The Annals of Mathematical Statistics

The Distribution of the Maximum Deviation Between two Sample Cumulative Step Functions

Frank J. Massey, Jr.

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Abstract

Let $x_1 < x_2 M \cdots < x_n$ and $y_1 < y_2 < \cdot < y_m$ be the ordered results of two random samples from populations having continuous cumulative distribution functions $F(x)$ and $G(x)$ respectively. Let $S_n(x) = K/n$ when $k$ is the number of observed values of $X$ which are less than or equal to $x$, and similarly let $S'_m(y) = j/m$ where $j$ is the number of observed values of $Y$ which are less than or equal to $y$. The statistic $d = \max | S_n(x) - S'_m(x) |$ can be used to test the hypothesis $F(x) \equiv G(x)$, where the hypothesis would be rejected if the observed $d$ is significantly large. The limiting distribution of $d \sqrt{mn}{m + n}$ has been derived [1] and [4], and tabled [5]. In this paper a method of obtaining the exact distribution of $d$ for small samples is described, and a short table for equal size samples is included. The general technique is that used by the author for the single sample case [2]. There is a lower bound to the power of the test against any specified alternative, [3]. This lower bound approaches one as $n$ and $m$ approach infinity proving that the test is consistent.

Article information

Source
Ann. Math. Statist., Volume 22, Number 1 (1951), 125-128.

Dates
First available in Project Euclid: 28 April 2007

Permanent link to this document
https://projecteuclid.org/euclid.aoms/1177729703

Digital Object Identifier
doi:10.1214/aoms/1177729703

Mathematical Reviews number (MathSciNet)
MR39952

Zentralblatt MATH identifier
0042.14107

JSTOR
links.jstor.org

Citation

Massey, Frank J. The Distribution of the Maximum Deviation Between two Sample Cumulative Step Functions. Ann. Math. Statist. 22 (1951), no. 1, 125--128. doi:10.1214/aoms/1177729703. https://projecteuclid.org/euclid.aoms/1177729703


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