Semi-Parametric Graphical Estimation Techniques for Long-Memory Data

This paper reviews several periodogram-based methods for estimating the long-memory parameter H in time series and suggests a way to robustify them. The high frequencies tend to bias the estimates. Using only low frequencies eliminates the bias but increases the variance. We hence suggest plotting the estimates of H as a function of a parameter which balances bias versus variance and, if the plot flattens in a central region, to use the flat part for estimating H. We apply this technique to the periodogram regression method, the Whittle approximation to maximum likelihood and to the local Whittle method. We investigate its effectiveness on several simulated fractional ARIMA series and also apply it to estimate the long-memory parameter H in computer network traffic. 1 Introduction Time series with long memory have been considered in many fields including hydrology, biology and computer networks. Unfortunately, estimating the long memory (long-range dependence) parameter H in a given d..

Theoretical and numerical comparisons of the parameter estimator of the fractional Brownian motion

International audienceThe fractional Brownian motion which has been defined by Kolmogorov \cite{k40} and numerous papers was devoted to its study since its study in Mandelbrot and Van Ness \cite{MvN:68} present it as a paradigm of self-similar processes. The self-similarity parameter, also called the Hurst parameter, commands the dynamic of this process and the accuracy of its estimation is often crucial. We present here the main and used methods of estimation, with the limit theorems satisfied by the estimators. A numerical comparison is also provided allowing to distinguish between the estimators