Robust Local Bootstrap for WeaklyStationary Time Series in the Presence ofAdditive Outliers

The aim of this paper is to propose a generalization of the local bootstrap for periodogram statistics to the case when weakly stationary time series are contaminated by additive outliers. In order to achieve robustness, we suggest to replace the classical version of the periodogram with the M-periodogram in the local bootstrap procedure. The robust bootstrap periodogram is implemented in the Whittle estimator to obtain confidence intervals for the parameters of a time series model. A finite sample size investigation was conducted to compare the performance of the classical local bootstrap with the one proposed in this paper, to estimate 95% confidence intervals for the parameters of autoregressive and of seasonal autoregressive time series. The results have shown that the robust estimator is resistant to additive outlier contamination and produces confidence intervals with coverage percentage closer to 95\% and with lower amplitudes than the ones obtained with the classical estimator, even for small percentages and magnitudes of outliers. It was also empirically demonstrated that when the expected number of outliers is kept constant, the coverage percentages of the confidence intervals of the robust estimators tend to 95\% as the sample size increases. An application to the daily mean concentration of the particulate matter with diameter smaller than \SI{10}{\micro\meter} (PM$_{10}$) was considered to illustrate the methodologies in a real data context. All the results presented here give strong motivation to use the proposed robust methodology in practical situations in which weakly stationary time series are contaminated by additive outliers.