Small Sample Distributions for Multi-Sample Statistics of the Smirnov Type

Abstract

Let \begin{equation*}\tag{1.1}X^{(i)}_1,X^{(i)}_2, \cdots, X^{(i)}_{n_i},\qquad i = 1, 2, \cdots, c,\end{equation*} be samples of $c$ independent random variables $X^{(i)}$ with continuous cumulative distribution functions $F^{(i)}$, and let \begin{equation*}\begin{align*}F^{\ast^{(i)}}(x) &= 0\qquad x &< X^{(i)}_1 \\ \tag{1.2}F^{\ast^{(i)}} (x) &= k/n_i\qquad X^{(1)}_k &\leqq x < X^{(1)}_{k+1}, 1 \leqq k < n_i \\ F^{\ast^{(i)}} (x) &= 1\qquad X^{(i)}_{ni} \leqq x\end{align*}\end{equation*} be the corresponding $c$ emprirical distribution functions. We define the statistics \begin{equation*}\tag{1.3} D(n_1, n_2, \cdots, n_c) = \sup_{\substack{x, i, j\\(i,j=1,2,\cdots,c)}} |F^{\ast(i)} (x) - F^{\ast(j)} (x)|\end{equation*} and \begin{equation*}\tag{1.4} D^+(n_1, n_2, \cdots, n_c) = \sup_{\substack{x,i,j\\(i<j;i,j=1,2,\cdots,c)}} \brack F^{\ast(i)} (x) - F^{\ast(j)} (x)\rbrack.\end{equation*} The well known Kolmogorov-Smirnov statistics $D(m, n)$ and $D^+(m, n)$ are special cases of (1.3) and (1.4), respectively, with $c = 2, n_1 = m, n_2 = n$. The exact small distribution, under the null hypothesis \begin{equation*}\tag{1.5} F^{(i)} = F^{(j)}\quad\text{for all} i,j = 1, 2, \cdots, c,\end{equation*} of the statistics defined by (1.3) and (1.4) for any number $c$ of samples, and for any sample sizes $n_1, n_2,\cdots,n_c$, can be obtained by solving simple difference equations which lend themselves to programming for machine computation. Using this procedure, tables of values of $P\lbrack D(n, n, n) \leqq r\rbrack,\quad P\lbrack D(n, n) \leqq r\rbrack,\quad P\lbrack D^+(n,n) \leqq r\rbrack$ were computed for selected values of $n$ between 1 and 40 and of $r = k/n,k = 1, 2, \cdots, n$. Furhermore, the inequalities \begin{equation*}\begin{align*} P\lbrack D(n, n,\cdots, n) \leqq r\rbrack \geqq 1 - \rbrack c(c - 1)/2\rbrack P\lbrack D(n,n) > r\rbrack \\ P\lbrack D(n, n, \cdots, n) \leqq r\rbrack \geqq 1 - \lbrack c(c - 1)(c - 2)/6\rbrack P\lbrack D(n, n, n) > r\lbrack\end{align*}\end{equation*} are noted, which may be useful for values of $c \geqq 4$ for which tables are not available.