Minimax Regret Simultaneous Confidence Bands for Multiple Regression Functions
提出一种最小最大遗憾准则来构造多元回归函数的置信带，在预测变量设置未知时最小化最大平均宽度损失，并给出简单线性回归和平衡多元回归的临界点公式与表格。
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 reasonable to want a minimax regret band, that is, a band minimizing maximum regret over all possible settings of the predictor variables. For a standard multiple regression model the unique taut minimax regret band is characterized and described in a simple manner. Formulas for computing critical points for minimax regret bands are given for two special cases: for simple linear regression, when the predictor variable is constrained to lie in an interval, and for (balanced) multiple regression with intercept, when the constraint region is one of those defined by Halperin and Gurian (1968). Tables of critical points are presented for balanced simple linear regression over a finite interval. The following application is presented to illustrate the use of these results. A fitted simple linear regression model based on data from Duncan (1974) is to be used to relate strengths of weldings to their diameters, and a simultaneous confidence band for expected welding strength is to be constructed over a practical range of welding diameters. For this problem, a comparison between confidence intervals defined by the 95% minimax regret band, the Scheffé-type band (Scheffé 1953, 1959), and the constant-width band is given.