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Cumulants and partition lattices III: Multiply-indexed arrays

Part of: Linear inference, regression

Published online by Cambridge University Press:  09 April 2009

T. P. Speed
T. P. Speed
Affiliation:
CSIRO Division of Mathematics and Statistics, Box 1965 GPO Canberra, A.C.T. 2601, Australia
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Abstract

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Earlier work of the author exploiting the role of partition lattices and their Mbius functions in the theory of cumulants, k-statistics and their generalisations is extended to multiply-indexed arrays of random variables. The natural generalisations of cumulants and k-statistics to this context are shown to include components of variance and the associated linear combinations of mean-squares which are used to estimate them. Expressions for the generalised cumulants of arrays built up as sums of independent arrays of effects as in anova models are derived in terms of the generalized cumulants of the effects. The special case of degree two, covering the unbiased estimation of components of variance, is discussed in some detail.

Keywords

cumulantk-statisticgroup actionanova modelcomponent of variance

MSC classification

Secondary: 62J10: Analysis of variance and covariance
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 40 , Issue 2 , April 1986 , pp. 161 - 182
Copyright
Copyright Australian Mathematical Society 1986

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