Hostname: page-component-cd9895bd7-p9bg8 Total loading time: 0 Render date: 2024-12-25T13:55:30.243Z Has data issue: false hasContentIssue false
  • English
  • Français

A fixed point approach to local minimax theory

Published online by Cambridge University Press:  24 October 2008

W. J. R. Eplett
W. J. R. Eplett
Affiliation:
Mathematical Institute, University of Oxford, Oxford OX1 3LB
Get access Rights & Permissions [Opens in a new window]

Abstract

The generalized Neyman-Pearson theorem for constructing robust hypothesis tests proved by Huber and Strassen is obtained here as an application of the Kakutani-Fan fixed point theorem. The same technique is applied to obtain the existence of locally minimax estimators.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 99 , Issue 2 , March 1986 , pp. 339 - 346
Copyright
Copyright © Cambridge Philosophical Society 1986

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

REFERENCES

[1]Beran, R.. Efficient robust estimates in parametric models. Z. Wahrsch. Verw. Gebiete 55 (1981), 91–108.CrossRefGoogle Scholar
[2]Choquet, G.. Theory of capacities. Ann. Inst. Fourier 5 (1953/1954), 131–292.CrossRefGoogle Scholar
[3]Eilenberg, S. and Montgomery, D.. Fixed point theorems for multivalued transformations Amer. J. Math. 68 (1946), 214–222.CrossRefGoogle Scholar
[4]Fan, K.. Fixed-point and minimax theorems in locally convex topological linear spaces. Proc. Nat. Acad. Sci. 38 (1952), 121–126.CrossRefGoogle ScholarPubMed
[5]Huber, P. J.. A robust version of the probability ratio test. Ann. Math. Statist. 36 (1965), 1753–1758.CrossRefGoogle Scholar
[6]Huber, P. J. and Strassen, V.. Minimax tests and the Neyman-Pearson lemma for capacities. Ann. Statist. 1 (1973), 251–263 (corrigendum: Ann. Statist. 2, 223–224).CrossRefGoogle Scholar
[7]Huber, P. J.. Robust Statistics (Wiley, 1980).Google Scholar
[8]Karlin, S.. Mathematical Methods and Theory in Games, Programming and Economics (Addison-Wesley, 1959).Google Scholar
[9]Taylor, A. E. and Lay, D. C.. Introduction to Functional Analysis (2nd edition) (Wiley, 1980).Google Scholar