## Estimation with quadratic loss.(English)Zbl 1281.62026

Proc. 4th Berkeley Symp. Math. Stat. Probab. 1, 361-379 (1961).
Following the introduction, in Section 2 of this paper is given a new proof by the authors of the result of [C. Stein, in: Proc. 3rd Berkeley Sympos. Math. Statist. Probability 1, 197–206 (1956; Zbl 0073.35602)] that the usual estimator of the mean of a multivariate normal distribution with the identity as covariance matrix is inadmissible when the loss is the sum of squares of the errors in the different coordinates if the dimension is at least three. An explicit formula is given for an estimator, still inadmissible, whose risk is never more than that of the usual estimator and considerably less near the origin. Other distributions and other loss functions are considered later in Section 2.
In Section 3 the general problem of admissibility of estimators for problems with quadratic loss is formulated and a sufficient condition for admissibility is given and its relation to the necessary and sufficient condition [C. Stein, Ann. Math. Stat. 26, 518–522 (1955; Zbl 0065.11703)] is briefly discussed.
In Section 4 theorems are given which show that under weak conditions Pitman’s estimator for one or two location parameters is admissible when the loss is taken to be equal to the sum of squares of the errors. Conjectures are discussed for the more difficult problem where unknown location parameters are also present as nuisance parameters, and Blackwell’s example is given.
In Section 5 a problem in multivariate analysis is given where the natural estimator is not even minimax although it has constant risk. These are related to the examples of one of the authors quoted by J. Kiefer [Ann. Math. Stat. 28, 573–601 (1957; Zbl 0080.13004)] and E. L. Lehmann [Testing statistical hypotheses. New York: John Wiley & Sons; London: Chapman & Hall (1959; Zbl 0089.14102), pp. 231 and 338].
In Section 6 some unsolved problems are mentioned.
The results of Section 2 were obtained by the two authors working together. The remainder of the paper is the work of C. Stein.
For the entire collection see [Zbl 0101.34803].

### MSC:

 62J07 Ridge regression; shrinkage estimators (Lasso) 62H12 Estimation in multivariate analysis 62C15 Admissibility in statistical decision theory