The Spacing of Observations in Polynomial Regression

Abstract

De la Garza ([1], [2]) has considered the estimation of a polynomial of degree $p$ from $n$ observations in a given range of the independent variable $x$. This range may conveniently be taken to be from $+1$ to $-1$. He showed that for any arbitrary distribution of the points of observation there was a distribution of the $n$ observations at only $p + 1$ points for which the variances (determined by the matrix $\mathbf{X}^T\mathbf{W X}$) were the same. He then considered how these $p + 1$ points should be distributed so that the maximum variance of the fitted value in the range of interpolation should be as small as possible. In the present note general formulae will be obtained for the distribution of the points of observation and for the variances of the fitted values in the minimax variance case, and the variances will be compared with those for the uniform spacing case.