Spacings Generated by Mixed Samples

Abstract

Suppose $X(1, 1), X(1, 2), \cdots, X(1, n_1), X(2, 1), \cdots, X(2, n_2), \cdots, X(k, 1), \cdots, X(k, n_k)$ are independent chance variables, $X(i, j)$ having the probability density function $f_i(x)$, for $j = 1, \cdots, n_i, i = 1, \cdots, k$. We assume that for each $i, f_i(x)$ is bounded and has at most a finite number of discontinuities. We denote $n_1 + n_2 + \cdots + n_k$ by $N$, and we assume that $n_i/N$ is equal to $r_i$, where $r_i$ is a given positive number. Let $Y_1 \leqq Y_2 \leqq \cdots \leqq Y_N$ denote the ordered values of the $N$ observations $$X(1, 1), \cdots, X(k, n_k).$$ Define $W_i$ as $Y_{i + 1} - Y_i$ for $i = 1, \cdots, N - 1$. For any given nonnegative $t$, let $R_N(t)$ denote the proportion of the values $W_1, \cdots, W_{N - 1}$ which are greater than $t/N$. Let $S(t)$ denote $$\int^\infty_{-\infty} (r_1f_1(x) + r_2f_2(x) + \cdots + r_kf_k(x)) \exp \{-t\lbrack r_1f_1(x) + \cdots + r_kf_k(x)\rbrack\} dx$$ and $V(N)$ denote $\sup_{t \geqq 0}|R_N(t) - S(t)|$. Then it is shown that $V(N)$ converges stochastically to zero as $N$ increases. This is a generalization of [1], where $k$ was equal to unity. The result is applied to find the asymptotic behavior of ranks in a $k$-sample problem.