Not all long-memory estimators are born equal: The case of nonstationary functional time series

We study a functional version of fractionally integrated time series that covers the nonstationary case when the memory parameter d is above 0.5. We project time series, with varying levels of nonstationarity, onto a finite-dimensional subspace. We obtain the eigenvalues and eigenfunctions that span a sample version of the dominant subspace through dynamic functional principal component analysis of the sample long-run covariance functions. Within the context of functional autoregressive fractionally integrated moving average models, we evaluate and compare finite-sample bias and mean-squared error among some time- and frequency-domain Hurst exponent estimators via Monte Carlo simulations. We apply the estimators to Canadian female and male life-table death counts.