Adaptive estimation of the fractional differencing coefficient

Semi-parametric estimation of the fractional differencing coefficient d of a long-range dependent stationary time series has received substantial attention in recent years. Some of the so-called local estimators introduced early on were proved rate-optimal over relevant classes of spectral densities. The rates of convergence of these estimators are limited to n^2/5, where 11 is the sample size. This paper focuses on the fractional exponential (FEXP) or broadband estimator of d. Minimax rates of convergence over classes of spectral densities which are smooth outside the zero frequency are obtained, and the FEXP estimator is proved rate-optimal over these classes. On a certain functional class which contains the spectral densities of FARIMA processes, the rate of convergence of the FEXP estimator is (n/log(n))^1/2, thus making it a reasonable alternative to parametric estimators. As usual in semiparametric estimation problems, these rate-optimal estimators are infeasible, since they depend on an unknown smoothness parameter defining the functional class. A feasible adaptive version of the broadband estimator is constructed. It is shown that this estimator is minimax rate-optimal up to a factor proportional to the logarithm of the sample size.