##### Department of Mathematics,

University of California San Diego

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### Math 295 - Mathematics Colloquium

## Rudolf Beran

#### University of California, Davis

## From Inadmissibility to Effective Regularization

##### Abstract:

Charles Stein (1956) discovered that, under quadratic loss, the usual unbiased estimator for the mean vector of a multivariate normal distribution is inadmissible if the dimension $n$ of the mean vector exceeds two. Contemporaries claimed that Stein's results and the subsequent James-Stein estimator are counter-intuitive, even paradoxical, and not very useful. This talk reexamines such assertions in the light of arguments presented, sketched, or foreshadowed in Stein's beautifully written 1956 paper. Among often overlooked aspects of the paper are the asymptotic geometry of quadratic loss in high dimensions that makes Stein estimation transparent; asymptotic optimality results associated with Stein estimators; the explicit mention of practical multiple shrinkage estimators; and the foreshadowing of confidence balls centered at Stein estimators. Implications of these ideas underlie effective modern regularization estimators, among them, penalized least squares estimators with multiple quadratic penalties, running weighted means, nested submodel fits, and more.

Host: Dimitris Politis

### April 22, 2010

### 4:00 PM

### AP&M 6402

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