Functional central limit theorems for self-normalized least squares processes in regression with possibly infinite variance data

Abstract

Based on an R 2 -valued random sample { ( y i , x i ) , 1 ≤ i ≤ n } on the simple linear regression model y i = x i β + α + ε i with unknown error variables ε i , least squares processes (LSPs) are introduced in D [ 0 , 1 ] for the unknown slope β and intercept α , as well as for the unknown β when α = 0 . These LSPs contain, in both cases, the classical least squares estimators (LSEs) for these parameters. It is assumed throughout that { ( x , ε ) , ( x i , ε i ) , i ≥ 1 } are i.i.d. random vectors with independent components x and ε that both belong to the domain of attraction of the normal law, possibly both with infinite variances. Functional central limit theorems (FCLTs) are established for self-normalized type versions of the vector of the introduced LSPs for ( β , α ) , as well as for their various marginal counterparts for each of the LSPs alone, respectively via uniform Euclidean norm and sup–norm approximations in probability. As consequences of the obtained FCLTs, joint and marginal central limit theorems (CLTs) are also discussed for Studentized and self-normalized type LSEs for the slope and intercept. Our FCLTs and CLTs provide a source for completely data-based asymptotic confidence intervals for β and α .