Sankhya: The Indian Journal of Statistics

1997, Volume 59, Series A, Pt. 2, 147--166

LARGE DEVIATIONS FOR JOINT DISTRIBUTIONS AND STATISTICAL APPLICATIONS

By

NARASINGHA R. CHAGANTY, Old Dominion University, Norfolk

SUMMARY. We obtain the large deviation principle (LDP) of a sequence of probability measures {\mu} on a product space $\sigma_1 X \sigma_2$ when the corresponding sequences of marginal and conditional distributions posses LDP's. This is the large deviation analogue of the results of Sethuraman [SankhyaA 23 1961, 379-386] for weak convergence. our large deviation result for probability measures on product spaces re-establishes the main theorem of Dinwoodie and Zabell[ Ann.Probab.20 1992, 1147-1166] as a simple consequence, and also generalises the LDP for product measures in Lynch and Sethuraman [ Ann.Probab.15 1987, 610-627]. Our main theorem is useful to establish the LDP for several statistical distributions. For example we show that, under bootstrapping, the ordinary empirical measure of a sample and the corresponding bootstrap empirical measure, jointly possess the LDP in weak topology. Other applications include the LDP for noncentral t-distributions and parametric bootstrap methods.

AMS (1991) subject classification. Primary 60F10, 62G09, 62F05, 62G30.

Key words and phrases. Large deviations, empirical measure, bootstrap, Bahadur slope, noncentral-t, Kullback-Leibler number.

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