The Annals of Mathematical Statistics

The Distribution of Galton's Statistic

Shulamith Gross and Paul W. Holland

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Let $X_{(1)} < \cdots < X_{(n)}$ and $Y_{(1)} < \cdots < Y_{(n)}$ be the order statistics of two independent random samples from the absolutely continuous distribution functions $F(x)$ and $G(y)$, respectively. Let $T_n$ be the proportion of pairs, $(X_{(i)}, Y_{(i)})$, for which $X_{(i)} \geqq Y_{(i)}$. Tests of the equality of $F$ and $G$ based on $T_n$ are among the oldest nonparametric procedures in the literature, going back at least to Galton's analysis of Darwin's data [3]. Hodges [5] showed the null distribution of $nT_n$ to be uniform over $0, 1,\cdots, n$. Bickel and Hodges [1] treated the asymptotic distribution of the Lehmann estimate based on the one-sample version of $T_n$. In this note we use very elementary methods to derive expressions for the distribution and moments of $T_n$ from which conditions for the consistency of tests based on $T_n$ follow immediately. More generally we can show that (unnormalized) $T_n$ always has an asymptotic distribution for any pair $(F, G)$. This distribution is degenerate at zero if $Y$ happens to be stochastically larger than $X$. We give informative expressions for the first two moments of this asymptotic distribution. Our technique is to express the distribution of $T_n$ in terms of integrals of certain multinomial probabilities.

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Ann. Math. Statist., Volume 39, Number 6 (1968), 2114-2117.

First available in Project Euclid: 27 April 2007

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Gross, Shulamith; Holland, Paul W. The Distribution of Galton's Statistic. Ann. Math. Statist. 39 (1968), no. 6, 2114--2117. doi:10.1214/aoms/1177698040.

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