Probability and Statistical Inference: Proceedings of the 2nd Pannonian Symposium on Mathematical Statistics, Bad Tatzmannsdorf, Austria, June 14–20, 1981 PDF
Preview Probability and Statistical Inference: Proceedings of the 2nd Pannonian Symposium on Mathematical Statistics, Bad Tatzmannsdorf, Austria, June 14–20, 1981
PROBABILITY AND STATISTICAL INFERENCE PROBABILITY AND STATISTIC'AL INFERENCE Proceedings oft he 2nd Pannonian Symposium on Mathematical Statistics, Bad Tatzmannsdoif, Austria, June 14 -20, 1981 Edited by WILFRIED GROSSMANN Institute of Statistics, University of Vienna, Austria GEORG CH. PFLUG Institute of Statistics, University of Vienna, Austria and WOLFGANG WERTZ Institute of Statistics, Technical University of Vienna, Austria D. Reidel Publishing Company Dordrecht: Holland / Boston: U.S.A. / London: England Ubmry of Congress Cataloging in Publication Data Pannonian Symposium on Mathematical Statistics (1981: Bad Tatzmannsdorf, Austria) Probability and statistical inference. Includes index. 1. Mathematical statistics-Congresses. 2. Probabilities- Congresses. I. Grossmann, Wilfried, 1948- . II. Pflug, Georg Ch.,1951- . III. Wertz, Wolfgang. IV. Title. QA276.AIP36 1981 519.5 82-5243 ISBN-I3: 978-94-009-7842-3 e-ISBN-I3: 978-94-009-7840-9 DOl: 10.1007/978-94-009-7840-9 Published by D. Reidel Publishing Company, P.O. Box 17, 3300 AA Dordrech t, Holland. Sold and distributed in the U.S.A. and Canada by Kluwer Boston Inc., 190 Old Derby Street, Hingham, MA 02043, U.S.A. In all other countries, sold and distributed by Kluwer Academic Publishers Group, P.O. Box 322, 3300 AH Dordrecht, Holland. D. Reidel Publishing Company is a member of the Kluwer Group. All Rights Reserved Copyright © 1982 by D. Reidel Publishing Company, Dordrecht, Holland Softcover reprint of the hardcover I st edition 1982 No part of the material protected by this copyright notice may be reproduced or utilized in any form or by any means, electronic or mechanical including photocopying, recording or by any informational storage and retrieval system, without written permission from the copyright owner TABLE OF CONTENTS Preface vii J. ADLER / Discrete Approximation of Markovprocesses by Markovchains J. ANDEL / An Autoregressive Representation of ARMA Processes 13 I. BAN and J. PERGEL / Characterization of a Type Multivariate Exponential Distributions 23 G. BANKOVI, J. VELICZKY, and M. ZIERMANN / Multivariate Time Series Analysis and Forecast 29 T. BEDNARSKI, S. GNOT, and T. LEDWINA / Testing Approximate Validity of Hardy-Weinberg Law in Population Genetics 35 P. BOD / On an Optimization Problem Related to Statistical Investigations 47 P. DEHEUVELS / A Construction of Extremal Processes 53 R. DUTTER and I. GANSTER / Monte Carlo Investigation of Robust Methods 59 D.M. ENACHESCU / Monte Carlo Methods for Solving Hyperbolic Equations 73 T. GERSTENKORN / The Compounding of the Binomial and Generalized Beta Distributions 87 J.K. GHORAI and V. SUSARLA / Empirical Bayes Estimation of Probability Density Function with Dirichlet Process Prior 101 W. GROSSMANN / On the Asymptotic Properties of Minimum Contrast Estimates 115 M. HUSKOvA and T. RATINGER / Contiguity in Some Nonregular Cases and its Applications 129 J. HUSTY / The Notion of Asympto~ically Least Favorable Configuration in Selection and Ranking Problems 143 vi TABLE OF CONTENTS L.B. KLEBANOV and J.A. MELAMED / One Method of Stable Estimation of a Location Parameter 157 F. KONECNY / Stochastic Integral Representation of Functionals from a Sequence of Martingales 171 P. KOSIK and K. SARKADI / Comparison of Multisample Tests of Normality 183 A. KRZYZAK and M. PAWLAK / Almost Everywhere Convergence of Recursive Kernel Regression Function Estimates 191 A. KRZYZAK and M. PAWLAK / Estimation of a Multivariate Density by Orthogonal Series 211 N. KUSOLITSCH / Longest Runs in Markov Chains 223 H. LAUTER / Approximation and Smoothing of Surfaces in (P+1)-Dimensional Spaces 231 A. LE§ANOVSKY / The Comparison of Two-Unit Standby Redundant System with Two and Three States of Units 239 F. MORICZ / A Probability Inequality of General Nature for the Maximum of Partial Sums 251 E. NEUWIRTH / Parametric Deviations in Linear Models 257 H. NIEDERREITER / Statistical Tests for Tausworthe Pseudo-Random Numbers 265 A. PAZMAN and J. VOLAUFOVA / Polynomials of Parameters in the Regression Model - Estimation and Design 275 G.Ch. PFLUG / The Limiting Log-Likelihood Process for Discontinuous Multiparameter Density Families 287 W. POLASEK / Two Kinds of Pooling Information in Cross-Sectional Regression Model 297 Z. PRASKOVA / Rate of Convergence for Simple Estimate in the Rejective Sampling 307 S.T. RACHEV / Minimal Metrics in the Random Variables Space 319 E. RONCHETTI / Robust Alternatives to the F-Fest for the Linear Model 329 L. RUTKOWSKI/Orthogonal Series Estimates of a Regression Function with Applications in System Identification 343 E. STADLOBER / Generating Student's T Variates by a Modified Rejection Method 349 ~. SUJAN / Block Transmissibility and Quantization 361 W. WERTZ / Invariantly Optimal Curve Estimators with Respect to Integrated Mean Error Risk 373 SUBJECT INDEX 385 PREFACE The interaction of various ideas from different researchers provides a main impetus to mathematical prosress. An important way to make communication possible is through international conferences on more or less spezialized The existence topics~ of several centers for research in probabil ity and statistics in the eastern part of central Europe - somewhat vaguely described as the Pannonian area - led to the idea of organizing Pannonian Symposia on Mathematical Statistics (PS~1S). The second such symposium was held at Bad Tatzmannsdorf, Burgenland (Austria), from 14 to 20 June 1981. About 100 researchers from 13 countries participated in that event and about 70 papers were delivered. Most of the papers dealt with one of the following topics: nonparametric estimation theory, asymptotic theory of estimation, invariance principles, limit theorems and aoplications. Full versions of selected papers, all presenting new results are included in this volume. The editors take this opportunity to thank the following institutions for their assistance in making the conference possible: the Provincial Government of Burgenland, the Austrian Ministry for Research and Science, the Burgenland Chamber of Commerce, the Control Data Corporation, the Austrian Society for Statistics and Informatics, the Landes hypothekenbank Burgenland, the Volksbank Oberwart, and the Community and Kurbad AG of Bad Tatzmannsdorf. We are also greatly indebted to all those persons who helped in editing this volume and in particular to the vii W. Grossmann et al. reds.), Probability and Statistical Inference, vii-viii. Copyright@ 1982 by D. Reidel Publishing Company. viii PREFACE specialists who performed invaluable work in the refereeing process: J. Andel (Praha), H. Bunke (Berlin), 1. Cziszar (Budapest), P. Deheuvels (Paris), H. Drygas (Kassel), B. Gyires (Debrecen), I. Katai (Budapest), E. Lukacs (Washington D.C.), M. Luptacik (Wien), E. Neuwirth (Wien), H. Niederreiter (Wien), W. Philipp (Cambridge), L. Schmetterer (Wien), W. Sendler (Trier), E. Stadlober (Graz), D. Szynal (Lublin), G. Tintner (Wien). Special thanks are also due to the publishing house for its patience in the cooperation and willingness to accept the many special wishes of the editors. The organizers W. Grossmann G. Pflug W. Wertz DISCRETE APPROXIMATION OF MARKOVPROCESSES BY MARKOVCHAINS Johannes ADLER Institut fUr Statistik und Wahrscheinlichkeitstheorie Technische UniversitRt Wien The concept of discrete convergence, introduced by Stummel [3] is the frame within the convergence of semigroups with discrete parameter to a semigroup with continous parameter can be studied, cf.Trotter [4] and Kurtz [2]. On the other hand, discrete convergence gives us the frame within we can define weak conver gence of a sequence of space-time discrete Harkovchains to a space-time continous Markovprocess. The connection is studied between discrete convergence of Harkov processes and discrete convergence of the corresponding semigroups and infinitesi0al operators. 1. DISCRETE CONVERGENCE OF MARKOV KERNELS For convenience we first recall some results from the theory of discrete convergence, which can be found in Stummel-Reinhardt [3]: A discrete limit space is a triple (, ~\ Xn, X~ d-lim) , whereas X, Xn are arbitrary sets 'nII",lXn denotes the cartesian product of the Xn and d-l~m denotes a mapping nCN with domain Df(d-lim) contained in Xn and range Rg(d-lim) equal to X. If we are given two discrete limit spaces (IT Xn, X, d-lim), (IT Yn , Y, d-lim) and for every nEN a mapping An:Xn~Yn and a mapping A:X~Y then we say An converges discretely to A, in symbols d-lim An=A, iff d-lim xn=x implies d-lim An(xn)=AX. The sequence (An) is said to be stable iff d-lim xn=d-lim x~ implies d-lim An(xn)=d-lim An(x~) in the sense that the existence of either side of the equc.tion implies the w. Grossmann et al. (eds.), Probability and Statisticalln[erence, 1-11. Copyright © 1982 by D. Reidel Publishing Company. 2 J.ADLER existence of the other side, and in that case equality holds. (An), A is sajd to be consistent iff for every xEX there exists a sequence (xn)EITXn such that holds A(x)=d-lim An(xn). Now we have the following results: The discrete limit of mappings is, if it exists, unique and d-lim An=A iff (An) is stable and (An),A is consistent. If Xn,X are metric spaces with corresponding metrics Pn and P then the following additional conditions must hold for the discrete limit space (TI Xn ' X, d-lim) to be a metric discrete limit space: (i) for every pair of sequences (xn), (x~) E ITXn such that (xn) or (x~) converges discretely, lim Pn(xn,x~)=0.ho11s if~ 9-li~ xn=9-lim xn,and (ii) d-lim xn=x, d-11m xn = x 1mp11es 11m Pn (xn,xn'>= =P (x,x') . In the paper of Stummel and Reinhardt [2] (n Xn,X,d-lim) is called a metric discrete limit space with discretely convergent metrics. We have the following results for metric discrete limit spaces (n Xn ' X, d-lim) and (IT Yn, Y, d-lim): Let An,A denote mappings from Xn to Yn respectively from X to Y, if d -lim An=A then A is continous. If (An) is stable and IT is a dense subset of X such that for every xED there exists (xn) E nXn with d-lim xn=x and d-lim An(xn)= =A(x),and if A is continous then d-lim An=A. If Xn , X are normed spaces and Rn:X~Xn is a sequence of linear and bounded mappings such that limlRnxl=lxl for every xEX then we get with the definition: d-lim xn=x iff lim I xn-Rnxi =0, a metric discrete limit space, which we call a normed discrete limit-space. For the normed discrete limit spaces (IT Xn , X, d-lim) and (IT Yn, Y, d-lim) the following is true: A sequence I. of continous linear mappings An:Xn~Ynis stable iff IAnl~M, whereas 1'1 does not depend on nEN; if d-lim An=A, whereas A denotes a mapping from X to Y then A is continous and linear, and in addition if IAnl~M holds, then IAI~M is true. With (E,B) we denote a locally compact space, of which the topology has a countable base, equipped with the a-field of Borel sets B. The sequence (En) of subsets of E is said to be a lattice for E iff the following conditions are ful filled: DISCRETE APPROXIMATION OF MARKOVPROCESSES BY MARKOVCHAINS 3 (i) the set of all clusterpoints of En is empty' (ii) the union of the En is dense in E, (iii) for every £>0 and every xEEk there is a NEN such that p(x,En)<£ for n~N whereas p denotes a fixed metric for E, generating the topology of E such that E is complete. It is easy to see that the last condition is equivalent to: (iii]) for every xEE there is a sequence (xn) with xnEEn and xn~x. Let Co(E) denote the Banachspace of all continous real functions on E that vanish at infinity; Co(En) is defined analogously, whereas the continuity condition is empty for Co(En) in view of the fact of En to be discrete. Rnf:=fIEn denotes the restriction of fECo(E) on En, so RnfECo(En). With these restric- tion operators we get the normed discrete limit- space (IT Co(En), Co(E) , d-lim), whereas d-lim fn = f holds by definition iff lim snp Ifn(x)-f(x) 1=0. n-+oo xEEn Lemma (1.1). The above normed limit-space can be characterized in the following way: Let (fn)EITCo(En) and fECo(E); then d-lim fn=f holds iff xnEEn, xn~xEE implies fn(xn)~f(x) . The necessity of the condition is obvious by continuity of f. Sufficiency: Suppose there is an £0>0 such that for every nEN there is a YnEEn with Ifn(Yn)-f(Yn) 1>£0' Now consider if necessary the one-point compactification of E whereas ~ denotes the additional point. By compactness there is a subsequence (Ynk) of (Yn) such that Ynk~x whereas xEE or x=~. En is a lattice for E and so there is a sequence (xn)EITEn converging to x (this is true even in case x=~) • Define a new sequence (zn): Ynk if there is a k such that n=nK t else; then we have znEEn and zn~x. If(x)-fn(zn) 1~lf(zn)-fn(zn) I-If(x)-f(zn) 1~£0/2 for n=nk~N,whereas N is choosen large enough such that If(x)-f(zn) ISco/2 holds for n~N. This gives the con tradiction. Denote b~ F the set of all linear, positive qperators P on Co(E), IPis1 such there is a sequence (fk)~o(E) with
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