Bandwidth choice, optimal rates and adaptivity in semiparametric estimation of long memory

Semiparametric estimation of long memory refers to periodogram based estimation of the shape of the spectral density f(λ) at low frequencies, where all but the lowest harmonics of the periodogram are discarded, so as to forego specification of the short range dynamic structure of the time series, and avoid bias incurred when the latter is misspecified. Such a procedure entails an order of magnitude loss of efficiency with respect to parametric estimation, but may be warranted when long series (earth scientific or financial) can be obtained. This paper presents strategies proposed for the choice of bandwidth, i.e. the number of periodogram harmonics used in estimation, with the aim of minimizing this loss of efficiency. Such strategies are assessed with respect to minimax rates of convergence, that depend on the smoothness of |λ|^-2d f(λ) (where d is the long memory parameter) in the neighbourhood of frequency zero. The plug-in strategy is discussed in the case where the degree of local smoothness is known a priori, and adaptive estimation of d is discussed for the case where the degree of local smoothness is unknown.