The Distribution of a Generalized $\mathrm{D}^+_n$ Statistic

Abstract

Let $F_n(x)$ be the empirical c.d.f. of $n$ independent random variables, each distributed according to the same continuous c.d.f. $F(x)$. The major object of this paper is to obtain in explicit form the probability law of the random variable $$D^+_n(\gamma) = \sup_{-\infty<x<\infty} \{F_n(x) - \gamma F(x)\}.$$ It is no loss of generality to suppose that $F(x)$ is the c.d.f. of the uniform distribution on [0, 1], so this assumption will be held throughout the paper. When $\gamma = 1$, then $D^+_n(1)$ is the usual one-sided goodness of fit statistic whose asymptotic distribution was first derived by Smirnov [6]. We obtain in several different forms (formulas 2.2 and 2.3) an expression for $$P(D^+_n(\gamma) < a) = P(F_n(x) \leqq a + \gamma^x, 0 \leqq x \leqq 1).$$ Formula (2.2) agrees with the one found by Birnbaum and Tingey [2] when $\gamma = 1$, which is the "classical" case. As a matter of fact, it seems to have been overlooked that this formula, for finite $n$, had already appeared in a paper by Smirnov [6]. The new formula (2.3) would seem to involve fewer computations for actual numerical evaluation. One rather remarkable fact which results from (2.3) is that \begin{equation*}P(F_n(x) \leqq \gamma x, 0 \leqq x \leqq 1) = \begin{cases}1 - \frac{1}{\gamma},\quad\gamma > 1\\ 0,\quad\gamma \leqq 1,\end{cases}\end{equation*} for any $n$. This was noted by Daniels [4] and was rediscovered by Robbins [5]. Using (2.3) it is easy to evaluate $\lim_{n\rightarrow\infty} P(F_n(x) \leqq a(n) + \gamma x)$ where $\gamma, (\gamma > 1)$ is fixed and $a(n) = d/n$, where $d$ is fixed. The limiting distribution when $\gamma > 1$ can be used to derive some facts about the Poisson Process which were recently discovered by Baxter and Donsker [1]. The methods used are elementary. To assist the reader, the results are all listed in Section 2 and Section 3 is devoted to giving proofs.