In this paper, we establish strong large deviation results for an arbitrary sequence of random variables, under some assumptions on the normalized cumulant generating function.
In other words, we give asymptotic expressions for the tail probabilities of the same kind as those obtained by Bahadur and Rao (1960) for the sample mean (these tail probabilities go to zero exponentially fast). We consider both the case where the random variables are absolutely continuous (or their distribution function has an absolutely continuous component) and the case where they are lattice-valued. Our proofs make use of arguments of Chaganty and Sethuraman (1993) who also obtained strong large deviation results and local limit theorems.
We illustrate our results with some statistical applications: the sample variance, the Wilcoxon signed-rank statistic and the Kendall's tau statistic.
R. Bahadur and R. Rao (1960). On deviations of the sample mean. Ann. Math. Statist., 31, 1015–1027.
N.R Chaganty and J. Sethuraman (1993). Strong large deviation and local limit theorems. Ann. Probab., 21, 1671–1690.
A. Dembo and O. Zeitouni (1998). Large deviations techniques and applications. Springer, New York. Second edition.
Keywords: Large deviations; Bahadur-Rao theorem; Wilcoxon signed-rank statistic; Kendall's tau statistic
Biography: Cyrille Joutard is currently an associate professor of statistics at the University of Montpellier in France.
His research interests include the study of large deviations in asymptotic statistics (nonparametric estimators, M-estimators, …), and the issues of model choice using hierarchical bayesian mixed-membership models