Some Renyi Type Limit Theorems for Empirical Distribution Functions

Abstract

Let $F_n(x)$ denote the empirical distribution function of a random sample of size $n$ drawn from a population having continuous distribution function $F(x)$. In Section 3 the limiting distribution of the supremum of the random variables $\{F_n(x) - F(x)\}/F_n(x), |F_n(x) - F(x)|/F_n(x), \{F_n(x) - F(x)\}/(1 - F(x)), |F_n(x) - F(x)|/(1 - F(x)), \{F_n(x) - F(x)\}/(1 - F_n(x)), |F_n(x) - F(x))|/(1 - F_n(x))$ is derived where sup is taken over suitable ranges of $x$ respectively. Relevant tests and some combinations of them are also discussed briefly in Section 3.