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Cumulants and Partition Lattices V. Calculating Generalized k-Statistics

Part of: Linear inference, regression

Published online by Cambridge University Press:  09 April 2009

T. P. Speed  and
H. L. Silcock
T. P. Speed
Affiliation:
Division of Mathematics and Statistics CSIRO Canberra 2601, Australia
H. L. Silcock
Affiliation:
Division of Mathematics and Statistics CSIRO Canberra 2601, Australia
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Abstract

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A method is developed for obtaining compact, easily computed and statistically interpretable expressions for the generalized k-statistics associated with multiply-indexed arrays of random variables such as those which arise in variance component analysis. These expressions will be used in the next paper in this series to give formulae for variances and covariances of estimates of components of variance.

Keywords

cumulantk-statisticanalysis of variance

MSC classification

Secondary: 62J10: Analysis of variance and covariance
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 44 , Issue 2 , April 1988 , pp. 171 - 196
Copyright
Copyright © Australian Mathematical Society 1988

References

Bailey, R. A., Praeger, C. E., Speed, T. P. and Taylor, D. E., Analysis of variance, forthcoming book.Google Scholar
Carney, E. J. (1967) Computation of variances and covariances of variance component estimates, (PhD Thesis, Iowa State University).Google Scholar
Carney, E. J. (1968) ‘Relationship of generalized polykays to unrestricted sums for balanced complete finite populations’, Ann. Math. Statist. 39, 634–656.CrossRefGoogle Scholar
Hooke, R. (1952) ‘Some applications of bipolykays to the estimation of cariance components and their moments’, Ann. Math. Statist. 27, 80–98.CrossRefGoogle Scholar
Kaplan, E. L. (1952) ‘Tensor notation and the sampling cumulants of k-statistics’, Biometrika, 39, 319–323.Google Scholar
Kendall, M. G. and Stuart, A. (1969), The advanced theory of statistics, Volume 1. (Third Edition, Griffin, London).Google Scholar
Speed, T. P. and Bailey, R. A. (1982) ‘On a class of association schemes derived from lattices of equivalence relations’, Algebraic Structures and their Applications, edited by Schultz, Philip, Praeger, Cheryl E. and Sullivan, Robert P. (Marcel Dekker, New York).Google Scholar
Speed, T. P. (1984) ‘On the Möbius function of Hom(P, Q)’, Bull. Austral. Math. Soc. 29, 39–46.CrossRefGoogle Scholar
Speed, T. P. (1985) ‘Dispersion models for factorial experiments’, Bull. International Statistical Institute. Proc. of 45th Session. 4, Amsterdam, 1985.Google Scholar
Speed, T. P. (1986 a) ‘Cumulants and partition lattices II. generalised k-statistics’, J. Austral. Math. Soc. Ser. A. 40, 34–53.CrossRefGoogle Scholar
Speed, T. P. (1986 b) ‘Cumulants and partition lattices III. multiply-indexed arrays’, J. Austral. Math. Soc. Ser. A. 40, 161–182.CrossRefGoogle Scholar
Speed, T. P. (1986 c) ‘Anova models with random effects: An approach via symmetry’, Essays in Time Series and Allied Processes: papers in honour of E. J. Hannan, Edited by Gani, J., and Priestley, M. B., pp. 355–368 (Sheffield: Applied Probability Trust).Google Scholar