LRD spectral analysis of multifractional functional time series on manifolds

TL;DR

This paper develops spectral analysis methods for estimating the second-order structure of multifractional functional time series on manifolds, accounting for spatially varying long-range dependence and mixed dependence scales.

Contribution

It introduces a semiparametric spectral framework for consistent estimation of long-memory operators on manifolds, extending previous models to handle multifractional and mixed dependence structures.

Findings

Estimation procedures are validated through simulations on multifractionally integrated SPHARMA processes.

Conditions for asymptotic unbiasedness of the integrated periodogram are established.

The methods effectively distinguish between SRD and LRD at different manifold scales.

Abstract

This paper addresses the estimation of the second-order structure of a manifold cross-time random field (RF) displaying spatially varying Long Range Dependence (LRD), adopting the functional time series framework introduced in Ruiz-Medina (2022). Conditions for the asymptotic unbiasedness of the integrated periodogram operator in the Hilbert-Schmidt operator norm are derived beyond structural assumptions. Weak-consistent estimation of the long-memory operator is achieved under a semiparametric functional spectral framework in the Gaussian context. The case where the projected manifold process can display Short Range Dependence (SRD) and LRD at different manifold scales is also analyzed. The performance of both estimation procedures is illustrated in the simulation study, in the context of multifractionally integrated spherical functional autoregressive-moving average (SPHARMA(p,q)) processes.