# Multivariate Symmetry Models

• R. J. Beran
• P. W. Millar
Chapter

## Abstract

Let Γ0 be a fixed, compact subgroup of the group Γ of orthogonal transformations on R d . A random variable x, with values in R d and distribution P, is Γ 0 -symmetric if x and γx have the same distribution for all γ ∈Γ0. In terms of P, this means P(A) = P(ΓA) for all Borel sets A and all γ ∈Γ0. Let x1,…,x n be iid random variables with values in R d . The Γ0-symmetry model asserts that the x1 have an unknown common distribution that is Γ0-symmetric. The Γ 0 -location model specifies that for some unknown η∈R d , the random variables x 1-η,…, x n-η have an unknown common distribution P which is Γ0-symmetric. This paper develops some methods of inference for these multivariate symmetry models. Unlike the one dimensional case, there are a large number of “symmetry” notions in R d ,d > 1; Section 2.3 provides a few simple, useful examples, which figure in subsequent development.

### Keywords

Covariance Convolution Cylin Proal

## Preview

### References

1. Alexander, K. S. (1984), ‘Probability inequalities for empirical processes and a law of the iterated logarithm’, Annals of Probability 12 1041–1067.
2. Anderssen, S. A. (1975), ‘Invariant normal models’, Annals of Statistics pp. 132–54.Google Scholar
3. Beran, R. (1984), ‘Bootstrap methods in statistics’, Jahresbericht der Deutschen Mathematischer Vereinigung pp. 14–30.Google Scholar
4. Beran, R. & Millar, P. (1989), ‘A stochastic minimum distance test for multivariate parametric models’, Annals of Statistics pp. 125–140.Google Scholar
5. Beran, R. J. & Millar, P. W. (1985), Rates of growth for weighted empirical processes, in L. Le Cam & R. A. Olshen, eds, ‘Proceedings of the Berkeley Conference in Honor of Jerzy Neyman and Jack Kiefer, Volume II’, Wadsworth, Belmont, CA, pp. 865–887.Google Scholar
6. Beran, R. J. & Millar, P. W. (1986), ‘Confidence sets for a multivariate distribution’, Annals of Statistics pp. 431–443.Google Scholar
7. Beran, R. J. & Millar, P. W. (1987), ‘Stochastic estimation and testing’, Annals of Statistics pp. 1131–1154.Google Scholar
8. Beran, R. J. & Millar, P. W. (1992), Tests of fit for logistic models, in K. V. Mardia, ed., ‘The Art of Statistical Science’, Wiley, pp. 153–172.Google Scholar
9. Bickel, P. & Freedman, D. (1981), ‘Some asymptotic theory for the bootstrap’, Annals of Statistics pp. 1196–1217.Google Scholar
10. Bickel, P. J. & Millar, P. W. (1992), ‘Uniform convergence of probabilities on classes of functions’, Statistica Sinica pp. 1–15.Google Scholar
11. Chow, E. D. (1991), Stochastic minimum distance tests for censored data, PhD thesis, University of California at Berkeley.Google Scholar
12. Chung, K. (1968), A Course in Probability Theory,Harcourt, Brace and World, New York.Google Scholar
13. Donoho, D. & Gasko, M. (1987), Multivariate generalizations of the median and trimmed mean, Technical Report 133, Statistics Department, University of California at Berkeley.Google Scholar
14. Dudley, R. M. (1978), ‘Central limit theorems for empirical measures’, Annals of Probability 6 899–929.Google Scholar
15. Eaton, M. (1983), Multivariate Statistics: A vector space approach, Wiley, New York.
16. Efron, B. (1979), ‘Bootstrap methods: another look at the jackknife’, Annals of Statistics pp. 1–26.Google Scholar
17. Huber, P. (1985), ‘Projection pursuit’, Annals of Statistics pp. 435–474.Google Scholar
18. Le Cam, L. (1972), Limits of experiments, in L. Le Cam,J. Neyman & E. L. Scott, eds, ‘Proceedings of the Sixth Berkeley Symposium on Mathematical Statistics and Probability’, Vol. I, University of California Press, Berkeley, pp. 245–261.Google Scholar
19. Le Cam, L. (1983), A remark on empirical measures, in P. J. Bickel, K. Doksum & J. L. Hodges, eds, ‘Festschrift for Erich Lehmann’, Wadsworth, Belmont, California, pp. 305–327.Google Scholar
20. Loranger, M. (1989), A stochastic test of ellipsoidal symmetry, PhD thesis, University of California at Berkeley.Google Scholar
21. Millar, P. (1985), Nonparametric applications of an infinite dimensional convolution theorem’, Zeitschrift für Wahrscheinlichkeitstheorie and Verwandte Gebiete pp. 545–556.Google Scholar
22. Millar, P. W. (1983), ‘The minimax principle in asymptotic statistical theory’, Springer Lecture Notes in Mathematics pp. 75–265.Google Scholar
23. Millar, P. W. (1988), Stochastic test statistics, in ‘Proceedings of the 20th Symposium on the Interface: Computer Science and Statistics’, American Statistical Association, pp. 62–68.Google Scholar
24. Millar, P. W. (1993), Stochastic search and the empirical process, Technical report, University of California at Berkeley.Google Scholar
25. Perlman, M. D. (1988), ‘Comment: group symmetry covariance models’, Statistical Science pp. 421–425.Google Scholar
26. Pollard, D. (1980), ‘The minimum distance method of testing’, Metrika pp.43–70Google Scholar
27. Pollard, D. (1982), ‘A central limit theorem for empirical processes’, Journal of the Australian Mathematical Society (Series A) pp. 235–248.Google Scholar
28. Singh, K. (1981), ‘On the asymptotic accuracy of Efron’s bootstrap’, Annals of Statistics pp. 1187–1195.Google Scholar
29. Watson, G. S. (1983), Statistics on Spheres, John Wiley and Sons, New York.