The Limiting Joint Distribution of the Largest and Smallest Sample Spacings

Abstract

$X_1, X_2, \cdots, X_n$ are independent chance variables, each with the same distribution. This common distribution assigns all the probability to the closed interval $\lbrack 0, 1\rbrack$, and has a density function $f(x)$ whose graph consists of any finite number of horizontal line segments. That is, there are $H$ non-degenerate subintervals $$I_1, I_2, \cdots, I_H, I_1 = \lbrack 0, z_1), I_2 = \lbrack z_1, z_2), \cdots, I_H = \lbrack z_{H-1}, 1 \rbrack,$$ and for each $x$ in $I_j, f(x) = a_j$. We assume that $a_j$ is positive for all $j$. Let $z_0$ denote zero, and $z_H$ denote unity. $M$ will denote $\min_ja_j, B$ will denote $$\sum_{j:a_j=M} (z_j - z_{j-1}),$$ and $S$ shall denote $\int^1_0 f^2(x) dx = \sum^H_{j=1} a^2_j(z_j - z_{j-1})$. Let $Y_1 \leqq Y_2 \leqq \cdots \leqq Y_n$ denote the ordered values of $X_1, \cdots, X_n$, and define $W_1 = Y_1, W_2 = Y_2 - Y_1, \cdots, W_n = Y_n - Y_{n-1}, W_{n + 1} = 1 - Y_n, U_n = \min (W_1, \cdots, W_{n + 1}), V_n = \max(W_1, \cdots, W_{n + 1})$. In [1] it is shown that if $f(x)$ is the uniform density function over $\lbrack 0, 1 \rbrack$, then $$\lim_{n\rightarrow\infty} P\big\lbrack U_n > \frac{u}{(n + 1)^2}, V_n < \frac{\log(n + 1) - \log v}{n + 1}\big\rbrack = \exp \{- (u + v)\},$$ for any positive numbers $u, v$. It is easy to see that the convergence must be uniform over any bounded rectangle in the space of $u$ and $v$. In this paper it is shown that if $f(x)$ is of the type described above, then $$\lim_{n\rightarrow\infty} P\Big\lbrack U_n > \frac{u}{(n + 1)^2}, V_n < \frac{\log(n + 1) + \log M - \log M - log v}{M(n + 1)}\Big\rbrack \\ \neq \exp \{-(Su + Bv)\},$$ for any positive values $u, v$. This result can be used to study the asymptotic power of various tests of fit based on $U_n$ and $V_n$ which have been proposed (see [1], p. 253).