Rafajłowicz Ewaryst - Stochastic Models, Statistics and Their Applications: Wrocław, Poland, February 2015

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Part I
Plenary Papers

Springer International Publishing Switzerland 2015

Ansgar Steland , Ewaryst Rafajowicz and Krzysztof Szajowski (eds.) Stochastic Models, Statistics and Their Applications Springer Proceedings in Mathematics & Statistics 10.1007/978-3-319-13881-7_1

1. Large Deviations of 2 Divergence Errors on Partitions

Lszl Gyrfi 1

(1)

Department of Computer Science and Information Theory, Budapest University of Technology and Economics, Stoczek u.2, 1521 Budapest, Hungary

Lszl Gyrfi

Email:

Abstract

We discuss Chernoff-type large deviation results for 2 divergence errors on partitions. In contrast to the total variation and the I-divergence, the 2-divergence has an unconventional large deviation rate. In this paper we extend the result of Quine and Robinson in Ann. Stat. 13:727742, from uniform distribution to arbitrary distribution.

This work was partially supported by the European Union and the European Social Fund through project FuturICT.hu (grant no.: TAMOP-4.2.2.C-11/1/KONV-2012-0013).

1.1 Introduction

We consider the problem of testing an unknown probability distribution. The test statistics are derived from dissimilarity measures of probability measures, like -divergences introduced by Csiszr []. The three most important -divergences in mathematical statistics and information theory are the total variation distance, the information divergence and the 2-divergence.

If and are probability measures on R d ( d 1), then the applications of large deviation results in statistical analysis mainly concern the comparison of test procedures using Bahadur efficiencies. We consider the problem of testing hypotheses

by means of test statistics T n = T n ( X 1,, X n ) where X 1, X 2, are independent and identically distributed random vectors along . Considering two tests rejecting H 0 for large values of the statistics T n ,1 and T n ,2, then (see e.g. Bahadur []) the efficiency e 1,2 of T n ,1 with respect to T n ,2 is calculated through the Bahadur exact slopes 2 b 1() and 2 b 2() for testing against an alternative : . The functions b 1 and b 2 are then given by

(1.1)

provided

(1.2)

(1.3)

and

(1.4)

for j =1,2. Such a limit assumption on the tail of the distribution of T n , j means that

Each of the divergence measures defined below, when applied to the distance between the null-hypothesis distribution and the empirical distribution n , both restricted to a partition, does lead to a test procedure. Hence the large deviation results given in the subsequent sections offer the possibility to calculate exact Bahadur slopes for these tests. Quine and Robinson [] for a survey on Bahadur efficiencies of different well-known tests.

1.2 The L 1 Error

If and are probability measures on R d ( d 1), then the total variation distance between and is defined by

where the supremum is taken over all Borel sets A .

We now consider some goodness of fit tests for H 0 given in the Introduction. Assume a sample of independent, R d valued random vectors X 1,, X n , distributed according to a probability measure , and let n denote the empirical measure.

Gyrfi and van der Meulen [] introduced the test statistic

based on a finite partition , ( n 2), of R d . These authors also showed that under H 0

Let and be the restrictions of and n to the partition n , then .

Beirlant et al. [] proved the following large deviation property: Assume that

(1.5)

and

(1.6)

Then for all 0<<2

(1.7)

where

(1.8)

Biau and Gyrfi [] gave an equivalent form of the rate function g :

Moreover, they proved that without having the non-atomic condition (), there is a general, tight upper bound

This upper bound implies strongly consistent tests for homogeneity and for independence (cf. Biau and Gyrfi []).

Remark 1.1

In Lemma 5.1 in Bahadur [] it was observed that

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