Annals of Mathematical Statistics

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

Lionel Weiss

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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 $V(n)$.

Article information

Source
Ann. Math. Statist., Volume 28, Number 3 (1957), 783-786.

Dates
First available in Project Euclid: 27 April 2007

Permanent link to this document
https://projecteuclid.org/euclid.aoms/1177706892

Digital Object Identifier
doi:10.1214/aoms/1177706892

Mathematical Reviews number (MathSciNet)
MR96327

Zentralblatt MATH identifier
0087.14801

JSTOR
links.jstor.org

Citation

Weiss, Lionel. The Asymptotic Power of Certain Tests of Fit Based on Sample Spacings. Ann. Math. Statist. 28 (1957), no. 3, 783--786. doi:10.1214/aoms/1177706892. https://projecteuclid.org/euclid.aoms/1177706892


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