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March, 1976 Some Properties of the Empirical Distribution Function in the NON-i.i.d. Case
M. C. A. van Zuylen
Ann. Statist. 4(2): 406-408 (March, 1976). DOI: 10.1214/aos/1176343417


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.


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M. C. A. van Zuylen. "Some Properties of the Empirical Distribution Function in the NON-i.i.d. Case." Ann. Statist. 4 (2) 406 - 408, March, 1976.


Published: March, 1976
First available in Project Euclid: 12 April 2007

zbMATH: 0329.60008
MathSciNet: MR394990
Digital Object Identifier: 10.1214/aos/1176343417

Primary: 60G17
Secondary: 62G30

Keywords: Empirical distribution function , order statistics

Rights: Copyright © 1976 Institute of Mathematical Statistics


Vol.4 • No. 2 • March, 1976
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