Hostname: page-component-745bb68f8f-lrblm Total loading time: 0 Render date: 2025-01-08T10:23:08.272Z Has data issue: false hasContentIssue false
  • English
  • Français

Testing for uniformity on a compact homogeneous space

Published online by Cambridge University Press:  14 July 2016

R. J. Beran
R. J. Beran*
Affiliation:
The Johns Hopkins University, Baltimore, Maryland
Get access Rights & Permissions [Opens in a new window]

Extract

This paper applies the invariance principle to the problem of testing a distribution on a compact homogeneous space for uniformity. The notion of using a reduction by invariance in such a situation is due to Ajne[1], who considers tests invariant under rotation on a circle. In his paper, he derives the distribution of the maximal invariant and gives the general form of the most powerful invariant test for uniformity on the circle.

Type
Research Papers
Information
Journal of Applied Probability , Volume 5 , Issue 1 , April 1968 , pp. 177 - 195
Copyright
Copyright © Sheffield: Applied Probability Trust 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1] Ajne, B. (1966) A simple test for uniformity of a circular distribution. To appear.Google Scholar
[2] Anderson, T. W. and Darling, D. A. (1952) Asymptotic theory and certain “goodness of fit” criteria based on stochastic processes. Ann. Math. Statist. 23, 193–212.CrossRefGoogle Scholar
[3] Blum, J. R., Kiefer, J. and Rosenblatt, M. (1961) Distribution free tests of independence based on the sample distribution function. Ann. Math. Statist. 32, 485–492.Google Scholar
[4] Donsker, M. D. (1952) Justification and extension of Doob's heuristic approach to the Kolmogorov-Smirnov theorems. Ann. Math. Statist. 23, 277–281.CrossRefGoogle Scholar
[5] Greenwood, J. A. and Durand, D. (1955) The distribution of length and components of the sum of n random unit vectors. Ann. Math. Statist. 26, 233–246.CrossRefGoogle Scholar
[6] Grenander, U. (1963) Probabilities on Algebraic Structures. John Wiley, New York.Google Scholar
[7] Hannan, E. J. (1965) Group representations and applied probability. J. Appl. Prob. 2, 1–68.Google Scholar
[8] Jones, R. H. (1963) Stochastic processes on a sphere. Ann. Math. Statist. 34, 213–218.CrossRefGoogle Scholar
[9] Kac, M. and Siegert, A. J. F. (1947) An explicit representation of a stationary Gaussian process. Ann. Math. Statist., 18, 438–442.Google Scholar
[10] Naimark, M. A. (1959) Normed Rings. Noordhoff, Groningen.Google Scholar
[11] Parzen, E (1962) Stochastic Processes. Holden-Day, San Francisco.Google Scholar
[12] Pontrjagin, L. (1939) Topological Groups. Princeton Univ. Press.Google Scholar
[13] Tricomi, F. G. (1957) Integral Equations. Interscience, New York.Google Scholar
[14] Watson, G. S. (1967) Another test for the uniformity of a circular distribution. Biometrika 54, 675–677.CrossRefGoogle ScholarPubMed
[15] Watson, G. S. (1965) Equatorial distributions on a sphere. Biometrika 52, 193–201.Google Scholar
[16] Yaglom, A. M. (1961) Second-order homogeneous random fields. Proc. Fourth Berkeley Symp. Math. Statist. and Prob. 2, 593–622.Google Scholar