Protection Against Assuming the Wrong Degree in Polynomial Regression

When choosing points of observation in the interval -1, 1 for purposes of estimating polynomial regression curves, we are often uncertain as to the degree p of the polynomial in the independent variable X. This paper deals with the influence of this uncertainty on the problem of allocating observations so as to minimize the maximum predictive variance over the interval. For example, if p = 1 the minimax allocation consists of placing half of the available observations at each end of the interval, but this allocation affords no chance whatever of detecting evidence that p = 2, say. Section 2 deals with the loss of precision incurred by using the minimax allocation for a degree higher than the true degree, with particular attention to the cases where the true degree p is equal to 1 or to 2. The results thereof are discussed in Section 3, the conclusion being that if we fear a possible alternative degree p1 one or two greater than the presumed degree po we should allocate observations as if p = pi , since if in fact p = po the loss of precision incurred by using p,-allocation will be relatively small. The point of departure for what follows will be the usual polynomial regression model of the form