RATE OPTIMAL SEMIPARAMETRIC ESTIMATION OF THE MEMORY PARAMETER OF THE GAUSSIAN TIME SERIES WITH LONG‐RANGE DEPENDENCE

Abstract

There exist several estimators of the memory parameter in long‐ memory time series models with the spectrum specified only locally near zero frequency. In this paper we give an asymptotic lower bound for the minimax risk of any estimator of the memory parameter as a function of the degree of local smoothness of the spectral density at zero. The lower bound allows one to evaluate and compare different estimators by their asymptotic behaviour, and to claim the rate optimality for any estimator attaining the bound. A log‐periodogram regression estimator, analysed by Robinson (Log‐periodogram regression of time series with long range dependence. Ann. Stat. 23 (1995), 1048‐‐72), is then shown to attain the lower bound, and is thus rate optimal.