Testing LRD in the spectral domain for functional time series in manifolds

TL;DR

This paper develops a spectral domain hypothesis test for detecting long-range dependence in functional time series on manifolds, using spectral density operators and random projections, with theoretical and simulation validation.

Contribution

It introduces a novel spectral domain test for LRD in manifold-valued functional time series, including asymptotic theory and practical implementation via random projections.

Findings

Test statistic follows asymptotic Gaussian distribution under null hypothesis.

The test is consistent and effective in detecting LRD in spherical functional data.

Simulation results confirm the theoretical properties and demonstrate good size and power.

Abstract

A statistical hypothesis test for long range dependence (LRD) is formulated in the spectral domain for functional time series in manifolds. The elements of the spectral density operator family are assumed to be invariant with respect to the group of isometries of the manifold. The proposed test statistic is based on the weighted periodogram operator. A Central Limit Theorem is derived to obtain the asymptotic Gaussian distribution of the proposed test statistic operator under the null hypothesis. The rate of convergence to zero, in the Hilbert--Schmidt operator norm, of the bias of the integrated empirical second and fourth order cumulant spectral density operators is obtained under the alternative hypothesis. The consistency of the test follows from the consistency of the integrated weighted periodogram operator under LRD. Practical implementation of our testing approach is based on the random projection methodology. A simulation study illustrates, in the context of spherical functional time series, the asymptotic normality of the test statistic under the null hypothesis, and its consistency under the alternative. The empirical size and power properties are also computed for different functional sample sizes, and under different scenarios.