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Adaptive estimation for Weakly Dependent Functional Times Series

Hassan Maissoro, Valentin Patilea, Myriam Vimond

Abstract

The local regularity of functional time series is studied under $L^p-m-$appro\-ximability assumptions. The sample paths are observed with error at possibly random design points. Non-asymptotic concentration bounds of the regularity estimators are derived. As an application, we build nonparametric mean and autocovariance functions estimators that adapt to the regularity and the design, which can be sparse or dense. We also derive the asymptotic normality of the mean estimator, which allows honest inference for irregular mean functions. Simulations and a real data application illustrate the performance of the new estimators.

Adaptive estimation for Weakly Dependent Functional Times Series

Abstract

The local regularity of functional time series is studied under appro\-ximability assumptions. The sample paths are observed with error at possibly random design points. Non-asymptotic concentration bounds of the regularity estimators are derived. As an application, we build nonparametric mean and autocovariance functions estimators that adapt to the regularity and the design, which can be sparse or dense. We also derive the asymptotic normality of the mean estimator, which allows honest inference for irregular mean functions. Simulations and a real data application illustrate the performance of the new estimators.
Paper Structure (27 sections, 20 theorems, 99 equations, 6 figures, 3 tables, 1 algorithm)

This paper contains 27 sections, 20 theorems, 99 equations, 6 figures, 3 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Functional Time Series
  3. Data
  4. The local regularity
  5. Weak dependence
  6. Estimation of the local regularity parameters
  7. The case of non-differentiable sample paths
  8. Regularity estimation for differentiable paths
  9. Adaptive mean and autocovariance functions estimators
  10. Adaptive mean function estimator
  11. Adaptive autocovariance function estimator
  12. The common design case
  13. Numerical study
  14. Simulation setting
  15. Local regularity estimation
  16. ...and 12 more sections

Key Result

Lemma 1

Assume that $X$ belongs to $\mathcal{X}(\delta+H_\delta,\boldsymbol L_\delta,J)$ for some $\delta\in\mathbb{N}^*$, $J$ an open sub-interval of $I$, $0 < H_\delta < 1$, and a bounded vector-valued function $\boldsymbol L_\delta\in \mathbb R^{\delta+1} _+$. Then, for any $d\in\{0,\dotsc,\delta-1\}$,

Figures (6)

  • Figure 1: Simulation parameters. Left: Logistic local exponent function $H_t$ used in FTS Model 2 and 3. Middle: The mean function $\mu$ used in FTS Model 1 and 2. Right: The empirical approximation of the lag$-1$ autocovariance function $\gamma_1(s,t)$ obtained from a large sample in FTS Model 2 when $\mu\equiv 0$.
  • Figure 2: Boxplots of $R=400$ pointwise estimates of $\widehat{H}_t$ and $\widehat{L}^2_t$, for $t\in\{0.2,0.4,0.7,0.8\}$ and four pairs $(N,\lambda)$, in FTS Model 2. The dashed horizontal lines indicate the true values of $H_t$ and $L^2_t$.
  • Figure 3: Empirical average of the risk function $\widehat{R}_\mu(t;h)$ at $t \in\{0.2, 0.4, 0.7, 0.8\}$ over $400$ independent functional time series generated according to FTS Model 2, with four setups $(N,\lambda)$.
  • Figure 4: Normal $Q-Q$ plots of $\sqrt{P_N(t;h_N)}\left(\widehat{\mu}_N(t;h_N) -\mu(t)\right) / \sqrt{\widehat{\mathbb{S}}_\mu (t) +\widehat{\Sigma}(t)}$ at $t=0.2$, with $h_N = \{h_\mu^*\}^{1.1}$. Results obtained with $400$ independent time series generated in the FTS Model 2.
  • Figure 5: Bandwidths selected by RP20 (left boxplot) and by our local approach for the mean estimation at $t\in\{0.2,0.4,0.7,0.8\}$; results from $400$ independent series generated in the FTS Model 2.
  • ...and 1 more figures

Theorems & Definitions (36)

  • Definition 1
  • Lemma 1
  • Definition 2
  • Example 1
  • Example 2: FAR(1) model
  • Definition 3
  • Example 3: FAR(1) model revisited
  • Example 4: Functional linear process
  • Example 5: Product Model
  • Example 6: Functional ARCH(1)
  • ...and 26 more