Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Testing the goodness of fit of a functional autoregressive model

W. González-Manteiga, M. D. Ruiz-Medina, M. Febrero-Bande

TL;DR

This work develops a distribution-free Goodness-of-Fit test for the linear autocorrelation structure in functional time series modeled by AR$\mathbb{H}$(1). It builds a Hilbert-space valued empirical process marked by functional residuals and indexed by $\mathbb{H}$-valued covariates, proving a functional central limit theorem that yields a time-changed $\mathbb{H}$-valued Wiener process with subordinator $P_{Y_0}$. The authors establish asymptotic results under both simple and composite null hypotheses, including consistency under simple nulls and asymptotic equivalence results for plug-in estimators of the autoregression operator. Practical performance is demonstrated via extensive simulations for both generic functional data and SP$\mathbb{H}$AR(1) on the sphere, with bootstrap-based critical values and random projection implementations. The approach provides a scalable, theoretically grounded framework for GoF testing in high-dimensional functional time series and can accommodate complex dependence structures, offering a valuable tool for specification testing in modern functional data analysis.

Abstract

The proposed Goodness-of-Fit (GoF) test for checking the linear autocorrelation model in a functional time series is based on an empirical process, whose residual marks and covariate index set are in a separable Hilbert space H. A functional central limit theorem is derived providing the convergence of the empirical process to a time-changed Wiener process evaluated in a separable Hilbert space H, with subordinator given by the marginal probability of the involved Autoregressive Hilbertian process (ARH(1) process). The large sample behavior of the test statistics is obtained under simple and composite null hypotheses. The consistency of the test is addressed under simple null hypothesis. The simulation study provided in the Appendix illustrates the finite-sample performance of the testing procedure under different families of alternatives.

Testing the goodness of fit of a functional autoregressive model

TL;DR

This work develops a distribution-free Goodness-of-Fit test for the linear autocorrelation structure in functional time series modeled by AR(1). It builds a Hilbert-space valued empirical process marked by functional residuals and indexed by -valued covariates, proving a functional central limit theorem that yields a time-changed -valued Wiener process with subordinator . The authors establish asymptotic results under both simple and composite null hypotheses, including consistency under simple nulls and asymptotic equivalence results for plug-in estimators of the autoregression operator. Practical performance is demonstrated via extensive simulations for both generic functional data and SPAR(1) on the sphere, with bootstrap-based critical values and random projection implementations. The approach provides a scalable, theoretically grounded framework for GoF testing in high-dimensional functional time series and can accommodate complex dependence structures, offering a valuable tool for specification testing in modern functional data analysis.

Abstract

The proposed Goodness-of-Fit (GoF) test for checking the linear autocorrelation model in a functional time series is based on an empirical process, whose residual marks and covariate index set are in a separable Hilbert space H. A functional central limit theorem is derived providing the convergence of the empirical process to a time-changed Wiener process evaluated in a separable Hilbert space H, with subordinator given by the marginal probability of the involved Autoregressive Hilbertian process (ARH(1) process). The large sample behavior of the test statistics is obtained under simple and composite null hypotheses. The consistency of the test is addressed under simple null hypothesis. The simulation study provided in the Appendix illustrates the finite-sample performance of the testing procedure under different families of alternatives.
Paper Structure (16 sections, 5 theorems, 65 equations, 1 figure, 8 tables)

This paper contains 16 sections, 5 theorems, 65 equations, 1 figure, 8 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Testing the autocorrelation model
  4. Asymptotic properties for simple hypothesis ($\Gamma=\Gamma_0$)
  5. Central Limit Theorem
  6. Functional Central Limit Theorem
  7. Consistency
  8. Asymptotic properties for composite hypothesis
  9. Discussion
  10. Auxiliary information
  11. Differences with asymptotic GoF under independent data
  12. Second--order properties of involved $\mathbb{H}$--valued martingale difference sequence $\{X_{i}(x),\ i\geq 1\}$
  13. Subordinators in the probability distribution identification of $W_{\infty}$
  14. Illustration of the performance of GoF
  15. Detecting independence and nonlinearities
  16. ...and 1 more sections

Key Result

Lemma 1

For each $x\in \mathbb{H},$ the sequence is an $\mathbb{H}$--valued martingale difference with respect to the filtration $\mathcal{M}_{0}^{Y}\subset \mathcal{M}_{1}^{Y}\dots \subset \mathcal{M}_{n}^{Y}\subset \dots,$ where $\mathcal{M}_{i-1}^{Y}=\sigma (Y_{t},\ t\leq i-1),$ for $i\geq 1.$

Figures (1)

  • Figure 1: Elements of the truncated orthonormal eigenfunction spherical basis (left--hand--side), and one realization of the functional values of the generated SP$\mathbb{H}$AR(1) process, projected into $\bigoplus_{k=0}^{3}\mathcal{H}_{k},$ at times $t=1,51, 101,151, 202,251, 301,351, 401,451,$ from a functional sample of size $n=500$ (right--hand--side)

Theorems & Definitions (10)

  • Lemma 1
  • Theorem 1
  • proof
  • Lemma 2
  • Theorem 2
  • Remark 1
  • proof
  • Theorem 3
  • proof
  • Remark 2