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Robust estimation for high-dimensional time series with heavy tails

Yu Wang, Guodong Li, Zhijie Xiao, Lihu Xu, Wenyang Zhang

Abstract

We study in this paper the problem of least absolute deviation (LAD) regression for high-dimensional heavy-tailed time series which have finite $α$-th moment with $α\in (1,2]$. To handle the heavy-tailed dependent data, we propose a Catoni type truncated minimization problem framework and obtain an $\mathcal{O}\big( \big( (d_1+d_2) (d_1\land d_2) \log^2 n / n \big)^{(α- 1)/α} \big)$ order excess risk, where $d_1$ and $d_2$ are the dimensionality and $n$ is the number of samples. We apply our result to study the LAD regression on high-dimensional heavy-tailed vector autoregressive (VAR) process. Simulations for the VAR($p$) model show that our new estimator with truncation are essential because the risk of the classical LAD has a tendency to blow up. We further apply our estimation to the real data and find that ours fits the data better than the classical LAD.

Robust estimation for high-dimensional time series with heavy tails

Abstract

We study in this paper the problem of least absolute deviation (LAD) regression for high-dimensional heavy-tailed time series which have finite -th moment with . To handle the heavy-tailed dependent data, we propose a Catoni type truncated minimization problem framework and obtain an order excess risk, where and are the dimensionality and is the number of samples. We apply our result to study the LAD regression on high-dimensional heavy-tailed vector autoregressive (VAR) process. Simulations for the VAR() model show that our new estimator with truncation are essential because the risk of the classical LAD has a tendency to blow up. We further apply our estimation to the real data and find that ours fits the data better than the classical LAD.

Paper Structure

This paper contains 23 sections, 10 theorems, 151 equations, 6 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Preliminary and Main Results
  3. Notations, Assumptions and Problem Setting
  4. Necessity of Truncation
  5. Our Truncated Estimation and Main Result
  6. Proof of Theorem \ref{['cor:M_n = m_n = log n']}
  7. The Strategy of The Proof.
  8. The Block Technique
  9. Auxiliary Lemmas
  10. Proof of Theorem \ref{['cor:M_n = m_n = log n']}
  11. Application to High-Dimensional Vector Autoregression
  12. Simulations
  13. Numerical Studies
  14. General description
  15. Simulations of VAR(1):
  16. ...and 8 more sections

Key Result

Theorem 2.7

Let $\bm{\theta}^*$ and $\hat{\bm{\theta}}$ be the minimizers of minimization problems e:PopLasso and e:C-min respectively. Under Assumptions A1:mixing, A2:moments, A3:net and A4:matrix, for any $\varepsilon \in (0,1/2)$, let the parameters $\delta$, $\lambda$ and $\gamma$ satisfy where $n$ is the sample size sufficiently large such that Then, the following inequality holds with probability at l

Figures (6)

  • Figure 1: Simulations values of risk $\hat{R}_{\psi_{\alpha},k}$, $\hat{R}_{ {\rm LAD},k}$ and $\hat{R}_{ {\rm Huber},k}$ for VAR(1) with $1 \le k \le N = 800$, and the logged prediction errors $\hat{E}^p_{\psi_{\alpha},L}$, $\hat{E}^p_{{\rm LAD},L}$ and $\hat{E}^p_{ {\rm Huber},L}$ with $L=10$. The noise $\varepsilon_k$ follows a Pareto distribution $\text{Pareto}(\mu)$ with $\mu = 1.2$, $1.5$ and $1.8$.
  • Figure 2: Simulations values of risk $\hat{R}_{\psi_{\alpha},k}$, $\hat{R}_{ {\rm LAD},k}$ and $\hat{R}_{ {\rm Huber},k}$ for VAR(1) with $1 \le k \le N = 800$, and the logged prediction errors $\hat{E}^p_{\psi_{\alpha},L}$, $\hat{E}^p_{{\rm LAD},L}$ and $\hat{E}^p_{ {\rm Huber},L}$ with $L=10$. The noise $\varepsilon_k$ follows a Fréchet distribution $\text{Fr\'echet}(\nu)$ with $\nu=1.2$, $1.5$ and $1.8$.
  • Figure 3: Simulations values of risk $\hat{R}_{\psi_{\alpha},k}$, $\hat{R}_{ {\rm LAD},k}$ and $\hat{R}_{ {\rm Huber},k}$ for VAR(2) with $1 \le k \le N = 800$, and the logged prediction errors $\hat{E}^p_{\psi_{\alpha},L}$, $\hat{E}^p_{{\rm LAD},L}$ and $\hat{E}^p_{ {\rm Huber},L}$ with $L=10$. The noise $\varepsilon_k$ follows a Pareto distribution $\text{Pareto}(\mu)$ with $\mu = 1.2$, $1.5$ and $1.8$.
  • Figure 4: Simulations values of risk $\hat{R}_{\psi_{\alpha},k}$, $\hat{R}_{ {\rm LAD},k}$ and $\hat{R}_{ {\rm Huber},k}$ for VAR(2) with $1 \le k \le N = 800$, and the logged prediction errors $\hat{E}^p_{\psi_{\alpha},L}$, $\hat{E}^p_{{\rm LAD},L}$ and $\hat{E}^p_{ {\rm Huber},L}$ with $L=10$. The noise $\varepsilon_k$ follows a Fréchet distribution $\text{Fr\'echet}(\nu)$ with $\nu=1.2$, $1.5$ and $1.8$.
  • Figure 5: (Average) Hill estimators for $\mathbf{Z}_t^1$, $\mathbf{Z}_t^2$ and $\mathbf{Z}_t$
  • ...and 1 more figures

Theorems & Definitions (20)

  • Definition 2.3
  • Definition 2.6
  • Theorem 2.7
  • Remark 2.1
  • Lemma 3.1
  • Lemma 3.2
  • Lemma 3.3
  • proof : Proof of Theorem \ref{['cor:M_n = m_n = log n']}
  • Remark 3.1
  • Lemma 4.3
  • ...and 10 more