The Annals of Statistics

Multivariate Location Estimation Using Extension of $R$-Estimates Through $U$-Statistics Type Approach

Probal Chaudhuri

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We consider a class of $U$-statistics type estimates for multivariate location. The estimates extend some $R$-estimates to multivariate data. In particular, the class of estimates includes the multivariate median considered by Gini and Galvani (1929) and Haldane (1948) and a multivariate extension of the well-known Hodges-Lehmann (1963) estimate. We explore large sample behavior of these estimates by deriving a Bahadur type representation for them. In the process of developing these asymptotic results, we observe some interesting phenomena that closely resemble the famous shrinkage phenomenon observed by Stein (1956) in high dimensions. Interestingly, the phenomena that we observe here occur even in dimension $d = 2$.

Article information

Ann. Statist., Volume 20, Number 2 (1992), 897-916.

First available in Project Euclid: 12 April 2007

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Zentralblatt MATH identifier


Primary: 62G05: Estimation
Secondary: 62G20: Asymptotic properties 62H12: Estimation 62E20: Asymptotic distribution theory 62H10: Distribution of statistics

$R$-estimates multivariate median Hodges-Lehmann estimate $U$-statistics generalized order statistics Bahadur representation Stein phenomenon


Chaudhuri, Probal. Multivariate Location Estimation Using Extension of $R$-Estimates Through $U$-Statistics Type Approach. Ann. Statist. 20 (1992), no. 2, 897--916. doi:10.1214/aos/1176348662.

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