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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
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