Shorter Confidence Bands in Linear Regression

原文摘要

In many linear regression problems, the values of the independent variable or variables may be subject to certain constraints. For example, the independent variables may necessarily be positive; as another example, the variables may not only all be positive but are powers of a single variable (e.g., polynomial regression on time). Previous writers considering the problem of obtaining confidence bands on a regression function for all values of the independent variable have not utilized such constraints; the usual basis for such bands has been the multiple comparison procedure of Scheffé which places no constraints at all upon the independent variables. Any procedure utilizing constraints will necessarily yield a uniform improvement over the method of Scheffé' (assuming both methods are applicable) in the sense of yielding narrower bands for a given confidence probability. In the present paper a non-trivial lower bound is obtained for the confidence probability associated with a multiple comparison procedure appropriate to the case where it can be assumed that each independent variable must be of specified sign; this includes, as a subclass, polynomial regression on a non-negative independent variable. This result gives a basis for a multiple comparison procedure less conservative than that of Scheffé when both are applicable. Implementation of the procedure requires the percentage points of a heretofore untabulated distribution. Tables of percentage points of this distribution appropriate to linear combinations of two, three, or four parameters are presented.

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