Confidence bands with a given maximal size for linear regression functions

Abstract

The size of the confidence bands for linear regression functions proposed in the literature is random. In this paper we construct confidence bands, whose size (width, content etc.) over a certain range for the regressor does not exceed a given value. We solve this problem by generalization of the known STEIN idea of two-stage confidence intervals. We propose two different two-stage procedures. The first procedure is based on an F-distribution and the margins of the bands are “surfaces of second order”. The second procedure is based on a multidimensional t-distribution and the margins are “hyperplanes”. We explain connections with problems of optimal design and with applications of linear and nonlinear programming considering estimated coefficients.