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.