Adaptive semiparametric estimation of the memory parameter

In Giraitis, Robinson, and Samarov (1997), we have shown that the optimal r ate for memory parameter estimators in semiparametric long memory models wi th degree of "local smoothness" beta is n(-r(beta)), r(beta) = beta/(2 beta + 1), and that a log-periodogram regression estimator (a modified Geweke a nd Porter-Hudak (1983) estimator) with maximum frequency m = m(beta) asympt otic to n(2r(beta)) is rate optimal. The question which we address in this paper is what is the best obtainable rate when beta is unknown, so that est imators cannot depend on beta. We obtain a lower bound for the asymptotic q uadratic risk of any such adaptive estimator, which turns out to be larger than the optimal nonadaptive rate n(-r(beta)) by a logarithmic factor. We t hen consider a modified log-periodogram regression estimator based on taper ed data and with a data-dependent maximum frequency m = m(<(beta)over cap>) , which depends on an adaptively chosen estimator <(beta)over cap> of beta, and show, using methods proposed by Lepskii (1990) in another context, tha t this estimator attains the lower bound up to a logarithmic factor. On one hand, this means that this estimator has nearly optimal rate among all ada ptive (free from beta) estimators, and, on the other hand, it shows near op timality of our data-dependent choice of the rate of the maximum frequency for the modified log-periodogram regression estimator. The proofs contain r esults which are also of independent interest: one result shows that data t apering gives a significant improvement in asymptotic properties of covaria nces of discrete Fourier transforms of long memory time series, while anoth er gives an exponential inequality for the modified log-periodogram regress ion estimator. (C) 2000 Academic Press.