Linear regression: robust heteroscedastic confidence bands that have some specified simultaneous probability coverage

Let $ M(Y\,|\,X) = \beta _0 + \beta _1X $ M(Y|X)=β0+β1X be some conditional measure of location associated with the random variable Y, given X, where the unknown parameters $ \beta _0 $ β0 and $ \beta _1 $ β1 are estimated based on the random sample $ (X_1, Y_1), \ldots, (X_n, Y_n) $ (X1,Y1),…,(Xn,Yn). When using the ordinary least squares (OLS) estimator and $ M(Y\,|\,X)=E(Y\,|\,X) $ M(Y|X)=E(Y|X), several methods for computing a confidence band have been derived that are aimed at achieving some specified simultaneous probability coverage assuming a homoscedastic error term and normality. There is an extant technique that allows heteroscedasticity, but a remaining concern is that the OLS estimator is not robust. Extant results indicate how a confidence interval can be computed via a robust regression estimator when there is heteroscedasticity and attention is focused on a single value of X. The paper extends this method by describing a heteroscedastic technique for computing a confidence interval for each $ M(Y\,|\,X=X_i) $ M(Y|X=Xi) ( $ i=1, \ldots, n $ i=1,…,n) such that the simultaneous probability coverage has some specified value. The small-sample properties of the method are studied when using the OLS estimators as well as three robust regression estimators.