A weak convergence result useful in robust autoregression

Abstract

Abstract This paper establishes the closeness, in a uniform sense, of a sequence of randomly weighted estimated residual empirical processes to a sequence of randomly weighted residual empirical processes via the weak convergence techniques. This result is used to obtain an asymptotic expansion of generalized M (G–M) estimators of autoregression parameters and the asymptotic uniform linearity (A.U.L.) of the sequence of ordinary residual empirical processes in a p -th order autoregression (AR( p )) model. The latter result is used to prove the A.U.L. of serial rank correlations of the residuals in an AR( p ) model and to yield a direct proof of the consistency of a class of estimators of the functional Γ( f ) ≔∫ f dϕ( F ), where f ( F ) is the error density (distribution function) in the AR( p ) model and ϕ is a nondecreasing bounded function on [0, 1]. These functionals appear in the asymptotic variances of various robust estimators of autoregression parameters. In summary, the paper offers a unified functional approach to some aspects of the robust estimation in AR( p ) models.