An LRD spectral test for irregularly discretely observed contaminated functional time series in manifolds

TL;DR

This work extends spectral-domain LRD testing for functional time series to irregular, discretely observed data on manifolds with additive noise, by integrating nonparametric series LS reconstruction into the test architecture. The authors establish mean-square consistency of the reconstruction, derive the asymptotic Gaussian distribution of the plug-in test statistic under $H_0$, and prove consistency under $H_1$ by quantifying second- and fourth-order spectral biases. The analysis leverages a delicate balance among the temporal sample size $T$, sieve dimension $k(T)$, bandwidth $B_T$, and spatial sampling $M(T)$, with memory parameters and smoothness playing crucial roles. Simulation results in the Appendix corroborate the theoretical findings, showing consistency and favorable finite-sample size and power under various sparse sampling regimes, thereby supporting practical applicability of the proposed LRD test in contaminated, irregularly observed manifold-valued functional data.

Abstract

A statistical hypothesis test for long range dependence (LRD) in functional time series in manifolds has been formulated in Ruiz-Medina and Crujeiras (2025) in the spectral domain for fully observed functional data. The asymptotic Gaussian distribution of the proposed test statistics, based on the weighted periodogram operator, under the null hypothesis, and the consistency of the test have been derived. In this paper, we analyze the asymptotic properties of this spectral LRD testing procedure, when functional data are contaminated, and discretely observed through random uniform spatial sampling.

Figures (4)

• Figure 1: Random spherical harmonics sieve basis of dimension $k(T)=15$ (three first lines at the top), contaminated discretely observed data (next two lines at the center), and its nonparametric series LS reconstruction (last two lines at the bottom)
• Figure 2: Example 1. Eigenvalues $\alpha (n,j),$$j=1,\dots,\Gamma (n,2),$$n=1,\dots, N(k(T)),$ of LRD operator $\mathcal{A},$ for $N(k(T))=5,$ and $k(T)= 15,$ with $L_{\alpha }= 0.4929,$ and $l_{\alpha }=0.2550$ (left--hand side). Sample projections of $\widehat{\mathcal{S}}_{B_{T}},$$B_{T}= T^{-1/4},$ into the eigenfunctions generating the tensor product eigenspaces $\mathcal{H}_{n}\otimes \mathcal{H}_{n},$$n=1,\dots,N(k(T))$ (four plots at the right--hand--side)
• Figure 3: Example 2. Eigenvalues $\alpha (n,j),$$j=1,\dots,\Gamma(n,2),$$n=1,\dots, N(k(T)),$ of LRD operator $\mathcal{A},$ for $N(k(T))=4,$ and $k(T)= 10,$ with $L_{\alpha }= 0.4950,$ and $l_{\alpha }=0.2629$ (left--hand side). Sample projections of $\widehat{\mathcal{S}}_{B_{T}},$$B_{T}= T^{-1/4},$ into the eigenfunctions generating the tensor product eigenspaces $\mathcal{H}_{n}\otimes \mathcal{H}_{n},$$n=1,\dots,N(k(T))$ (four plots at the right--hand--side)
• Figure 4: Example 3. Eigenvalues $\alpha (n,j),$$j=1,\dots, \Gamma (n,2),$$n=1,\dots, N(k(T)),$ of LRD operator $\mathcal{A},$ for $N(k(T))=4,$ and $k(T)= 10,$ with $L_{\alpha }= 0.4743,$ and $l_{\alpha }= 0.2678$ (left--hand side). Sample projections of $\widehat{\mathcal{S}}_{B_{T}},$$B_{T}= T^{-1/4},$ into the eigenfunctions generating the tensor product eigenspaces $\mathcal{H}_{n}\otimes \mathcal{H}_{n},$$n=1,\dots,N(k(T))$ (four plots at the right--hand--side)

Theorems & Definitions (12)

• Remark 1
• Theorem 1
• Remark 2
• Remark 3
• Proposition 1
• Lemma 1
• Corollary 1
• Lemma 2
• Proposition 2
• Theorem 2