
tests under local alternatives
Our treatment of the power of conditional goodness-of-fit tests @, and Cp.is closely related to the continuous case already analysed in the literature. For these reasons we restrict ourselves to the Kolmogorov-Smirnov test in order to show which type of modifications are required. The continuous case was successfully studied in Milbrodt and Strasser (1990). Here the reader finds references about the asymptotic power functions of goodness-of-fit tests. Moreover, we refer to the early paper of Chibisov (1965), where the likelihood ratio (4.3) and the power function was treated within a restricted continuous case. The key for our results is the asymptotic equivalence of both Gn and (Pn to unconditional tests which can be treated under local alternatives. Throughout, the modern approach via tangent vectors is used, cf. Pfanzagl and Wefelmeyer (1982) and references therein. The model (4.1) was earlier proposed by Janssen and Milbrodt (1993). Let 9 + Ps denote an &-differentiable curve at 9 = 0, see Strasser (1985, Section 75), whose tangent gulf’:= {gE&(P,): s g dPo =0} 1s g iven by the L,(P,,)-derivative of 9 -+ 2 (dPg/dPO)“*. Starting with regression coefficients (3.3), we introduce the model of the joint distribution of XI, . . . ,X, as
2(X
x,1=
l,...,
6 Pcni=:Pn,F,p s~q’(po) >
(4.1)
i=l
where F denotes the distribution function of PO. The parametrization allowed within the asymptotic setting since
I6
P,,,-
6
i=l
i=l
p:,,
by F and g is
-+O II
in the variational distance whenever Pg and P$ admit the same tangent g. This is due to local asymptotic normality (LAN). Note that L2-differentiability implies the LAN of the experiment &I=(~“,~~, cf. Strasser
{P”,F,& s~L:“‘(Po))),
(1985, Section dP” F d=U)-; log dP n,F,O
(4.2)
79.2), given by the approximation
g s
g2dF+op,,F,o(l),
(4.3)
413
A. Janssen 1 Two-sample goodness-of--fit tests when ties are present
where I,,( .) denotes
the central
L,(g)(x):=
i
sequence x=(x1,
C7(xi)9
cni
,X,)T.
(4.4)
i=l
Let gl,gELr’
(P,). Then Le Cam’s third lemma
s
in distribution
under
g1 0
s 1
1
Jxg1)~
implies
F-l (+A-,(du)+
0
gloF-‘(u)goF-‘(u)du
(4.5)
0
P,, F,4. Note that the right-hand
local alternatives
side of (4.5)
has mean
s s
gPF-‘g°F-‘d&o,l)=
and variance
h°F-')2d~,co,l,=
These arguments Gaussian
motivate
s
s
(4.6)
glg2dP0
(4.7)
g:dP,,.
that the limit experiment
GF of E, (4.2) is given by the
shift
GF:=(CCO, 11, WCC& 111,{QF, g: s~L’,“‘(J’o)l), with QF,g defined log
by
dQ~,~ _ %,o
(4.8)
’
’ g20F-l(u)du,
g#(u)B,(du)-;
s o
s
(4.9)
0
where QF, o = Q is the distribution of the Brownian bridge Bo. This model is a submodel of the case where the uniform distribution AI(o, 1J (= PO) is restricted to the tangent set (g 0 F-‘: gall’}, cf. Strasser (1985, Section 82.23). Moreover, we have convergence of experiments E,+GF in the sense of Strasser (1985, Section 80.6). For these reasons one expects that the asymptotic power of the Kolmogorov-Smirnov test can be calculated within the limit model CF. These arguments can be made rigorous. Theorem 4.1. Let (P”(3.12), & (3.19) and & (3.39) denote the two-sided asymptotic a~(0,1) Kolmogorov-Smirnov tests dejined through (3.10). Then we have
level
a(s) := n+m lim EP”,~,# cp.= n-m lim EP,,,~,~$J,,=n-+m lim EP,,F,g(Pn (4.10) where c, denotes the (1 - cc)-quantile of (2.24).
A. Janssen / Two-sample goodness-o@
414
tests when ties are present
Proof. The present proof follows standard arguments of Hajek and Sidak (1967 Chap. VI, Section 4.4), see also Milbrodt and Strasser (1990). Consider first -F(t), LE R. Then (4.5) is equal to 91=1,-m,,, F (0 Bo(F(r))+ which is actually i
g”F-‘(u)du, s
(4.11)
0
the limit distribution crd
of (4.12)
l(-mrt](Xi)
i=l
device establishes the convergence of their finite under Pn,F,S. The CramCrWold dimensional marginal distributions. In addition tightness of (4.12) remains valid under contiguous alternatives if we consider s~F(R)n(0, 1) and replace t =F’(s). Thus we may apply the functional f-
(4.13)
sup If(s ssF(R)
which completes
the proof.
Remark 4.1. The limit
0
experiment
GF (4.8) can be written
in an equivalent
form
given by ~F=(CO([W),~(CO([W)),{~F,g:gEL(20'(PO)})
on the set of continuous
functions
C,(R) on If%vanishing
at infinity.
The distributions
OF4 are defined by the shift model
&~,,:=~((Bo(F(x))+~~_~,~,
gdF)xeR)
of the Brownian bridge. Obviously, the corresponding log-likelihood as in (4.9). The right-hand side of (4.10) then reads as
Next we briefly sketch some applications continuous case. We restrict our attention regression coefficients (2.9).
ratio is the same
of Theorem 4.1 and some extensions of the to the two-sample testing problem given by
Example 4.1 (Continuation of Example 2.1). (a) The conditional two-sample Kolmogorov-Smirnov tests &, (3.19) and (Pn (3.39) given by IKS are asymptotically admissible in the following sense. Assume that qn denotes a further asymptotic level LXtest for Ho (2.1) and suppose
lim inf&,, n-30
Fr
g
r?,2 B(s)
(4.14)
A. Janssen / Two-sample
goodness-of--fit
tests when ties are present
415
for each gEL$‘) (P,) and each PO. Then we have
in
probability. The proof follows the lines of Strasser (1985, p. 436). Pn,F,g (b) (strict asymptotic unbiasedness of (Pn and I$,,). For each gELi’) (P,)\(O), we have p(g) > CI,where p is as in (4.10).
(c) (consistency). For g#O and t,+co, we have fl(t”g)+l as y1--tcc. (d) As further consequence one immediately generalizes the principal component decomposition for the power of Kolmogorov-Smirnov tests of Milbrodt and Strasser (1990) if ties are present. Note that for each gEL$“(P,) we have ~(tg)=cl++a(g)t2+o(t2)
(4.15)
as t+O,
where a(g)=a(g, c()30 denotes the curvature of the power function along the ray {tg: PER} at t=O. It can be shown that for fixed PO the curvature g-a(g) admits a principal component decomposition in the sense of Milbrodt and Strasser (1990, Theorem 2.8).
5. Main proofs The proof of Theorem 3.1 is based on an conditional invariance principle. denote i.i.d. uniformly distributed random Throughout let Ui, U,, . . . : Q-(0,1) variables with standard empirical process cn(m,t). Let M denote (the probability one set) M=
W: SUP Iti,(O,t)--tl+O,
Ui(W)#Uj(O)
for all i#j
rsm, 11
For REM introduce
the process
SH Y,,, on [0,1] into C[O, l] by
Y,,,(w,c5):= i 1[O,s](Ui:n(u))C,,~i(~)+~Rn(S),
(5.1)
i=l
with remainders making (5.1) continuous and Y,_, is piecewise linear in between. Lemma 5.1. For.$xed
o~A4
YE,s (QA6) 5 weakly in distribution
in s such that R,(O) = R,( 1) = 0, R,( Ui : ,,) = 0
we have
Bo(4
on C[O, l] under uniformly distributed
(5.2) permutations
(a,i(G))i,,.
416
A. Janssenl
Two-sample goodness-of--fit
Proof. Let REM be fixed. According (1968, Section 24), we have
tests when ties are present
to Hajek and Sidak (1967, p. 186) or Billingsley
weakly in C[O, 11, where Z,,, is given in (3.15). Check that Ui : n(w)
yn,s=~“,ii”(o,s,. Let (Pi : [0, l] + [0, l] denote
the one-to-one continuous transformation given by i n, which is linear on [ Ui _ 1 : ,, (CO), Ui : n (co)]. Conserp,(O)=O, $%(I)= 1, cPn(Ui:, (m))= / quently, we have
yn, s=
&I, qpn (s) .
Since sup(l~,(s)---sl:s~[O,
(5.5) l]}-0,
it is easy to see that (5.6)
llfnO(Pn-fll+O> whenever /fn - f 11+O in C [0, 11. If we now combine 0 Billingsley (1968 p. 34) complete the proof.
(5.3))(5.6) standard
arguments
of
Proof of Theorem 3.1 (part (a)). The proof is given in two steps. Step 1: I = ZKs. Consider a fixed element OEM and Xi = F- 1(Ui). The basic identity (3.17) implies sup lZ!I’,’ I= sup lr7:;: I sssupp(ni.) SPS”PP (4)
if we take (2.11) into account.
fH
Now we can apply the continuous
function
sup Im’+‘I SOP(R)
and the invariance principle (5.2) proves the desired results. Note that G is continuous according to the appendix. Step 2: Z=Ztq), q bounded. Lemma 5.2 below shows weak convergence of &,(o)+HF on [0, l] for all o lying in a probability one set N. Let oeN be fixed and
A. Janssenl
consider
Two-sample
fn, fE{gEC[O,l],
goodness-of--fit
g(O)=g(l)=O}.
tests when ties are present
Then
417
it is easy to see that
llfn-flI+O
implies (5.7)
jffqd&+f’qdH,. The invariance
principle
(5.3) together
with Theorem
5.5 of Billingsley
(1968) now
imply (.G~~‘)‘q(s)d&,(~,s)+ which completes
BbfW2q(s)dH&),
s
s
the proof of part (a).
0
Lemma 5.2. Under H, we have weak convergence
of
A,jHF
(5.8)
on [0,1) almost surely, where HF is de$ned
Proof. Let US use again Xi = F 5.1, and set A:=
by (3.21).
’ (Ui). Define N:= Mn@,
where M is as in Lemma
.
o: supI&(F(x)I+O XER
Let S(q) denote the set of continuity points (5.8) will be established for fixed oeN. (1) First let us verify that
of a monotone
function
cp. Throughout,
lim inf &, [0, y] 3 HF [0, y) “-02 holds except for a countable distribution the inverse
subset
(5.9) of (0,l).
Note
implies F^;’ (y)-+F-’ (y) for all ~ES(F-‘), of t H f,(o, t). Moreover,
that convergence
choose z
Since o~h?,
we have by (5.10)
lim inf riz, [0, y] > F(z), n+m where F(z)fF(F-’
(y)-)
as zfF_’
(5.11) (y). Note
that in addition
HFCO,Y)=E,~~~,~,(U:F~F-~(~)
This statement,
together
in
denotes
(5.10)
&J-o~Yl3F^,(R(Y)-).
For ~ES(F-‘)
of t+F,(t)
y~(0, l), where F^;’
F-‘(u)
with (5.1 l), implies
assertion
(5.9).
418
A. Janssen / Two-sample goodness-of@
tests when ties are present
(2) The converse inequality of (5.9) for limsup can Consider random variables I’, on (0,l) with distribution fixed woN (cf. Lemma 5.1). Consequently,
V,$+
be obtained as follows. function TV fi,,(~, t) for
U, and FOF-‘(I’,)*
FOF’(Ui)
converge both in distribution since F 0 F- ’ is IIIco,1j almost everywhere Check that F- ‘(V,) has distribution function t H $,,(a~,t). Thus c!Z’(Fd,‘lA,,,,
~,)=Z(FOF-~(I’~))+H,
weakly on [0, l] as n-03. ^ & = y(F, Choose observe
now that
(5.12)
On the other hand, note that
^ o F, ’ 11,co,1)).
continuity
continuous.
points
(5.13)
y, y+ E~(0, l), s>O,
of x H HFIO, x].
In addition
I F^,(x)-F(x)1 de for all x and II large enough. large n &cO,Yl=~,(,,l,(~:
If we take (5.12) and (5.13) into account,
F^,(F^,‘(u))dy)
dA,(o, l)(u: F(F^,l(u))d~+~)~H~CO,~+~l. Altogether, (5.9) and (5.14) yield convergence of (0,l). 0
q(s) dH&) dQ =
s
VWM4Ms)
The present proof is based on Theorem sequence of seminorms (3.32) as
h&f)
=
(f
f’ mink, 4 dH
(5.14)
of &,[O, y]+HFIO,y]
Proof of Theorem 3.1 (part (b)). First, it can be checked finite almost surely. Observe that by Fubini’s theorem
Is&(s)~
we have for
for a dense set
that JB,,(s)2q(s)dHF(s)
dH&) < 00
3.2. We may choose
.
is
(5.15)
the approximating
11-7
,
(5.16)
with bounded weight function min(q, k) for kEN. Note that assumption (3.33) is valid according to part (a) of the theorem. In order to apply Theorem 3.2, it is now enough to prove unconditional convergence, see condition (C). This is done in Lemma 5.3. The convergence of the quantile functions (3.27) can be obtained as follows. Note that supp(G’+‘) is a closed interval and that the inverse G(+)-’ is continuous on (0,l). Thus convergence in distribution implies here pointwise convergence of their inverse functions. Monotonicity arguments yield uniform convergence on compact sets.
A. Janssen/ Two-sample goodness-of-fit tests when ties are present
Lemma
5.3. For each weight function
s
(X!CA2 q(s) d&,(s) --%
in distribution
under nondegenerate
419
q given by (3.24), we have
s
(&,(s)‘+‘)~ q(s)dH,(s)
distributions
(5.17)
FEH~.
Proof.
Throughout consider continuity points 6 and l-6, 0~ 6 < 1, of the distribution function of HF (3.21). Introduce qs(s):=q(s) 1C6,1_6j (s). According to Billingsley (1968, Theorem 4.2), it remains to check the following three conditions. (1) For 6 > 0 we have convergence in distribution of
s
(X!Z2qs(s)
d% (s) *
(Bb+‘(s))2qd(s) dH,(s). s
(2) We have
s
(@,+’ (s))2qa(s) dHF (s) ---%
in distribution
(Bb”(s))2q(s)
dH,(s)
as 6 JO.
(3) lunli_m supE
(X:::)’
(q(s)-q4s(s))d@,(s)
(S Assertion (1) follows from the conditional convergence theorem for the bounded weight function qs, see Remark 3.1(b). Note that qs is HF almost surely continuous and (5.7) carries over. It should be mentioned that (1) also follows directly along the lines of the proof of Theorem 3.1(a). Moreover, observe that condition (2) is verified provided ;gE U holds.
Obviously,
(Bd+)(~))~(q(s)-qqs(s)dH,(s)
(5.18)
=0 1
it remains
to check
the conditions
(3) and (5.18) for two-sided
tests. Note that E
B:(s)(q(s)-&))dHF(s)=
s
E(B;(s)(q(s)-&))dH,(s)
converges to zero as 6 JO which implies (5.18). Similarly, the third condition can be verified which is done below. As in Section 3, we get that the conditional variance of X,,, given the order statistics is
420
A. Janssen / Two-sample goodness-of--fit tests when ties are present
which is equal to [n/(nthe order statistics, E
U
l)] (~(1 -s))
for s=j/n~supp(&,).
If we now condition
under
we have
X,2,,(q(s)-qqd(~))d~,(~)
=SS~(s(‘-r))(q(S)-q6(S))dl(s)d~(X1:”,
s
where the upper bound
X”,,)
~,((0,1)\(6,1_6))d~(X,:.,...,X,:.), converges
to (5.19)
KHF((O, l)\@, 1 - 6)). This limit follows from Lemma
5.2 and the strong law of large number,
which yields
&l({l})+HF((l)) almost surely for a proper treatment of the upper endpoint. Since (5.9) becomes arbitrary small for 6 10 the conditions (l)-(3) are established and the proof is complete. 0 Proof of Theorem 3.2. The proof is carried apply to GJ (. , w). Notice that for each x
out for G,( . , co). Analogous
arguments
(5.20)
Gk, n(x, w) 3 G,(x, 0). Let B denote the intersection of the sets of continuity points Throughout let XEB be fixed. The assumption implies as k-co
Gck’(x)- G(x)+0
(5.21)
On the other hand, we have Gk, ,,(x, o) - Gck)(x) 7 keN. This result combined
and
observe
(cni)i
as n+ cc for each
proves (5.22)
P
of (5.22) to negative that
0 in probability
with (5.20) and (5.21) immediately
(G,(x,o)-G(x))+-0. The extension
of G and Gk for each k.
(Dni(u))i
(Xi : n)i = (F- ’ (UC:n))i. Returning (2.22), that OH (X,,,(c.$ &,(~))
parts
runs
which
to our definition
and
as follows. turns
out
Choose
Xi=F-‘(Ui)
to be independent
(3.4) and (3.14) we see, similarly
(w, &)H (znJ~,
G), h,(w))
of to
A. Janssen J Two-sample goodness-of-jt tests when ties are present
have the same distribution we have
since fi, only depends
on order statistics.
421
By Lemma
3.1,
s
G,(x,O)dP(o)=P(I(~,,(X,,,),)Qx)~G(x),
which converges
according
to our assumptions.
Thus (5.23)
{G,(vJ-G(x)}dP(w)+O
s
holds for fixed XEB. On the other hand, we obtain
s s
(G,(x, co)- G(x))+ dP(o)+O
by (5.22). Combining
(5.24)
(5.23) and (5.24), we have
IG,(x,co-G(x)IdP(o)+O.
Now we may choose a countable
s
dense subset
{xiEB: igN}
of [0, co). Then
itI IG,(xi,~)-G(xi)l/2’dP(~)~O.
For each subsequence $I
there exists a further
subsequence
nk such that
IG,,(xi,~)-G(xi)//2’~0
holds almost surely. Along that subsequence we get G,,(x, w)- G(x)-+0 almost surely again for all continuity points x of G. This gives us convergence of d(G,(. , w), G( .)) to zero in probability. 0 Proof of Theorem 3.3. First consider
sup {lz::l)SSSUPPbw
sup
the semi-norm
IKs. Then, we have
{IBL+‘(s)l} d ll&(~)-G,. II.
(5.25)
SES”PPl~“)
According to (5.3) and Skorokhod’s embedding theorem (Shorack and Wellner (1986, p. 47), one can find versions of B,( .) and Z,, such that the right-hand side of (5.25) converges almost surely to zero. Together with (3.25) and (3.26) one gets assertions (3.40) and (3.41). The same arguments apply to I (4) for bounded weight functions. The device of the proof of Theorem 3.2 can now be used to establish the result for unbounded weight functions q (3.24). Consider again the approximating sequence Ik,H (5.16) of semi-norms and let G, T+)n denote their conditional distribution functions (3.38). Then we have, similarly to (5.20),
422
A. Janssen 1 Two-sample goodness-of-@
We already
proved
tests when ties are present
(3.40) and (3.41) for k. Hence
Theorem 3.2 can be adapted. Lemma 3.2. 0
The proof of statement
the arguments
of the proof
of
(3.42) is the same as that proof of
Appendix Let I : C [0, l] -[O, co] denote a measurable semi-norm and let Q = _.Y((B,(s),)) be the distribution of the standard Brownian bridge. It is well known that each measurable linear subspace Vc C[O, l] has either Q-measure zero or one, cf. Kallianpur (1970). A survey of results of that type is given in Janssen (1984). Evidently, Ni:= {f: I(f)=O} and N2:= {f: I(f) < co} are O-l sets. Throughout, we are concerned with the distribution of the semi-norm I. Lemma A.l(a) is due to Hoffmann-Jorgensen et al. (1979) who applied convexity arguments of Bore11 (1974). Let I be nontrivial, following lemma.
i.e. Q(N,uN;)=O.
Then, we have the
Lemma A.l. Let H:‘)(t):= Q(I((Bb+’ (s)),) < t) denote the distribution functions of I under B,, and BJ. Choose xbf):= sup{t: Hj+)(t)=O}. Then we have: (a) log H,(t) is concave on (x,,, co) and HI is absolutely continuous on that interval, (b) if1 is positive increasing (3.30) and Q(f: Z(lfl) < 00) = 1 the same result holds for H; on (x0’, co). This H:
result
implies
may have typically
Remarks condition
A2. (a) The
absolute a jump
continuity
distribution
QV((Bo(s),)@O,r))>O
of HI under
mild
conditions
whereas
at zero. HI
is absolutely
continuous
whenever
the (Al)
holds for all t >O. Obviously, (Al) is valid for nontrivial 11./I-continuous semi-norms on C[O, 11. (b) However, continuity of I does not imply H: (0) =0 and absolute continuity of Ht. For instance choose I(f) = If(y) )for some y@O, 1). More generally consider ZE {Zks, Ztq’} (3.10), (3.11) and nondegenerate FEH~. Then there exists y~supp(H~)n(O, 1). Thus
interval with lower endpoint (c) The support of H :+) is an (usually unbounded) xr’. (For reasons of concavity H:” cannot be constant on whole intervals between xy’ and the upper endpoint, since log Hi+‘(t)+0 as t+co). Hence the quantile function Hi+‘- ’ is continuous on (0,l).
A. Jamsen/
Two-sample
goodness-of-fit
tests when ties are present
423
Proof. The present proof relies on an obvious extension of Theorem 1.2 of subspace with Hoffman-Jorgensen et al. (1979). Let C,, c C [0, l] be a measurable Q((B,(s)),EC~)= 1. Consider a convex function q: Co-+[O, co), i.e. q(Af+(l -A)g)< Then t c, log Q(q((B,(s)),) < t) is concave nq(f)+(l-n)q(g), OGA< 1 and JgEC,. and
dP(q((Bo(s))s))
is absolutely
continuous
on
the
interval
(to, co), t,:=sup{t:
QM(&WMdt)=Ol. To prove this, consider
s, te[to,
l”{f: q(f)bt}+(l-A){g:
co). Then convexity q(g)ds}c{f:
of q implies
q(f)dIt+(l-+}.
If we proceed as in Hoffmann-Jorgensen et al. (1979), (l.ll), the statement follows from the results of Bore11 (1974). Obviously, this result gives assertion (a). To prove part (b) define q(f):=I(f +). Now it is easy to check that q is convex iff I is positive increasing, which is done below. Assume (3.30). Then (,?f+(l -i)g)+
take 0 d f< g and consider f= 4 (f-g)
+3 (g +f).
Convexity
of q yields
I(f)GfI((.!-g)+)+t1(g+f)d&U(g)+1(f)), since I((f-g)+)=O. Next we take care that q remains finite on Co= {j r(lfl)< 00). Note that (3.30) implies Z(f’)
Acknowledgement I am grateful to D.M. Mason Lemma 5.3.
who showed me the &method
(l)-(3)
of the proof of
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