On the Probability of Large Deviations of Functions of Several Empirical CDF'S

Abstract

In [14], Sanov proved that if $F_N$ is the empirical cumulative distribution function (cdf) of a sample drawn from a population whose true cdf is $F_0$ and $\Omega$ is a set of cdf's which satisfies certain regularity conditions and does not contain $F_0$, then $P\{F_N \epsilon \Omega\}$ is roughly $\exp \{-N \inf_{F\epsilon \Omega} \int \ln (dF/dF_0) dF\}$. This theory is extended to the $c$-sample case and to the case where the set of cdf's in question depends on $N$. These extensions are used to estimate the probability of a large deviation of those statistics which are, or can be approximated by, uniformly continuous functions of the empirical cdf's. As an example, the main result is applied to the Wilcoxon statistic, and the resulting formula is used to compute the exact Bahadur efficiency of the Wilcoxon test relative to the $t$-test.