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Manifold functional multiple regression model with LRD error term

Diana P. Ovalle-Muñoz, M. Dolores Ruiz-Medina

Abstract

This paper considers the problem of manifold functional multiple regression with functional response, time--varying scalar regressors, and functional error term displaying Long Range Dependence (LRD) in time. Specifically, the error term is given by a manifold multifractionally integrated functional time series (see, e.g., Ovalle--Muñoz \& Ruiz--Medina, 2024)). The manifold is defined by a connected and compact two--point homogeneous space. The functional regression parameters have support in the manifold. The Generalized Least--Squares (GLS) estimator of the vector functional regression parameter is computed, and its asymptotic properties are analyzed under a totally specified and misspecified model scenario. A multiscale residual correlation analysis in the simulation study undertaken illustrates the empirical distributional properties of the errors at different spherical resolution levels.

Manifold functional multiple regression model with LRD error term

Abstract

This paper considers the problem of manifold functional multiple regression with functional response, time--varying scalar regressors, and functional error term displaying Long Range Dependence (LRD) in time. Specifically, the error term is given by a manifold multifractionally integrated functional time series (see, e.g., Ovalle--Muñoz \& Ruiz--Medina, 2024)). The manifold is defined by a connected and compact two--point homogeneous space. The functional regression parameters have support in the manifold. The Generalized Least--Squares (GLS) estimator of the vector functional regression parameter is computed, and its asymptotic properties are analyzed under a totally specified and misspecified model scenario. A multiscale residual correlation analysis in the simulation study undertaken illustrates the empirical distributional properties of the errors at different spherical resolution levels.
Paper Structure (11 sections, 2 theorems, 39 equations, 17 figures)

This paper contains 11 sections, 2 theorems, 39 equations, 17 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Multiple Functional regression model
  4. GLS functional parameter estimation
  5. Distributional characteristics of GLS estimator
  6. Functional spectral based plug--in estimation of the functional regression parameter
  7. Asymptotic properties of the GLS functional parameter estimator
  8. Simulation study
  9. Spherical multiscale residual analysis under totally specified model
  10. Results under misspecified model
  11. Conclusions and open research lines

Key Result

Lemma 1

(See Gine1975 and Andrews99) For every $n\in \mathbb{N}_{0},$ the following addition formula holds: Here, $\omega_{d}=\int_{\mathbb{M}_{d}}d\nu(\mathbf{x}),$ and $\delta(n,d)$ denotes the dimension of the eigenspace $\mathcal{H}_{n}$ associated with the eigenvalue $\lambda_{n}=-n\varepsilon(n\varepsilon+\alpha+\beta+1)$ of the Laplace Beltrami operator, which is given, for every $n\in \mathbb{N}_

Figures (17)

  • Figure 1: Fourier coefficients ${\beta}_{n,j},$$n=1,2,\dots,30,$ for $j=1,2,3,4,5,$ are respectively displayed in blue star--dashed line, red dashed line, orange dotted line, green squares-dashed line and black cross--dashed line.
  • Figure 2: The regression parameters $\beta_{j}(\mathbf{x}), \mathbf{x}\in \mathbb{S}_{2},j=1,2,\dots,5,$ projected into the direct sum of eigenspaces $\mathcal{H}_{n}$, $n=1,2,\dots,30,$ of the Laplace Beltrami operator on $L^{2}(\mathbb{S}_{2}, d\nu , \mathbb{R})$.
  • Figure 3: The first $30$ eigenvalues $\alpha (n,\theta_{0}),$$n=1,\dots,30,$ of the LRD operator $\mathcal{A}_{\theta_{0}}.$ The considered DPBS of eigenvalues is plotted at the left--hand side, and the IPBS of eigenvalues is plotted at the right hand--side.
  • Figure 4: REM $\hat{E}[Y_{t}(\mathbf{x})],$$\mathbf{x}\in \mathbb{S}_{2},$ based on $R=100$ repetitions, at times $t=0,62,124,187,249,311,374,436,499,$ projected into the direct sum of the eigenspaces $\mathcal{H}_{n}$, $n= 1,2,\dots,30$ of the Laplace Beltrami operator on $L^{2}(\mathbb{S}_{2}, d\nu, \mathbb{R})$, under DPBS of eigenvalues of LRD operator. The functional sample size generated is $T=500$.
  • Figure 5: RTPEM $\hat{E}\left[\hat{Y}_{t}(\mathbf{x})\right],$$\mathbf{x}\in \mathbb{S}_{2},$ based on $R=100$ repetitions, at times $t=0,62,124,187,249,311,374,$$436,499,$ projected into the direct sum of the eigenspaces $\mathcal{H}_{n}$, $n= 1,2,\dots,30$ of the Laplace Beltrami operator on $L^{2}(\mathbb{S}_{2}, d\nu, \mathbb{R})$, under DPBS of eigenvalues of LRD operator. The functional sample size generated is $T=500$.
  • ...and 12 more figures

Theorems & Definitions (2)

  • Lemma 1
  • Lemma 2