On Sequential Empirical Distribution Functions for Random Variables with Mean Zero

Abstract

The classical sequential empirical distribution function incorporates all subsamples of a sample of independent and identically distributed random variables and is therefore well suited to construct tests for detecting a distributional change occurring somewhere in the sample. If the independent and identically distributed variables are replaced by the residuals of appropriate time series models tests for a distributional change in the unobservable errors (or innovations) of these models are obtained; see Bai (Annals of Statistics, 22:2051–2061, 1994) for the discussion of ARMA models. These errors are often assumed to have mean zero, an information which is not taken into account by the classical sequential empirical distribution function. Based upon ideas from empirical likelihood, see Owen (London/Boca Raton: Chapman & Hall/CRC, 2001), we consider a modified sequential empirical distribution function for random variables with mean zero which does exploit this information.