Nonparametric Estimation in the Model of Moving Average
The subject of robust estimation in time series is widely discussed in literature. One of the approaches is to use GM-estimation. This method incorporates a broad class of nonparametric estimators which under suitable conditions includes estimators robust to outliers in data. For the linear models the sensitivity of GM-estimators to outliers have been studied in the work by Martin and Yohai [5], and influence functionals for this estimator were derived. In this paper we follow this direction and examine the asymptotical properties of the class of M-estimators, which is narrower than the class of GM-estimators, but gives more insight into asymptotical properties of such estimators. This paper gives an asymptotic expansion of the residual weighted empirical process, which allows to prove asymptotic normality of these estimators in case of non-smooth objective functions. For simplicity MA(1) model is considered, but it will be shown that even in this case mathematical techniques used to derive these asymptotic properties appear to be rather complicated.However, the approach used in this paper could be applied to GM-estimators and to more realistic models.