## Multivariate location estimation using extension of $$R$$-estimates through $$U$$-statistics type approach.(English)Zbl 0762.62013

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 J. B. S. Haldane [Biometrika., Cambridge 35, 414-415 (1948; Zbl 0032.03601)] and a multivariate extension of the well-known Hodges-Lehmann estimate [J. L. Hodges and E. L. Lehmann, Ann. Math. Statistics 34, 598-611 (1963; Zbl 0203.211)]. 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 C. Stein [Proc. 3rd Berkeley Sympos. Math. Statist. Probability 1, 197-206 (1956; Zbl 0073.356)] in high dimensions. Interestingly, the phenomena that we observe here occur even in dimension $$d=2$$. (Author’s abstract.)
An interesting contrast to the James-Stein result is that the author achieves shrinking of asymptotic variance for each component of his estimator. It appears also that the sample median (sample size $$n)$$ as an estimator of the mean of a normal with variance can be improved by generating an independent sample of size $$n$$ from standard normal.

### MSC:

 62H12 Estimation in multivariate analysis 62G05 Nonparametric estimation 62G20 Asymptotic properties of nonparametric inference

### Citations:

Zbl 0032.03601; Zbl 0203.211; Zbl 0073.356
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