The Annals of Mathematical Statistics

The Convergence of Certain Functions of Sample Spacings

Lionel Weiss

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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.

Article information

Source
Ann. Math. Statist., Volume 28, Number 3 (1957), 778-782.

Dates
First available in Project Euclid: 27 April 2007

Permanent link to this document
https://projecteuclid.org/euclid.aoms/1177706891

Digital Object Identifier
doi:10.1214/aoms/1177706891

Mathematical Reviews number (MathSciNet)
MR96326

Zentralblatt MATH identifier
0089.13501

JSTOR
links.jstor.org

Citation

Weiss, Lionel. The Convergence of Certain Functions of Sample Spacings. Ann. Math. Statist. 28 (1957), no. 3, 778--782. doi:10.1214/aoms/1177706891. https://projecteuclid.org/euclid.aoms/1177706891


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