The Asymptotic Power of Certain Tests of Fit Based on Sample Spacings

Abstract

Suppose $X_1, X_2, \cdots, X_n$ are independent and identically distributed chance variables, each with density $f(x)$, where $\int^1_0 f(x) dx = 1, f(x)$ has a finite number of discontinuities, and there are two constants $A, B(0 < A < B < \infty)$ such that $A \leqq f(x) \leqq B$ for all $x$ in $\lbrack 0, 1\rbrack$. Let $Y_0$ denote zero, $Y_{n + 1}$ denote unity, and let $Y_1 \leqq Y_2 \leqq \cdots \leqq Y_n$ be the ordered values of $X_1, X_2, \cdots, X_n$. Define $R_i$ as $Y_i - Y_{i - 1}$ for $i = 1, \cdots, n + 1$. Let $r$ be any positive number greater than unity, and let $V(n)$ denote $\sum^{n + 1}_{i = 1} T^r_i$. The following theorem was proved in [1]. THEOREM A. If $f(x) = 1$ for $x$ in $\lbrack 0, 1\rbrack$, then the distribution of $$\frac{n^{r-1/2}V(n) - \sqrt n\Gamma(r + 1)}{\sqrt{\Gamma(2r + 1) - (r^2 + 1)\lbrack\Gamma(r + 1)\rbrack^2}}$$ approaches the standard normal distribution as $n$ increases. In the present paper, we prove the following generalization of Theorem A: THEOREM 1: The distribution of $$\frac{n^{r-1/2}V(n) - \sqrt n\Gamma(r + 1) \int^1_0 f^{1 - r}(x) dx}{\sqrt{\lbrack\Gamma(2r + 1) - 2r\Gamma^2(r + 1)\rbrack \int^1_0 f^{1 - 2r}(x) dx - \big\lbrack(r - 1)\Gamma(r + 1) \int^1_0 f^{1-r}(x) dx \big\rbrack^2}}$$ approaches the standard normal distribution as $n$ increases. Theorem 1 can be used to compute the asymptotic power of certain tests of fit based on $V(n)$.