The Convergence of Certain Functions of Sample Spacings

Abstract

Suppose $g(u_1, \cdots, u_k)$ is a continuous function of its arguments, homogeneous of order $r$, monotonic nondecreasing in each of its arguments, which is positive whenever each of its arguments is positive, and is such that for any given $K(0 < K < \infty)$, there is a number $R(K)(0 < R(K) < \infty)$ such that $g(u_1, \cdots, u_k) < K$ and $u_1 \geqq 0, \cdots, u_k \geqq 0$ imply that $u_1 + \cdots + u_k < R(K)$. Let $U_1, \cdots, U_k$ be chance variables with joint density $e^{-(u_1 + \cdots + u_k)}$ for $u_1 \geqq 0, \cdots, u_k \geqq 0$, and zero elsewhere. For any $t$, define $U(t)$ as $P\lbrack g(U_1, \cdots, U_k) \geqq t\rbrack$. We note that $U(t)$ is a continuous distribution function, with $U(0) = 0$. Let $\rho(v)$ be a bounded nonnegative function with a finite number of discontinuities, defined for $0 \leqq v \leqq 1$. Suppose $X_1, X_2, \cdots, X_n$ are independently and identically distributed chance variables, each with density $f(x), f(x)$ being bounded, and having a finite number of discontinuities and oscillations. $F(x)$ denotes $\int^x_{-\infty} f(x) dx$. Define $Y_1 \leqq Y_2 \leqq \cdots \leqq Y_n$ as the ordered values of $X_1, \cdots, X_n$, and define $T_i$ as $Y_{i + 1} - Y_i(i = 1, \cdots, n - 1)$. Let $R_n(t)$ denote the proportion of the value $$\rho\big(\frac{1}{n}\big)g(T_1, \cdots, T_k), \quad\rho\big(\frac{2}{n}\big) g(T_2, \cdots, T_{k + 1}), \cdots,$$ $$\rho\big(\frac{n - k}{n}\big) g(T_{n - k}, \cdots, T_{n - 1})$$ which are less than or equal to $t/n^r$. Let $\overline{U}\lbrack\lbrack tf^r(x)\rbrack / \{\rho\lbrack F(x)\rbrack\}\rbrack$ be defined as follows. If $f(x) = 0$, $$\overline U\lbrack\lbrack tf^r(x)\rbrack / \{\rho\lbrack F(x)\rbrack\}\rbrack = 0$$ regardless of the value of $t$. If $x$ is such that $f(x) > 0$ and $\rho\lbrack F(x)\rbrack = 0$, then $\overline U\lbrack\lbrack tf^r(x) \rbrack / \{\rho\lbrack F(x)\rbrack\}\rbrack = 1$ regardless of the value of $t$. If $f(x) > 0$ and $\rho\lbrack F(x) \rbrack > 0$, then $\overline U\lbrack\lbrack tf^r(x) \rbrack / \{\rho\lbrack F(x) \rbrack\}\rbrack = U\lbrack\lbrack tf^r (x) \rbrack / \{\rho \lbrack F(x) \rbrack\}\rbrack$. Let $S(t)$ denote $$\int^\infty_{-\infty} \overline{U}\lbrack\lbrack t \cdot f^r(x)\rbrack / \{\rho\lbrack F(x)\rbrack\}\rbrack f(x) dx$$, and let $V(n)$ denote $\sup_{t \geqq 0}|R_n(t) = S(t)|$. Then $V(n)$ converges to zero stochastically as $n$ increases. This generalizes the result of [1], where $k = 1, g(u_1) = u_1, \rho(v) = 1$. The present result may be used to construct tests of fit in the presence of unknown location and scale parameters.