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Cumulants and partition lattices IV: a.s. convergence of generalised k-statistics

Part of: Linear inference, regression Stochastic processes Parametric inference

Published online by Cambridge University Press:  09 April 2009

T. P. Speed
T. P. Speed
Affiliation:
C.S.I.R.O., Box 1965, G.P.O., Canberra, A.C.T. 2601, Australia
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Abstract

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Generalised k-statistics associated with multi-indexed arrays of random variables satisfying a generalised form of exchangeability are studied. By showing that they form multi-indexed reversed martingales and that the associated family of σ-fields possesses certain conditional independence properties, conditions for the a.s. convergence of generalised k-statistics are obtained. When the arrays of random variables are sums of independent arrays of independent effects, as is the case with the standard random effects anova models, the limits are identified as the associated generalixed cumulants.

Keywords

cumulantk-statisticpartial exchangeabilityanova modelsvariance component estimationasymptotics

MSC classification

Secondary: 60G42: Martingales with discrete parameter 62F12: Asymptotic properties of estimators 62J10: Analysis of variance and covariance
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 41 , Issue 1 , August 1986 , pp. 79 - 94
Copyright
Copyright © Australian Mathematical Society 1986

References

Aldous, David J. (1981), ‘Representations for partially exchangeable arrays of random variables’, J. Multivar. Anal. 11 581–598.CrossRefGoogle Scholar
Aldous, David J. (1985), ‘Exchangeability and related topics’, École d'Été de Probabilités de Saint-Flour XIII 1983, eds., Aldous, D. J., Ibragimov, I. A. and Jacod, J., Lecture Notes in Mathematics 1117, Springer, New York, 1–198.Google Scholar
Bailey, R. A., Praeger, C. E., Rowley, C. A. and Speed, T. P. (1983), ‘Generalised wreath products of permutation groups’, Proc. London. Math. Soc. (3) 47, 69–82.CrossRefGoogle Scholar
Dubins, Lester E. and Pitman, Jim (1979), ‘A pointwise ergodic theorem for the group of rational rotations’, Trans. Amer. Math. Soc. 251, 299–308.CrossRefGoogle Scholar
Gut, Allan (1976), ‘Convergence of reversed martingales with multidimensional indices’, Duke Math. J. 43, 269–275.CrossRefGoogle Scholar
Miller, John James (1973), Asymptotic properties and computation of maximum likelihood estimates in the mixed model of the analysis of variance (Ph.D. Thesis, Stanford University).Google Scholar
Miller, John J. (1977), ‘Asymptotic properties of maximum likelihood estimates in the mixed model of the analysis of variance’, Ann. Statist. 5, 746–762.CrossRefGoogle Scholar
Meyer, P. A. (1966), Probabilities and potentials (Blaisdell, Waltham, Mass.)Google Scholar
Praeger, C. E., Rowley, C. A. and Speed, T. P. (1985), ‘A note on generalised wreath product groups’, J. Austral. Math. Soc. (Ser. A) 39, 415–420.CrossRefGoogle Scholar
Rényi, Alfred (1970), Foundations of probability (Holden-Day, San Francisco).Google Scholar
Speed, T. P. (1984), ‘On the Mobius function of Hom(P, Q)’, Bull. Austral. Math. Soc. 29, 39–46.CrossRefGoogle Scholar
Speed, T. P. (1986a), ‘Cumulants and partition lattices II: Generalised k-statistics’, J. Austral. Math. Soc. (Ser. A) 40, 34–53.CrossRefGoogle Scholar
Speed, T. P. (1986b), ‘Cumulants and partition lattices III: Multiply-indexed arrays’, J. Austral. Math. Soc. (Ser. A) 40, 161–182.CrossRefGoogle Scholar