Central limit theorem for the log-regression wavelet estimation of the memory parameter in the Gaussian semi-parametric context

摘要

We consider a Gaussian time series, stationary or not, with long memory exponent d is an element of R. The generalized spectral density function of the time series is characterized by d and by a function f*(lambda) which specifies the short-range dependence structure. Our setting is semi-parametric in that both d and f* are unknown, and only the smoothness of f* around lambda = 0 matters. The parameter d is the one of interest. It is estimated by regression using the wavelet coefficients of the time series, which are dependent when d not equal 0. We establish a Central Limit Theorem (CLT) for the resulting estimator (d) over cap. We show that the deviation (d) over cap - d, adequately normalized, is asymptotically normal and specify the asymptotic variance.