Minimax Regret Simultaneous Confidence Bands for Multiple Regression Functions
Abstract When a simultaneous confidence band for a multiple regression function is to be constructed, one encounters the problem of choosing a band from among those available. The following notion of optimality for bands is introduced. Suppose that the band is constructed with the purpose of giving confidence intervals for evaluations x i βt, of the regression function at future settings x i (i = 1, …, n) of the predictor variables. The settings are assumed to be unknown at the time the band is to be chosen, but they are assumed to lie within a given bounded set, which is referred to as the constraint set. Once a particular band is chosen and the predictor variable settings become known, the “regret” for the procedure can be defined as the difference between the average width of the n intervals and the smallest possible width attainable by a band constructed using knowledge of the predictor variable settings. Thus the regret measures the cost of not knowing the settings when the band is chosen. It is reas...