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An Exponential Inequality for U-Statistics with Applications to Testing*

Published online by Cambridge University Press:  27 July 2009

Peter J. Bickel
Affiliation:
Department of Statistics, University of California, Berkeley, California 94720
Ya'acov Ritov
Affiliation:
Department of Statistics, The Hebrew University, 91905 Jerusalem, Israel

Abstract

We present a new exponential inequality for degenerate U-statistics. The bound on the tail probability is quadratic for small to medium values of the deviation and linear for larger values. We apply this bound to a family of test statistics and provide the key step in an optimality result for adaptive tests [1].

Type
Research Article
Copyright
Copyright © Cambridge University Press 1995

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References

1.Bickel, P.J. & Ritov, Y.( 1991). Testing for goodness of fit: A new approach. In Saleh, E. (ed.), Nonparametric statistics and related topics. Amsterdam: Elsevier, pp. 5157.Google Scholar
2.De La Pena, V.H. (1992). Decoupling and Khintchines inequalities for U-statistics. Annals of Probability 21: 18771892.Google Scholar
3.Eichelsbacher, P. & Löwe (1993). Large deviation principle for m-variate von-Mises-statistics and U-statistics. Unpublished.Google Scholar
4.Kallenberg, W.C.M., Oosterhoff, J., & Schriever, B.F. (1985). The number of classes in x 2. Journal of the American Statistical Association 80: 959968.CrossRefGoogle Scholar
5.Mann, H.B. & Wald, A. (1942). On the choice of the number of intervals in the application of the chi-square test. Annals of Mathematical Statistics 13: 306317.CrossRefGoogle Scholar
6.Neyman, J. (1959). Optimal asymptotic tests of composite hypotheses. In Grenander, U. (ed.), Probability and Statistics, The Herlad Cramer Volume. Stockholm: Almquist and Wicksell, pp. 213234.Google Scholar
7.Nolan, D. & Pollard, D. (1987). U-processes: Rates of convergence. Annals of Statistics 15: 788799.CrossRefGoogle Scholar
8.Nolan, D. & Pollard, D. (1988). Functional limit theorem for U-processes. Annals of Probability 16: 12911298.CrossRefGoogle Scholar
9.Rayner, J.C.W. & Best, D.J. (1989). Smooth tests of goodness of fit. New York: Oxford University Press.Google Scholar
10.Serfling, R.J. (1980). Approximation theorems of mathematical statistics. New York: Wiley.CrossRefGoogle Scholar