Local Whittle estimation of long‐range dependence for functional time series

This article studies stationary functional time series with long‐range dependence, and estimates the memory parameter involved. Semiparametric local Whittle estimation is used, where periodogram is constructed from the approximate first score, which is an inner product of the functional observation and estimated leading eigenfunction. The latter is obtained via classical functional principal component analysis. Under the restrictive condition of constancy of the memory parameter over the function support, and other conditions which include rather unprimitive ones on the first score, the estimate is shown to be consistent and asymptotically normal with asymptotic variance free of any unknown parameter, facilitating inference, as in the scalar time series case. Although the primary interest lies in long‐range dependence, our methods and theory are relevant to short‐range dependent or negative dependent functional time series. A Monte Carlo study of finite sample performance and an empirical example are included.