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Abstract This article presents a method for approximating the coverage probability of simultaneous confidence bounds for multiple regression functions when the vector of predictor variables is constrained to lie in a polyhedral convex set. The method is useful because it allows one to construct simultaneous confidence intervals with prescribed coverage probability for the regression function evaluated at various settings of the predictor variables, which are narrower than bounds obtained without using the predictor variable constraints. For a family of two-sided simultaneous confidence bounds that includes Scheffé-type and constant-width bounds, the probability of coverage is related to the distribution of the maximum Euclidean norm of the projection onto a polyhedral cone for a pair of random vectors with known joint distribution. An analogous relation holds for one-sided bounds. If an algorithm for computing the projection onto the cone is available, then these results enable one to use the Monte Carlo method to approximate the coverage probabilities of bounds. Particular attention is given to the case in which lower and upper bounds can be specified for each of the predictor variables so that the constraint region for bounding the regression function is rectangular. An efficient algorithm for calculating projections onto the appropriate cone for rectangular constraint regions, which facilitates coverage probability approximation of Schefffé-type bounds, is presented. This algorithm is used to calculate approximate critical points of Scheffé-type bounds for some specific designs and constraint regions and we show that substantial improvements for one-sided bounds and modest improvements for two-sided bounds over the conservative bounds of Casella and Strawderman (1980) may be obtained.
Daniel Q. Naiman (Sun,) studied this question.