The regression smoother LOWESS: A confidence band that allows heteroscedasticity and has some specified simultaneous probability coverage

Abstract

Many nonparametric regression estimators (smoothers) have been proposed that provide a more flexible method for estimating the true regression line compared to using some of the more obvious parametric models. A basic goal when using any smoother is computing a confidence band for the true regression line. Let M(Y|X) be some conditional measure of location associated with the random variable Y, given X and let x be some specific value of the covariate. When using the LOWESS estimator, an extant method that assumes homoscedasticity can be used to compute a confidence interval for M(Y|X = x). A trivial way of computing a confidence band is to compute confidence intervals for K covariate values, each having probability coverage 1 - α. But an obvious concern is that the simultaneous probability coverage can be substantially smaller than 1 - α. A method is suggested for dealing with this issue that allows heteroscedasticity and simultaneously performs better than the Bonferroni method or the Studentized maximum modulus distribution.