經驗分佈之比的極限分佈

<正> §1.引言 令x為一隨機變數,其分佈函數為F(x).對於x作n次相互獨立的试驗,便得n個結果x_1,x_2,…,x_n.我們也可以把x_1,x_2,…,x_n看作是遵循同一個分佈函數F(x)的相互獨立隨機變數.現在把x_1,x_2,…,x_n依其值由小到大的次序排列,我們得到

ON THE LIMITING DISTRIBUTION OF THE RATIO OF TWO EMPIRICAL DISTRIBUTIONS

In this paper the limiting distribution of the ratio of two empirical distributions is obtained, and it may be characterized as an analog to Smirnov's test of the difference between two empirical distributions of two independent samples. Recently Renyi has obtained the limiting distribution of the ratio of an empirical distribution and its theoretical distribution. We have given here another proof of Renyi's theorem.Let F_n(x) denote the empirical distribution of a sample of size n on the random variable x with distribution function F(x), i.e. where x_1≤x_2≤…≤x_n is the variational series of the sample x_1, x_2, .., x_n. We have proved: Theorem 1. Let S_m(x) and T_n(x) be the empirical distributions of two independent samples of the same random variable x with continuous distribution function F (x). Put N =mn/(m+n). Suppose that m → ∞ , n → ∞ so that m/n→d≤1, where d is a constant. Then for every fixed aTheorem 2. (Renyi). Let F_n(x) be the empirical distribution of a sample of size n on the random variable x with continuous distribution function F(x), then for every fixed a