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

ABSTRACT

Let $M\left(Y|X\right)={\beta }_0+{\beta }_{1}X$ be some conditional measure of location associated with the random variable Y, given X, where the unknown parameters ${\beta }_0$ and ${\beta }_1$ are estimated based on the random sample $(X_1,Y_1),\dots ,(X_n,Y_n)$. When using the ordinary least squares (OLS) estimator and $M\left(Y|X\right)=E\left(Y|X\right)$, 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 $M(Y|X=X_i)$ ($i=1,\dots ,n$) 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.

KEYWORDS:

• Prediction intervals
• heteroscedasticity
• robust regression
• analysis of covariance
• Well Elderly 2 study

Disclosure statement

No potential conflict of interest was reported by the author.