New Trends in Probability and Statistics. Vol. 3 Multivariate Statistics and Matrices in Statistics: Proceedings of the 5th Tartu Conference, Tartu-Pühajärve, Estonia, 23–28 May, 1994 [Reprint 2020 ed.] 9783112314210, 9783112303054

Table of contents :
CONTENTS
PREFACE
I. ASYMPTOTIC DISTRIBUTIONS AND EXPANSIONS
ERROR BOUNDS FOR ASYMPTOTIC EXPANSIONS OF THE DISTRIBUTIONS OF THE CLASSIFICATION STATISTIC W AND RELATED STATISTICS
SOME RESULTS ON THE EMPIRICAL SPACINGS PROCESS AND ITS BOOTSTRAPPED VERSION
ON THE CORNISH-FISHER EXPANSION IN FINITE POPULATION
II. GENERAL LINEAR REGRESSION MODELS
ON PROPORTIONALITY OF REGRESSION COEFFICIENTS IN MISSPECIFIED GENERAL LINEAR REGRESSION MODELS
MORE ON PARTITIONED POSSIBLY RESTRICTED LINEAR REGRESSION
SIMPLER TESTS OF LINEAR INEQUALITY CONSTRAINTS IN THE STANDARD LINEAR MODEL
INCREASING THE CORRELATIONS WITH THE RESPONSE VARIABLE MAY NOT INCREASE THE COEFFICIENT OF DETERMINATION: A PCA INTERPRETATION
DISCUSSION ON THE PAPER OF C. M. CUADRAS "INCREASING THE CORRELATIONS WITH THE RESPONSE VARIABLE MAY NOT INCREASE THE COEFFICIENT OF DETERMINATION: A PCA INTERPRETATION"
TESTING LACK OF FIT IN MIXED EFFECT MODELS FOR LONGITUDINAL DATA
III. TESTS IN MULTIVARIATE STATISTICS
A PROJECTION NT-TYPE TEST FOR SPHERICAL SYMMETRY OF A MULTIVARIATE DISTRIBUTION
SOME APPLICATIONS OF DIRECTIONAL STATISTICS TO ASTRONOMY
SIMPLE METHODS FOR FITTING CIRCLES OR POINTS TO SPHERICAL DATA
IV. MULTIVARIATE NONPARAMETRIC MODELS AND THEIR APPLICATIONS
THE USE OF HELLINGER DISTANCE IN GRAPHICAL DISPLAYS OF CONTINGENCY TABLE DATA
BIVARIATE GENERALIZATIONS OF THE MEDIAN
DISTANCE-BASED REGRESSION IN SOME HELIOPHYSICAL DATA ANALYSIS
SOME APPLICATIONS OF MULTIVARIATE ANALYSIS IN ENVIRONMENTAL RESEARCH
METHODS FOR LONGITUDINAL SOCIAL NETWORK DATA: REVIEW AND MARKOV PROCESS MODELS
OPTIMAL INDUSTRIAL CLASSIFICATION
V. DISCRIMINATION AND CLASSIFICATION
SMALL SAMPLE PROPERTIES OF RIDGE-ESTIMATE OF THE COVARIANCE MATRIX IN STATISTICAL AND NEURAL NET CLASSIFICATION
TRIMMED k-MEANS AND THE CAUCHY MEAN-VALUE PROPERTY
ON SOME PROPERTIES OF k-VARIANCE
RANDOM MULTIGRAPHS, CLASSIFICATION AND CLUSTERING
VI. MATRICES IN STATISTICS
SHORTED MATRICES AND THEIR APPLICATIONS IN LINEAR STATISTICAL MODELS: A REVIEW
ARRAYS OF MULTIVARIATE STATISTICS AND THEIR REPRESENTATION
ON THE CALCULATIONS WITH THE INDEXIZED SETS OF DATA
ON STATISTICAL APPLICATIONS OF MATRIX MAJORIZATION

Citation preview

New Trends in Probability and Statistics - Volume 3 ' i Multivariate Statistics» and Matrices in Statistics

NEW TRENDS IN PROBABILITY AND STATISTICS Volume 3 Multivariate Statistics and Matrices in Statistics Proceedings of the 5th Tartu Conference, Tartu - Puhajarve, Estonia, 23-28 May 1994

Editors E.-M. Tiit, T. Kollo and H. Niemi

///VSR/// VILNIUS, LITHUANIA

UTRECHT, THE NETHERLANDS TOKYO, JAPAN

VSP BV P.O. Box 346 3700 AH Zeist The Netherlands

TEV Ltd. Akademijos 4 Vilnius Lithuania

©1995 VSP BV

First published in 1995 ISBN 90-6764-195-2 (VSP) ISBN 9986-546-03-6 (TEV)

All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without the prior permission of the copyright owner.

Typeset by TEV Ltd., Vilnius Printed in Lithuania by Spindulys, Kaunas

CONTENTS Preface I. ASYMPTOTIC DISTRIBUTIONS AND EXPANSIONS Error Bounds for Asymptotic Expansions of the Distributions of the Classification Statistic W and Related Statistics Y. Fujikoshi Some Results on the Empirical Spacings Process and Its Bootstrapped Version S. Rao Jammalamadaka and Jing Qin On the Cornish-Fisher Expansion in Finite Population I. Traat

II. GENERAL LINEAR REGRESSION MODELS On Proportionality of Regression Coefficients in Misspecified General Linear Regression Models U. Potter More on Partitioned Possibly Restricted Linear Regression H. J. Werner and C. Yapar Simpler Tests of Linear Inequality Constraints in the Standard Linear Model R. W. Farebrother Increasing the Correlations with the Response Variable may not Increase the Coefficient of Determination: A PCA Interpretation C. M. Cuadras Discussion on the Paper of C. M. Cuadras: "Increasing the Correlations with the Response Variable may not Increase the Coefficient of Determination: A PCA Interpretation" E.-M. Tiit Testing Lack of Fit in Mixed Effect Models for Longitudinal Data M. Palta and R. P. Qu

VI

Contents

III. TESTS IN MULTIVARIATE STATISTICS A Projection NT-type Test for Spherical Symmetry of a Multivariate Distribution Li-Xing Zhu, Kai-Tai Fang and Jin-Ting Zhang

109

Some Applications of Directional Statistics to Astronomy P. E. Jupp

123

Simple Methods for Fitting Circles or Points to Spherical Data R. W. Farebrother

135

IV. MULTIVARIATE NONPARAMETRIC MODELS AND THEIR APPLICATIONS The Use of Hellinger Distance in Graphical Displays of Contingency Table Data C. Radhakrishna

Rao

143

Bivariate Generalizations of the Median A. Niinimaa

163

Distance-Based Regression in some Heliophysical Data Analysis A. Bartkowiak

181

Some Applications of Multivariate Analysis in Environmental Research M. Greenacre and R. Fieler

197

Methods for Longitudinal Social Network Data: Review and Markov Process Models T. A. B. Snijders

211

Optimal Industrial Classification J. S. Chipman and P. Winker

229

V. DISCRIMINATION AND CLASSIFICATION Small Sample Properties of Ridge-Estimate of the Covariance Matr : x in Statistical and Neural Net Classification S. Raudys and M. Skurikhina

237

Trimmed /.--Means and the Cauchy Mean-Value Property ./. A. Cuesta-Albertos. /\. Cordaliza and C. Matrán

247

Or) some Piopenics of A'-Variance K. Pama and./.

I,ember

Random Multigraphs, Classification and Clustering B. Harris, li. Codehardt and A. florsch

267 279

Contents

VII

VI. MATRICES IN STATISTICS Shorted Matrices and Their Applications in Linear Statistical Models: A Review S. K. Mitra, S. Puntanen and G. P. H. Styan

289

Arrays of Multivariate Statistics and Their Representation E. Kaarik and E.-M. Tiit

313

On the Calculations with the Indexized Sets of Data O. Karma

325

On Statistical Applications of Matrix Majorization J. Hauke and A. Markiewicz

335

PREFACE

In Tartu, an old university city in Estonia, the tradition of organizing international conferences in multivariate statistics and its applications began in the early seventies. The working language of the conferences was Russian because the majority of the participants came from different regions of the Soviet Union. After Estonia regained independence, the conference tradition continued. In 1994, the same organizing committee, the academic staff of the Department of Mathematical Statistics, tried to organize a truly international conference on the same topic - multivariate statistics. The organizers expected this conference to be a meeting point for scientists from the West and East, but, unfortunately, only a few guests from the East participated in the conference. In any case, the conference turned out to be a success. About 70 participants from 18 countries travelled to Tartu, where the conference was opened. The following days were spent in the picturesque village of Piihajarve. The atmosphere of the conference was "friendly and stimulating", quoting our most distinguished guest Professor C. R. Rao. After the official conference, a one-day satellite workshop followed on the matrices in statistics, which was organized by Professor G. Styan. The majority of participants were interested in publishing the proceedings of the conference. Most of the papers were sent to the Editorial Board before the intended deadline. A selection of papers was made based upon referee reports and revisions. This collection of papers reflects the volume and range of the problems considered in the conference. The papers are clusterized into 6 parts, which are close to, but not identical with, the sections of the conference. The clusterization process has been rather statistical, where the clusters might overlap. According to some traditions in ordering the clusters and the papers in clusters (in the multivariate space) as the first characteristic the scale 'theoretical - practical' has been used, where the ordering number is stochastically increasing with the practical use of the paper. Using the rules of democracy and equality of all our guests, we did not separate the papers of the invited speakers in to special chapters, but improved the existing chapters by including the texts of invited speakers as introductions. On behalf of the organizing committee of the conference, I would like to express our gratitude to the invited speakers, honourable professors C. R. Rao, K.-T. Fang, Y. Fujikoshi, I. Olkin and G. Styan, who determined the creative atmosphere of the conference.

Preface

X

T h e local organizing c o m m i t t e e is very thankful to p r o f e s s o r s C. M . Cuadras, O . F r a n k , H. N e u d e c k e r , H. N i e m i , S . R a u d y s and G . Styan for their participation in the International O r g a n i z i n g C o m m i t t e e and their help during the w h o l e organizing p r o c e s s o f the c o n f e r e n c e . M a n y thanks to all participants, especially to the authors o f papers, those w h o g a v e s o f t w a r e demonstrations and poster presentations, and also to the a c c o m p a n y i n g guests, w h o helped to create a friendly a t m o s p h e r e during the conference. I wish to e x p r e s s the deepest gratitude o f the Editorial B o a r d to all our r e f e r e e s w h o efficiently g a v e many valuable r e c o m m e n d a t i o n s to our authors and helped to c o m p i l e the p r o c e e d i n g s . 1 hope that this c o l l e c t i o n o f papers preserves and passes onto the readers the stimulating and creative a t m o s p h e r e o f the 5th Tartu C o n f e r e n c e and e n c o u r a g e s them to participate in the 6th Tartu C o n f e r e n c e on Multivariate Statistics. E . - M . Tiit

Part One

ASYMPTOTIC DISTRIBUTIONS AND EXPANSIONS

Multivar. Statist., pp. 3 - 1 5 E.-M. Tiit et al. (Eds) © 1995 VSP/TEV

ERROR BOUNDS FOR ASYMPTOTIC EXPANSIONS OF THE DISTRIBUTIONS OF THE CLASSIFICATION STATISTIC W AND RELATED STATISTICS YASUNORI FUJIKOSHI Department of Mathematics, Hiroshima University, Higashi-Hiroshima, 724,..Japan ABSTRACT The classification statistic W is used to classify an observation as coming from one of two multivariate normal populations with common covariance matrix and different means when all these parameters are estimated from a sample from each of the populations. Let A be the Mahalanobis distance between the populations. Then it is known that the limiting distribution of (W - \ A 2 )/A as the sample sizes approach infinity is the standard normal distribution if the observation is from the first population. Further, an asymptotic expansion for its distribution has been obtained. The purpose of this paper is to obtain explicit bounds for the errors in these approximations. Similar results are obtained for the distributions of the Studentized classification statistic and other related statistics.

1. INTRODUCTION One of the important interests in discriminant analysis is to classify an observation as coming from one of two observations, based on a sample from each of the two populations. Suppose that we have a sample a ^ 1 ' , . . . , from FIi: 2 ) and sample x f \ . . . , x f y from Yl2: N P { ^ 2 \ Y ) where 1 f/ ) / 2 is positive definite and all the parameters are unknown. Let x be a new observation which comes from one of FIi and II2. Then the observation x may be classified into ITi or 112 by using the classification statistic, W = (xW - a ^ ' S " 1

(z« +

,

where x ^ are the sample mean vectors,

nS

=

f ^

¿=1

3=1

( x f - ¿W)

(XW

_

¿(0)',

(1.1)

4

Y. Fujikoshi

and n = N\ + N2 — 2. The distribution of W is needed, to know the probability of misclassification or the probability of correct classification. The distributions of related statistics are also needed for the statistical inference in the discriminant analysis. In general, the exact distributions are hard to obtain, but their asymptotic approximations including asymptotic expansions have been obtained. For a review of these results, see (Siotani, 1982). The purpose of this paper is to study error bounds for asymptotic approximations of the distributions. Usually the error terms have been known as the ones of the first, second or third order with respect to (ATj-1, A^ - 1 , n - 1 ) . However, such error estimators are not sufficient, since the results are obtained under the assumption if and N j are sufficiently large. More precisely, such results give no precise information about how N\ and should be large in order to make the actual errors less than a given constant. It is hoped to obtain error estimators under given values of N\ and N2. Unfortunately there is little research done on explicit error bounds (for a review, see (Fujikoshi, 1993)). In Section 2 we give a fundamental result in obtaining an error bound. In Section 3 the results are applied to obtain error bounds as well as validities for asymptotic approximations of the distribution of W under given values of and N2. Similar results are obtained for the Studentized classification statistic and other related statistics, which are discussed in Section 4.

2. A FUNDAMENTAL RESULT Explicit error bounds have been studied for the distributions of scale mixtures. For a review, see (Fujikoshi, 1993). In this paper we consider an extension of a scale mixture defined by X = V Z - U , (2.1) where Z and (V, U) are independent variables such that V > 0 with probability one. We assume that Z is distributed as the standard normal distribution with the cumulative distribution function $ and the density function . Then we can write the distribution function of X as P(X

= F(x)

= E

+ CO)] •

( v m

(2-2)

Now we will make an approximation for i>(v - 1 (a; + u)) which is useful in the following typical situation: V and U depend on a parameter n, V = 1 + O p ( n - 1 / 2 ) , U = O p ( n " 1 / 2 ) and the first few moments of V2 (or V~2) are more computable than the ones of V (or V~]). The following two types of approximations for (v -1 z) have been introduced by Fujikoshi (1985,1987) and Shimizu (1987):

*s ( x ) { v

i=i

j=i

2 S

J

- l )

3

u \

~ '

where = (3!/dxl){as,j(x) 0 , where

1)-1I

d=

n' ~d

A and n—p—2 1

B, 2 n —p —1

(n — p)(n - p — l ) ( n — p — 3 ) .

trAtrB trAB

n).

Then

10

K Fujikoshi

Let D2 and D2 be the statistics defined by ( 3 . 1 ) and respectively. Then the statistics D2 and D2 are distributed like 1L \ Di _ Nn . 21 ) ^ — nxN2 Y2> LEMMA 3 . 2 .

2) £>2 U = Nni ATjATJ .Yly(12 + Y4) ' where ri~X2p(r2), Y2~x2n-P+1, N ~ X2N-P+2,

T2 =

(3.5),

Y3~4_lt (NIN2/N)A2,

and Yj's are mutually independent. For a proof of ( 1 ) and ( 2 ) in Lemma 3 . 1 , see (Das Gupta, 1 9 6 8 ) . The third result can be obtained by modifying the method as in (Fujikoshi, 1 9 8 7 ) . The first result (1) of Lemma 3.2 is essentially the same as Hotelling's T2-statistic. The second result ( 2 ) can be obtained by the same way as in (Fujikoshi, 1 9 9 3 ) . Using Lemmas 3.1 and 3.2, we can evaluate each term in the upper bound G in (3.8) as follows:

2 E (U v )

'

=

"

2

m(m — 2)

*

A~2

l(-k + -k) p ( p + 2 )

+

2NIN2

1

+

4

„2 n - ( P + 2)-(p — -2)1i n-

to + 1

E{(y2-l)2} = E | ( f )

\ 2

m+

+

l)(p

+

3)

(39)

1

|-l-2E(^

2

-l),

n 4 { 1 + 2 (p + 2)r~ 2 + p(p + 2 ) r " 4 m(m — 2 ) ( m — 4 ) ( m — 6 ) „2 \

m+ 1

(m + 1)(TO - 1) / '

where m = n - p - 1 and r 2 = (A^A^/iV^A2. We note that the inequality (3.8) with the above moments is valid for all Ni and N2 so that n —p-1 > 0. Using Theorem 2.1 with oo and N 2 oo. Similarly we can write Pi(®: A) = E(v j t/){i>(V~ 1 (x +

U))},

(4.2)

where (4.3) D

12

K Fujikoshi

Therefore we can apply the results (3.8) and (3.9) with V and U given by (4.3) to the approximations of P\(x: A) and their error bounds. In this case, from Lemma 3.2 we can write

which does not depend on A. The computations of the moments of V2 in (4.4) are much simpler than the one in the case of W statistic. On the other hand, the computations of exact moments of U are not simple. For the computation of G, we can use \U\ < {(xW

- ^ ' S -

and hence

1

- M(1))}1/2

^

-

T

(4.5)

i1/2

r =U(n

P-l)}

0 and n - p - 1 > 0, respectively. On the other hand, the corresponding conditions for the classification statistic W are, n - p -1 > 0 and n - p - 15 > 0, respectively. 4 . 2 . The expectation

of Qd

=

(-D/2)

The distribution of QD = (-D/2) plays an important role in estimating the probabilities of misclassification. McLachlan (1973) gave an asymptotic expansion for the expectation of QD in the following form: EW,} - • ( - i a)

(-\A) { 4

+ 3^[A2-(y-10}, (4.9) where £ = - A / 2 and V = &/D. Therefore, using Theorem 2.1 with U = O, we have a general approximation - E J t ,

E { (V2S - 1 ) j } .

3=1

(4.10)

In this case we have to use a general approximation with S = — 1, since the moments of V~2 are computable, but those of V2 are not. Letting k — 2, we obtain ^ : D

2

(4.11)

Further, letting k = 4, we obtain E

{QD}

(4.12)

- (5The moments of ( ^ - 1) in (4.11) and (4.12) are given as follows:

m(m —

2) n

nz

+ 2 |2 + - p(p + 3) J-

E

A2

n m(m — 2)(m^ 4 j [ è {

T~1

+ p{p + 2 ) r "

2 ( 3 p + 1 1 )

+ - ( p + l)(p + 3)(p + 5 ) | n J + - ( 2(3p + 10) + - p(p + 3Xp + 5) [. r " 2 n ^ n + 3(p + 2 ) | 4 + i p ( p + 5 ) J r - 4 + p(p + 2)(p +

4)T~ 6

(4.13)

14

Y. Fujikoshi „4 'PL

, A2

- 1

— 2)(m — 4)(m — 6)

m(m

J - / l 2 + - f 3 p 2 + 28p + (

x

71 ^

n

\

+ ~L ( p + l ) ( p + 3)(p + 5)(p n

+

7)} J

+ -" Ui 2 + - (n p + 3)(6p + 35) n

i

+ i K P

+

.-2 7)^r

+ 2)(p + 3)(p +

6 ( 8 + l n

(p +

2)(5p

+ 38)

1

+ - 1 p ( p + 2)(p + 5)(p + 7) + 4(p + 2)(p + 4)^6 + i + p{p

p(p

+ 2)(p + 4)(p + 6)r -

.-4 >T

+ 7)|r'

8

where m = n - p - 1 and r 2 = ( N \ N 2 / N ) / A 2 . We note that the inequality (4.11) and (4.12) are valid under the conditions n — p — 3 > 0 and n — p — 7 > 0, respectively. Further, the orders of the resultant upper bounds in (4.11) and (4.12) are Oi and O2 with respect to ( N ^ 1 , N 2 ~ i , n ~ 1 ) , respectively. REFERENCES Anderson, T. W. (1973). An asymptotic expansion of the distribution of the studentized classification statistic W. Ann. Statist. 1, 964-972. Anderson, T. W. (1984). An Introduction New York. Das Gupta, S. (1968). 387-400.

to Multivariate

Statistical Analysis.

2nd ed. Wiley,

Some aspects of discrimination function coefficients. Sankhya,

A 30,

Fujikoshi, Y. (1985). An error bound for an asymptotic expansion of the distribution function of an estimate in a multivariate linear model. Ann. Statist. 13, 827-831. Fujikoshi, Y. (1987). Error bounds for asymptotic expansions of scale mixtures of distributions. Hiroshima Math. J. 17, 301-324. Fujikoshi, Y. (1993). Error bounds for approximations to the distributions of the standardized and studentized estimates in a multivariate linear model. J. Statist. Plann. Inf. 36, 165-174. Fujikoshi, Y. (1993). Error bounds for asymptotic approximations of some distribution functions. In: Multivariate Analysis'. Future Directions, pp. 181-208, C. R. Rao (Ed.), North-Holland, Amsterdam. MsLachlan, G. J. (1973). An asymptotic expansion of the expectation of the estimated error rate in discriminant analysis. Austral. J. Statist. 15, 210-214.

Error Bounds for Asymptotic

Expansions

15

Okamoto, M. (1963). An asymptotic expansion for the distribution of the linear discriminant function. Ann. Math. Statist. 34, 1286-1301. Shimizu, R. (1987). Error bounds for asymptotic expansion of the scale mixtures of the normal distribution. Ann. Inst. Statist. Math. 39, 611-622. Shimizu, R. and Fujikoshi, Y. (1994). Sharp error bounds for asymptotic expansions of the distribution functions of scale mixtures. Submitted for publication. Siotan, M. (1982). Large sample approximations and asymptotic exepansions of classification statistic. In: Handbook of Statistics, Vol. 2, pp. 47-60, P. R. Krishnaiah and L. N. Kanal (Eds), North-Holland, Amsterdam.

Multivar. Statist., pp. 1 7 - 3 3 E . - M . Tiit et al. (Eds) © 1995 V S P / T E V

SOME RESULTS ON THE EMPIRICAL SPACINGS PROCESS AND ITS BOOTSTRAPPED VERSION S. RAO JAMMALAMADAKA1 and JING QIN2 University of California, Santa Barbara 2

University of Maryland

ABSTRACT This paper considers the CLT and SLLN for the empirical spacings process, indexed by functions. A bootstrapped version of this process is shown to work and a strong approximation rate established for this bootstrapped version.

1. INTRODUCTION AND PRELIMINARIES Let X\, Xi,..., Xn_\ be independently and identically distributed (i.i.d.) random variables (r.v.s) with a common continuous distribution function (d.f.) F with support on R1. An important basic question of interest is whether these observations are from a specified distribution function — the goodness of fit problem. A simple probability integral transformation on these random variables, lets us equate the specified distribution to the uniform distribution on [0,1]. Thus from now on, we shall assume that this reduction has been done and under the null hypothesis of interest, the observations have a U(0,1) distribution. A broad class of procedures for testing this null hypothesis are based on the spacings, namely Di-.n — (Xi)U

Xi—ifn),

i — 1,2,..., 71,

(1.1)

where 0 = Xo,n ^ X^ n ^ • • • ^ X n _ l j n ^ X „ „ = 1 are the order statistics from U(0,1) distribution. See for instance (Pyke, 1965, 1972; Rao and Sethuraman, 1975; Shorack, 1972; Aly et al., 1984). If Zi, i = 1 , 2 , . . . , is a sequence of i.i.d. exponential random variables with mean 1, i.e., with d.f. F(z)= l-e-*, (1.2) then it is well known (see, e.g. (Pyke, 1965)) that (1.3)

18

S. R. Jammalamadaka

and J. Qin

where Zn = X)"=i Zi/n and ~ denotes equivalence in distribution. Indeed there is a probability space (Q, A, V) on which both Dvn and Zt can be defined, so that ~ in (1.3) can be replaced by = almost surely. Let n

Fn(t)

=

n_1

^

I(Zi

(1.4)

< t)

¿=i and Fn(t)

= n"1

I

^ t j = Fn(Zt),

(1.5)

denote the empirical d.f. (e.d.f.) of the exponential sequence and the normalized spacings sequence {nDi:n}"= j respectively. We call an(t)

= y/ii(Fn(t)

- F(t)),

t > 0,

(1.6)

the empirical spacings process. Denote by «/„(•) = V S ( P n O ) - - P O )

(1-7)

¿>«0 = V^(Pn(•) - P ( 0 ) ,

(1.8)

and where Pn,Pn, and P are probability measures corresponding to Fn,Fn, and F respectively. Limit theory for empirical processes has grown enormously over the last few decades, as evidenced by the voluminous book by Shorack and Wellner (1986). Recently the study of empirical processes indexed by sets or functions has become very important and the results are quite general in scope. See, for instance (Pollard, 1984; Sheehy and Wellner, 1992), etc. In Section 2, we explore the central limit theorem (CLT) and the strong law of large numbers (SLLN) for the spacings process, indexed by functions. It is easily checked that the usual bootstrap method fails for the spacings process, so in Section 3, we introduce a resampling scheme for which the bootstrap approximation is shown to work. This allows one to get critical values for any spacings test statistics, which are known to be notoriously slow in converging to their limiting distributions. Finally in Section 4, a bootstrap strong approximation rate is derived.

2. SPACINGS PROCESSES INDEXED BY VC FUNCTIONS For 1 < s < oo and for some probability measure Q on Rd, we denote by s Ls(Rd, Q) the space of measurable real functions g on Rd with (J