Time series with long-range dependence have been recently the focus of much attention. They appear in a number of applications, for example, in the analysis of traffic in computer networks. In finite variance time series, long-range dependence is characterized by a covariance function which decreases slowly to 0 as the lag increases. The decrease is so slow that the corresponding spectral density blows up at very low frequencies, a phenomenon also known as ''long-range dependence'', ``long memory'' or ``1/f noise''. There are many empirical procedures which are graphical in nature but rigorous estimation procedures are few. The better ones use a semi-parametric approach because long-range dependence does not involve the short range dependence structure. The most successful methods have been Fourier-based. One approach involves a regression on the logarithm of the periodogram. Another is the local Whittle approach. It is a pseudo-likelihood approach which uses the periodogram at low frequencies only. The investigator proposes to use a pseudo-likelihood approach based not on the Fourier spectrum but on wavelets. The advantage of wavelets on Fourier is that there is no need to difference the time series if these are not stationary and, in general, they are much more robust against departure from stationarity. Since wavelets are associated with scaling it is, moreover, natural to attempt to use wavelets in order to estimate the intensity of long-range dependence. Wavelets have been successfully used as an alternative to the regression on the logarithm of the periodogram but wavelets have not been used as an alternative to the Fourier-based local Whittle approach. The investigator proposes to do so. This would provide a robust semi-parametic pseudo-likelihood based method to estimate the intensity of long-range dependence.
The subject of time series analysis has sparked considerable research activity over the past several decades. Essentially concerned with measurements over time of various kinds of phenomena --- from yearly income, exchange rates, to the level of a river --- the goal is to develop suitable models and obtain accurate predictions for such measurements, the core ingredient being the notion of time dependence. Benoit Mandelbrot in the sixties suggested using models involving long-range dependence. Long-range dependence, to put it concisely, affects phenomena in which correlations between the present and the past decay slowly with time and thus they cannot be easily ignored. Time series with long-range dependence and their variations have been used in hydrology, geophysics and biophysics, and more recently, in finance and in analyzing traffic in computer networks. It is thus important to be able to estimate effectively the intensity of long-range dependence. The purpose of this research is to develop a new methodology to achieve this goal. It is wavelet-based and hence insensitive to the effects of trends and other deviations from the model.
