Lectures on the theory of estimation of many parameters

This article is based on the series of lectures given by Prof. Charles Stein of Stanford University at LOMI AN SSSR in the fall of 1976. The first three lectures are concerned with the estimation of the mean vector of a multivariate normal distribution with quadratic loss function. James-Stein estimators are considered and their relation to Bayesian estimators is discussed. The problem of estimating the covariance matrix of the normal distribution and the estimation of the entropy of a multinomial distribution are considered in the following two lectures. The final lecture discusses several problems related to the estimation of multivariate parameters and poses some unsolved problems.Published in Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 74, pp. 4–65, 1977.In conclusion of this series of lectures, I would like to acknowledge the assistance of M. Ermakov, A. Borodin, and A. Makshanov in translating my lectures into Russian.