Spline confidence bands for variance functions

Abstract

Asymptotically exact and conservative confidence bands are obtained for possibly heteroscedastic variance functions, using piecewise constant and piecewise linear spline estimation, respectively. The variance estimation is as efficient as an infeasible estimator when the conditional mean function is known, and the widths of the confidence bands are of optimal order. Simulation experiments provide strong evidence that corroborates the asymptotic theory while the computing is extremely fast. A slower bootstrap band is also proposed, with much higher accuracy. As illustrations, the bootstrap spline band has been applied to test for heteroscedasticity in fossil data and in motorcycle data. Keywords: B-spline; heteroscedasticity; infeasible estimator; knots; nonparametric regression; variance function AMS Subject Classification: 62G08; 62G10; 62G15 1.