Some remarks about the gasser-sroka-jennen-steinnietz variance estimator

This article proposes some simplifications of the residual variance estimator of Gasser, Sroka, and jennen-Steinmetz (GSJ, 1986) which is often used in conjunction with nonparametric regression. The GSJ estimator is a quadratic form of the data, which depends on the relative spacings of the design points. When the errors are independent, identically distributed Gaussian variables, and the true regression curve is flat, theestimate is distributed as a weighted sum of x² variables. By matching the first two moments, the distribution can be approximated by a x^2with degrees of freedom determined by the coefficients of the quadratic for. Computation ofthe estimated degrees of freedom requires computing the trace of the square of an n x n matrix, where n is the number of design points. In this article, (n-2)/3 is shown to be a conservative estimate of the approximate degrees of freedom, and (n-2)/2 is shown to be conservative for many designs. In addition, a simplified version of the estimator is shown to be asymptotically equivalent, under many conditions.