Nonlinear log-periodogram regression for perturbed fractional processes

摘要

This paper studies fractional processes that may be perturbed by weakly dependent time scries. The model for a perturbed fractional process has a components framework in which there may be components of both long and short memory. All commonly usedestimates of the long memory parameter (such as log periodogram (LP) regression) may be used in a components model where the data are alfectcd by weakly dependent perturbations, but these estimates can suffer from serious downward bias. To circunncnt this problem, the present paper proposes a new procedure that allows for the possible presence of additive perturbations in the data. The new estimator resembles the LP regression estimator but involves an additional (nonlinear) term in the regression thattakes account of possible perturbation elTects in the data. Under some smoothness assumptions at the origin, the bias of the new estimator is shown to disappear at a faster rate than that of the LP estimator, while its asymptotic variance is inflated only by a multiplicative constant. In consequence, the optimal rate of convergence to zero of the asymptotic MSE of the new estimator is faster than that of the LP estimator. Some simulation results demonstrate the viability and the bias-reducing feature ofthe new estimator relative to the LP estimator in finite samples. A lest for the presence of perturbations in the data is given.