Stein's example

Abstract: It has long been customary to measure the adequacy of an estimator by the smallness of its mean squared error. The least squares estimators were studied by Gauss and by other authors later in the nineteenth century. A proof that the best unbiased estimator of a linear function of the means of a set of observed random variables is the least squares estimator was given by Markov [12], a modified version of whose proof is given by David and Neyman [4]. A slightly more general theorem is given by Aitken [1]. Fisher [5] indicated that for large samples the maximum likelihood estimator approximately minimizes the mean squared error when compared with other reasonable estimators. This paper will be concerned with optimum properties or failure of optimum properties of the natural estimator in certain special problems with the risk usually measured by the mean squared error or, in the case of several parameters, by a quadratic function of the estimators. We shall first mention some recent papers on this subject and then give some results, mostly unpublished, in greater detail.

Abstract: Stein's estimator for k normal means is known to dominate the MLE if k ≥ 3. In this article we ask if Stein's estimator is any good in its own right. Our answer is yes: the positive part version of Stein's estimator is one member of a class of “good” rules that have Bayesian properties and also dominate the MLE. Other members of this class are also useful in various situations. Our approach is by means of empirical Bayes ideas. In the later sections we discuss rules for more complicated estimation problems, and conclude with results from empirical linear Bayes rules in non-normal cases.

Abstract: In 1961, James and Stein exhibited an estimator of the mean of a multivariate normal distribution having uniformly lower mean squared error than the sample mean. This estimator is reviewed briefly in an empirical Bayes context. Stein's rule and its generalizations are then applied to predict baseball averages, to estimate toxomosis prevalence rates, and to estimate the exact size of Pearson's chi-square test with results from a computer simulation. In each of these examples, the mean square error of these rules is less than half that of the sample mean.

Abstract: 1. Introduction, Coverage, Philosophy, and Computation. 2. The Linear Model. 3. Least-Squares Estimation: Batch Processing. 4. Least-Squares Estimation: Singular-Value Decomposition. 5. Least-Squares Estimation: Recursive Processing. 6. Small Sample Properties of Estimators. 7. Large Sample Properties of Estimators. 8. Properties of Least-Squares Estimators. 9. Best Linear Unbiased Estimation. 10. Likelihood. 11. Maximum-Likelihood Estimation. 12. Multivariate Gaussian Random Variables. 13. Mean-Squared Estimation of Random Parameters. 14. Maximum A Posteriori Estimation of Random Parameters. 15. Elements of Discrete-Time Gauss-Markov Random Sequences. 16. State Estimation: Prediction. 17. State Estimation: Filtering (The Kalman Filter). 18. State Estimation: Filtering Examples. 19. State Estimation: Steady-State Kalman Filter and Its Relationships to a Digital Wiener Filter. 20. State Estimation: Smoothing. 21. State Estimation: Smoothing (General Results). 22. State Estimation for the Not-So-Basic State-Variable Model. 23. Linearization and Discretization of Nonlinear Systems. 24. Iterated Least Squares and Extended Kalman Filtering. 25. Maximum-Likelihood State and Parameter Estimation. 26. Kalman-Bucy Filtering. A. Sufficient Statistics and Statistical Estimation of Parameters. B. Introduction to Higher-Order Statistics. C. Estimation and Applications of Higher-Order Statistics. D. Introduction to State-Variable Models and Methods. Appendix A: Glossary of Major Results. Appendix B: Estimation of Algorithm M-Files. References. Index.