On the Probability of Large Deviations and Exact Slopes

Abstract

The purpose of this paper is to investigate a certain probability of a large deviation for a sequence of random variables $\{W_n\}$ which have moment-generating functions. We will assume that the mean of $W_n$ is given by $n\mu_n$ and the variance by $n\sigma_n^2$, where $\{\mu_n\}$ and $\{\sigma_n^2\}$ are covergent sequences. We seek the limit, as $n \rightarrow \infty$, of the expression $n^{-1} \ln P\lbrack W_n > na_n \rbrack,$ where $\{a_n\}$ is a convergent sequence with $\lim a_n > \lim \mu_n$. It is shown that, if the moment-generating function of $W_n$ satisfies certain limiting conditions, the above expression has a limit which depends on certain limits of this moment-generating function and its derivative. This result can be used in the computation of exact slopes for test statistics whose moment-generating function is known under the null hypothesis. Some applications are given.