July 13, 2018 Statistics: A Journal of Theoretical and Applied Statistics periodogram˙robust˙statistics˙revision3 To appear in Statistics: A Journal of Theoretical and Applied Statistics Vol. 00, No. 00, Month 20XX, 1–21 M -periodogram for the analysis of long-range dependent time series F.A. Fajardo a , V. A. Reisen b,e , C. L´ evy-Leduc c˚ and M.S. Taqqu d a DEST-Universidade Federal do Espirito Santo, Brazil; b DEST and PPGEA-Universidade Federal do Espirito Santo, Brazil; c UMR MIA-Paris, AgroParisTech, INRA, Universit´ e Paris-Saclay, 75005, Paris, France; d Department of Mathematics, Boston University; e Laboratoire des Signaux et Syst` emes, CNRS, Universit´ e Paris-Saclay, France. (Received 00 Month 20XX; final version received 00 Month 20XX) This paper focuses on time series with long-memory and suggests using an alternative periodogram, called M-periodogram, which is obtained by relating the periodogram to a re- gression problem and then using an M-estimator for the coefficients of the regression model. The asymptotic properties of this novel M-periodogram are established and its empirical properties are investigated for finite samples under different scenarios. Furthermore, in ad- dition to being an appealing alternative periodogram for long-memory time series, it is also resistant to additive outliers. We investigate the robustness performance of the estimator through simulation. As a practical application, the paper investigates the effect of atypical observations in air pollution data, namely, daily Particulate Matter (PM10) observations. Besides the importance of modeling and forecasting this pollutant, the PM10 series presents, in general, interesting features such as seasonal poles, asymmetry, and also high levels of pollution which can be regarded as atypical observations in the context of this work. Keywords: Time Series, robust estimation, long memory, outliers, pollution. 1. Introduction The classical periodogram is a powerful tool in the spectral analysis of time series. It is well-known, however, that it is very sensitive to the presence of outliers (Fox (1972)), and therefore, it becomes useless in situations where the real data is contaminated by atypical observations. Since additive outliers are quite common in practice, defining a new periodogram which is not sensitive to the presence of additive outliers has a real practical interest. Several approaches have been proposed in the time series literature in order to deal with outliers. Under a Gaussian assumption for the core process, that is, the uncon- taminated time series, Tatum and Hurvich (1993a) and Tatum and Hurvich (1993b) proposed interesting ways to construct high breakdown methods for handling additive contamination based on robust trigonometric regression to obtain a robustified discrete Fourier transform. Their approaches consist in obtaining a filtered version of the data to reconstruct the “core process”, namely the original underlying process without the out- liers. References related to computing robust spectral estimators include, for example, Katkovnik (1998), Li (2008) and references therein. More generally, Kleiner et al. (1979), Martin and Thomson (1982) and more recently Spangl and Dutter (2013), Hagemann ˚ Corresponding author. Email: celine.levy-leduc@agroparistech.fr