Some Properties of the Empirical Distribution Function in the NON-i.i.d. Case

Abstract

For $N = 1, 2, \cdots$ let $X_{1N}, X_{2N}, \cdots, X_{NN}$ be independent rv's having continuous df's $F_{1N}, F_{2N}, \cdots, F_{NN}$. For the set $X_{1N}, X_{2N}, \cdots, X_{NN}$, let us denote by $X_{1:N} \leqq X_{2:N} \leqq \cdots \leqq X_{N:N}$ the order statistics, by $\mathbb{F}_N$ the empirical df and by $\bar{F}_N$ the averaged df, i.e. $\bar{F}_N(x) = N^{-1}\sum^N_{n=1} F_{nN}(x)$ for $x\epsilon(-\infty, \infty)$. It is shown that for each $\varepsilon > 0$ there exists a $0 < \beta(= \beta_\varepsilon) < 1$, independent of $N$, such that for $N = 1, 2, \cdots$, $(a) P(\mathbb{F}_N(x) \leqq \beta^{-1}\bar{F}_N(x), \text{for} x \epsilon (-\infty, \infty)) \geqq 1 - \varepsilon,$ $(b) P(\mathbb{F}_N(x) \geqq \beta\bar{F}_N(x), \text{for} x \epsilon \lbrack X_{1:N}, \infty)) \geqq 1 - \varepsilon.$ Moreover, these assertions hold uniformly in all continuous df's $F_{1N}, F_{2N}, \cdots, F_{NN}$. The theorem can be used to prove asymptotic normality of rank statistics and of linear combinations of functions of order statistics in the case where the sample elements are allowed to have different df's.