M-estimators in linear models with long range dependent errors

Abstract

This paper discusses the asymptotic behavior of a class of M-estimators in linear models when errors are Gaussian, or a function of Gaussian random variables, that are long range dependent. The asymptotics are discussed when the design variables are either i.i.d. or long range dependent, independent of the errors, or known constants. It is observed that in the latter two cases, the leading r.v.'s in the approximation of the class M-estimators of the regression parameter vector corresponding to the skew symmetric scores and symmetric errors is proportional to the least squares estimator in the Gaussian errors case. Moreover, if the design variables are either i.i.d. or the known constants then the limiting distributions are normal. But if the design variables are long range dependent then the limiting distributions are nonnormal.