The Choice of the Degree of a Polynomial Regression as a Multiple Decision Problem

On the basis of a sample of observations, an investigator wants to determine the appropriate degree of a polynomial in the index, say time, to represent the regression of the observable variable. This multiple decision problem is formulated in terms used in the theory of testing hypotheses. Given the degree of polynomial regression, the probability of deciding a higher degree is specified and does not depend on what the actual polynomial is (expect its degree). Within the class of procedures satisfying these conditions and symmetry (or two-sidedness) conditions, the probabilities of correct decisions are maximized. The optimal procedure is to test in sequence whether coefficients are 0, starting with the highest (specified) degree. The procedure holds for other linear regression functions when the independent variates are ordered. The problem and its solution can be generalized to the multivariate case and to other cases with a certain structure of sufficient statistics.