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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 regression 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 addition 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 () observations. Besides the importance of modelling and forecasting this pollutant, the 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. [ABSTRACT FROM AUTHOR]

In recent years, methods to estimate the memory parameter using wavelet analysis have gained popularity in many areas of science. Despite its widespread use, a rigorous semi-parametric asymptotic theory, comparable with the one developed for Fourier methods, is still lacking. In this article, we adapt to the wavelet setting, the classical semi-parametric framework introduced by Robinson and his co-authors for estimating the memory parameter of a (possibly) non-stationary process. Our results apply to a class of wavelets with bounded supports, which include but are not limited to Daubechies wavelets. We derive an explicit expression of the spectral density of the wavelet coefficients and show that it can be approximated, at large scales, by the spectral density of the continuous-time wavelet coefficients of fractional Brownian motion. We derive an explicit bound for the difference between the spectral densities. As an application, we obtain minimax upper bounds for the log-scale regression estimator of the memory parameter for a Gaussian process and we derive an explicit expression of its asymptotic variance. [ABSTRACT FROM AUTHOR]