Estimating the Common Parameter of Several Distributions

Abstract

As described in Sect. 1.3, the consulting problem was that of estimating the common mean of two normal distributions, when the data of the two samples are given, and the corresponding variances are unknown. There are two types of problems. One is an estimation problem, when the samples from the two distributions are given. In this case we are in the estimation phase. Prior to this phase there is a design phase, in which one has to determine how many observations to draw from each distribution. Obviously, if we know which distribution has the smaller variance, then the best is to draw all the observations from that distribution. If this information is not available, the design problem becomes very difficult if the objective is to minimize the total number of observations, under the constraint that the variance (or mean-squared error) of the estimator satisfies certain condition. This design problem is similar to the well-known “two-armed bandit problem.” The present chapter is devoted to the estimation problem, when two or more samples of equal size are given from the corresponding distributions. For a sequential design problem, when the value of one variance is known, see the paper of Zacks (Naval Res Logist Q 21:569–574, 1974).