Adaptive estimation of the fractional differencing coefficient

Abstract

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 $n$ 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 semi-parametric 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.