Multivariate chi-square distribution for non-homogeneous polynomial scheme

Abstract

We consider a statistic whose components are the χ 2 statistics, constructed on the base of the frequencies of outcomes of non-homogeneous polynomial samples of growing volumes. Conditions are formulated which guarantee the existence of a limiting distribution of this statistics, and its Laplace transform is presented. In the homogeneous case, the Laplace transform obtained is reduced to the form which has been known before.This work was supported by the Russian Foundation for Basic Research, grants 96-01-00531, 96-15-96092. FORMULATION OF THE PROBLEMIn [1], a multivariate chi-square distribution was obtained, which was a limit distribution of the set of chi-square statistics based on frequencies of outcomes of growing polynomial samples. The distribution obtained was used in [1] to calculate the error probabilities of the r-fold chi-square test of sequential type.The results of [1] are cited in [2], where other approaches to the definition of multivariate chi-square distributions were presented. Some generalizations and applications of the r-fold chi-square test were given in [3][4][5].In the change-point problem, to evaluate the characteristics of tests used, we need to know the distribution of the corresponding statistics in the case of non-homogeneous polynomial scheme. In this paper, we formulate the conditions which guarantee the existence of a limiting distribution of the set of chi-square statistics in that case, and give its Laplace transform.We consider a sequence of independent (non-homogeneous polynomial) trials with m outcomes 1 , 2, . . . , m. Let p\ (t) , . . . , p m (t) be the probabilities of these outcomes in the ith trial, t = 1 , 2, . . . We fix r, r > 2, natural numbers 1 < n\ < . . . < n r , and assume that for n\ -> <»(1)

Sequential two-fold Pearson chi-squared test and tails of the Bessel process distributions

“…In this paper we study joint distributions of the values of Pearson statistics arising in the sequential χ 2 test (see [1], [6]- [8]). Consider a scheme of independent trials with N outcomes (N ≥ 2).…”