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 constructed from approximate first score, which an inner product of observation estimated leading eigenfunction. The latter obtained via classical principal component analysis. Under restrictive condition constancy over function support, other conditions include rather unprimitive ones on estimate shown to be consistent asymptotically normal asymptotic variance free any unknown parameter, facilitating inference, as in scalar case. Although primary interest lies our methods theory are relevant short-range dependent or negative series. A Monte Carlo study finite sample performance empirical example included.