The Annals of Statistics

Reduced $U$-Statistics and the Hodges-Lehmann Estimator

B. M. Brown and D. G. Kildea

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A reduced $U$-statistic (of order 2) is defined as the sum of terms $f(X_i, X_j),$ where $f$ is a symmetric function, $(X_1, \cdots, X_N)$ are independent and identically distributed (i.i.d.) random variables (rv's), and $(i,j)$ are drawn from a restricted, though balanced, set of pairs. (A $U$-statistic corresponds to summation over all $(i, j)$ pairs.) A limit normal distribution is found for the reduced $U$-statistic, and it follows that estimates based on reduced $U$-statistics can have asymptotic efficiencies comparable with those based on $U$-statistics, even though the number of steps in computing a reduced $U$-statistic becomes asymptotically negligible in comparison with the number required for the corresponding $U$-statistic. As an illustration, a short-cut version of the Hodges-Lehmann estimator is defined, and its asymptotic properties derived, from a corresponding reduced $U$-statistic. A multivariate limit theorem is proved for a vector of reduced $U$-statistics, plus another result obtaining asymptotic normality even when $(i, j)$ are drawn from an unbalanced set of pairs. The present results are related to those of Blom.

Article information

Ann. Statist., Volume 6, Number 4 (1978), 828-835.

First available in Project Euclid: 12 April 2007

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier


Primary: 60F05: Central limit and other weak theorems
Secondary: 60G05: Foundations of stochastic processes 60G20: Generalized stochastic processes 60G25: Prediction theory [See also 62M20]

$U$-statistics Hodges-Lehmann estimator asymptotic efficiency convergence of moments


Brown, B. M.; Kildea, D. G. Reduced $U$-Statistics and the Hodges-Lehmann Estimator. Ann. Statist. 6 (1978), no. 4, 828--835. doi:10.1214/aos/1176344256.

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