An Extension of Massey's Distribution of the Maximum Deviation Between Two- Sample Cumulative Step Functions

Abstract

Let $x_1 < x_2 < \cdots < x_n$ and $y_1 < y_2 < \cdots < y_m$ be the ordered results of two random samples from populations having continuous cumulative distribution functions $F(x)$ and $G(x)$ respectively. Let $S_n(x) = k/n$, where $k$ is the number of observations of $X$ which are less than or equal to $x$, and $S'_m(x) = j/m$, where $j$ is the number of observations of $Y$ which are less than or equal to $x$. The statistics \begin{align*}d_r &= \underset{x \leqq x_r}\max | S_n(x) - S'_m(x) |, \\ d'_r &= \underset{x \leqq \max(x_r,y_r)}\max | S_n (x) - S'_m(x) |, r \leqq \min (m, n), \\\end{align*} can be used to test the hypothesis $F(x) = G(x)$. For example, using $d_r$ we would reject the hypothesis if the observed $d_r$, that is, the maximum absolute deviation between the two step functions at or below the $r$th observation of a given sample, is significantly large. In this paper, the distributions of $d_r$ and $d'_r$ under the hypothesis $F(x) = G(x)$ are obtained and tabulated. Some possible applications are discussed and a numerical example in life testing is given.