Robust local bootstrap for weakly stationary time series in the presence of additive outliers

This paper proposes a generalization of the local bootstrap for periodogram statistics when weakly stationary time series are contaminated by additive outliers. To achieve robustness, we suggest replacing 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 seasonal autoregressive time series. The results have shown that the robust estimator is resistant to additive outlier contamination and produces confidence intervals with coverage percentages closer to 95% and lower amplitudes than the ones obtained with the classical estimator, even for small percentages and magnitudes of outliers. It was also empirically observed 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 particulate matter with a diameter smaller than 10μm\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\upmu \textrm{m}$$\end{document} (PM10\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$_{10}$$\end{document}) was considered to illustrate the methodologies in a real data context. All the results presented here strongly motivate using the proposed robust methodology in practical situations where additive outliers contaminate weakly stationary time series.