Central limit theorem for quadratic forms for sparse tables
type: Item
instance of: scholarly article
title: Central limit theorem for quadratic forms for sparse tables
zbMATH Open document ID: 0623.62014
DOI: 10.1016/0047-259X(87)90090-X
author: Prabir Burman
published in: Journal of Multivariate Analysis
publication date: 1987-01-01T00:00:00Z
review text: Decompose the sample space into \(k_ n\) disjoint subsets \(A_ t\), and let \(n_ t\) be the number of n observations in which \(A_ t\) occurs. Let \(k_ n\to +\infty\) with n. Here, n is either a large preassigned number (multinomial sampling), or a Poisson variable whose expectation tends to infinity. (Poisson sampling). The author studies the asymptotic behaviour of some weighted mean \(T_ n\) of general quadratic forms in \(n_ t-np_ t\), \(p_ t=P_ n(A_ t)\), \(1\leq t\leq k_ n\). For both types of sampling, \(T_ n-E(T_ n)\) is shown to be asymptotically normal. The author reduces the case of Poisson sampling to the multinomial case by showing that, under some conditions, \[ T_ n-E(T_ n)-[T^*_ n-E(T^*_ n)]\to 0\quad in\quad probability, \] where \(T_ n\) and \(T^*_ n\) are the weighted means for the two types of sampling.
Mathematics Subject Classification ID: 62E20
Mathematics Subject Classification ID: 62H17
Mathematics Subject Classification ID: 62D05
zbMATH DE Number: 4011644
zbMATH Keywords: Pearson's chi-square
zbMATH Keywords: central limit theorem
zbMATH Keywords: sparse tables
zbMATH Keywords: weighted quadratic mean
zbMATH Keywords: observed counts
zbMATH Keywords: asymptotic normality
zbMATH Keywords: multinomial sampling
zbMATH Keywords: Poisson sampling
zbMATH Keywords: weighted mean
zbMATH Keywords: quadratic forms
reviewed by: Janos Galambos
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 4011644