Convergence of polynomial least squares end-point and mid-point estimators

In a recent corcespondence, Leskiw and Miller [1] obtained the variance of a least squares polynomial estimator, as function of the polynomial order, for a large number of equally spaced data points, when the estimate is for the end-point. A simpler proof is given, and the result is extended to the mid-point. A mid-point estimator may be of special interest since it exhibits the lowest variance.