Optimal Spacing and Weighting in Polynomial Prediction

Abstract

A solution is given to the problem of how to determine at which points in the interval $\lbrack -1, 1\rbrack$ observations should be taken and what proportion of the observations should be taken at each such point so as to minimize the variance of the predicted value of a polynomial regression curve at a specified point beyong the interval observations. The solution obtained states that the points are to be chosen to be Chebychev points and the number of observations are to be selected proportional to the absolute value of the corresponding Lagrange polynomial at the specified point. The preceding Chebychev solution becomes the minimax solution for the interval $(-1, t),$ provided $t > t_1 > 1$ where $t_1$ is a value satisfying a certain equation. Under the customary normality assumptions, the Chebychev solution to the prediction problem is used to construct a confidence band for a polynomial curve that will possess minimum width at any specified point beyond the interval of observations.