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Trend estimation for time series with polynomial-tailed noise

Michael H. Neumann, Anne Leucht

TL;DR

This work tackles nonparametric trend estimation for time series observed at uneven time points with a bounded total variation $TV(m_0;\{x_t\})$. It introduces a nonlinear Haar-type wavelet estimator adapted to non-dyadic and irregular designs, with thresholding rules tailored to polynomial-tailed noise. The authors establish near-optimal sparse-recovery rates under heavy-tailed errors and extend the framework to partially linear models by separating a linear component from the nonlinear wavelet part, supported by simulations and a real data application. The results demonstrate robust performance against kernel methods, highlighting improved adaptability to jumps and irregular sampling in practical time-series settings.

Abstract

For time series data observed at non-random and possibly non-equidistant time points, we estimate the trend function nonparametrically. Under the assumption of a bounded total variation of the function and low-order moment conditions on the errors we propose a nonlinear wavelet estimator which uses a Haar-type basis adapted to a possibly non-dyadic sample size. An appropriate thresholding scheme for sparse signals with an additive polynomial-tailed noise is first derived in an abstract framework and then applied to the problem of trend estimation.

Trend estimation for time series with polynomial-tailed noise

TL;DR

This work tackles nonparametric trend estimation for time series observed at uneven time points with a bounded total variation . It introduces a nonlinear Haar-type wavelet estimator adapted to non-dyadic and irregular designs, with thresholding rules tailored to polynomial-tailed noise. The authors establish near-optimal sparse-recovery rates under heavy-tailed errors and extend the framework to partially linear models by separating a linear component from the nonlinear wavelet part, supported by simulations and a real data application. The results demonstrate robust performance against kernel methods, highlighting improved adaptability to jumps and irregular sampling in practical time-series settings.

Abstract

For time series data observed at non-random and possibly non-equidistant time points, we estimate the trend function nonparametrically. Under the assumption of a bounded total variation of the function and low-order moment conditions on the errors we propose a nonlinear wavelet estimator which uses a Haar-type basis adapted to a possibly non-dyadic sample size. An appropriate thresholding scheme for sparse signals with an additive polynomial-tailed noise is first derived in an abstract framework and then applied to the problem of trend estimation.

Paper Structure

This paper contains 12 sections, 8 theorems, 95 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Assumptions and a preview of our main results
  3. Optimal reconstruction of sparse signals from data with polynomial-tailed noise
  4. Main results
  5. A Haar-type basis for unevenly spaced data
  6. A nonlinear wavelet estimator of the trend function
  7. A partially linear model
  8. Simulations and data examples
  9. Simulations
  10. A real data example
  11. Proofs of the main results
  12. A few auxiliary results

Key Result

Proposition 3.1

Suppose that (eq3.1a), (eq3.12a), and (eq3.12b) are fulfilled. Then the unique Bayes estimator $T^*(Y_k)$ is given by $T^*(-2\lambda)\,=\,-\lambda$, $T^*(-\lambda)\,=\,-\lambda/2$, $T^*(0)\,=\,0$, $T^*(\lambda)\,=\,\lambda/2$, $T^*(2\lambda)\,=\,\lambda$, and its Bayes risk is equal to This result can be used to obtain a lower bound to a related minimax risk. Suppose in addition that $q^{-4/3}=O(

Figures (6)

  • Figure 1.1: Red: monthly overseas arrivals in Australia (in millions) with structural breaks due to the COVID pandemic, black / blue: Nadaraya Watson estimator with Epanechnikov and rectangular kernel, bandwidth chosen by Scott's rule of thumb.
  • Figure 5.1: Red: function $f$ (left) and $g$ (right), black: data generated according to $Y_t=f(t/n)+\varepsilon_t$ (left) and $Y_t=g(t/n)+\varepsilon_t$ (right).
  • Figure 5.2: left: MSE of the wavelet estimator of $f$ with $K=0.1$, middle: MSE of the NW estimator of $f$ (rectangular kernel with $b=0.009$) right: MSE of the NW estimator of $f$ (Epanechnikov kernel with $b=0.007$).
  • Figure 5.3: left: MSE of the wavelet estimator of $g$ with $K=0.045$, middle: MSE of the Nadaraya-Watson estimator of $g$ (rectangular kernel with $b=0.006$) right: MSE of the Nadaraya-Watson estimator of $g$ (Epanechnikov kernel with $b=0.007$).
  • Figure 5.4: red: original data, blue: (partially linear) wavelet approximations with different choices of $K$
  • ...and 1 more figures

Theorems & Definitions (15)

  • Proposition 3.1
  • Theorem 3.1
  • Lemma 4.1
  • Theorem 4.1
  • Remark 1
  • Theorem 4.2
  • Proposition 4.1
  • proof : Proof of Proposition \ref{['P3.1']}
  • proof : Proof of Theorem \ref{['T3.1']}
  • proof : Proof of Theorem \ref{['T4.2']}
  • ...and 5 more