Sequential Inference Procedures of Stein's Type for a Class of Multivariate Regression Problems

Abstract

In this paper the case of multivariate regression with stochastic predictors is considered, the joint distribution of the predictors being unknown, and the conditional distribution of the predict and given the predictors being normal with an unknown standard deviation. Sequential procedures of Stein's [13] type, terminating with probability one, are developed to obtain tests, confidence regions, and point estimates for the regression parameters. For the tests, the power function does not depend on the unknown distribution of the predictors or any nuisance parameters; for the confidence regions, the "span" is fixed and known; and for the point estimates, the expected loss, for a particular type of loss function, is a known constant. The procedure is subsequently modified to get more useful and "efficient" tests and estimates. Some study of the distribution and expectation of the sample size is also made for the sequential procedures developed.