## The Annals of Statistics

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

Probal Chaudhuri

#### Abstract

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

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

Dates
First available in Project Euclid: 12 April 2007

https://projecteuclid.org/euclid.aos/1176348662

Digital Object Identifier
doi:10.1214/aos/1176348662

Mathematical Reviews number (MathSciNet)
MR1165598

Zentralblatt MATH identifier
0762.62013

JSTOR
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. https://projecteuclid.org/euclid.aos/1176348662