經驗分佈對理論分佈的比值

<正> 設F(x)是隨機變数X的分佈函數,x_1,x_2,…,x_n是對X的n次相互獨立觀測的結果.將n個數據按照數值從小到大排列起來,以x_k代表其中的第k個,我們把原來的結果寫成

ON THE RATIO OF AN EMPIRICAL DISTRIBUTION FUNCTION TO THE THEORETICAL DISTRIBUTION FUNCTION

This paper contains various distributional results about the ratio F_n(x) :F(x), where F(x) is a distribution function, and F_n(x) is the corresponding empirical distribution function formed of a sample of size n. The main results are:(A) Concerning stability. It is proved that, as n→∞, where a_n is any sequence of positive numbers tending to infinity.(B) Concerning exact distributions. We have derived exact formulae for the following distribution functions: where 0