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Gaussian Approximation for Lag-Window Estimators and the Construction of Confidence bands for the Spectral Density

Jens-Peter Kreiss, Anne Leucht, Efstathios Paparoditis

TL;DR

The paper develops a Gaussian-approximation framework for the maximum deviation of a lag-window spectral density estimator across all positive Fourier frequencies, enabling asymptotically valid simultaneous confidence bands via a multiplier bootstrap. It proves a high-dimensional Gaussian approximation with explicit rates under weak-dependent time-series assumptions and provides a practical covariance-estimation bootstrap to form bands that adapt to local spectral-variability. Simulation studies illustrate near-nominal coverage and improved finite-sample behavior over coarser-grid Gumbel-based methods, highlighting sensitivity to bandwidth choices and dependence structure. Overall, the work enables refined, nonparametric, frequency-wise inference for spectral density in stationary time series on a dense grid of Fourier frequencies.

Abstract

In this paper we consider the construction of simultaneous confidence bands for the spectral density of a stationary time series using a Gaussian approximation for classical lag-window spectral density estimators evaluated at the set of all positive Fourier frequencies. The Gaussian approximation opens up the possibility to verify asymptotic validity of a multiplier bootstrap procedure and, even further, to derive the corresponding rate of convergence. A small simulation study sheds light on the finite sample properties of this bootstrap proposal.

Gaussian Approximation for Lag-Window Estimators and the Construction of Confidence bands for the Spectral Density

TL;DR

The paper develops a Gaussian-approximation framework for the maximum deviation of a lag-window spectral density estimator across all positive Fourier frequencies, enabling asymptotically valid simultaneous confidence bands via a multiplier bootstrap. It proves a high-dimensional Gaussian approximation with explicit rates under weak-dependent time-series assumptions and provides a practical covariance-estimation bootstrap to form bands that adapt to local spectral-variability. Simulation studies illustrate near-nominal coverage and improved finite-sample behavior over coarser-grid Gumbel-based methods, highlighting sensitivity to bandwidth choices and dependence structure. Overall, the work enables refined, nonparametric, frequency-wise inference for spectral density in stationary time series on a dense grid of Fourier frequencies.

Abstract

In this paper we consider the construction of simultaneous confidence bands for the spectral density of a stationary time series using a Gaussian approximation for classical lag-window spectral density estimators evaluated at the set of all positive Fourier frequencies. The Gaussian approximation opens up the possibility to verify asymptotic validity of a multiplier bootstrap procedure and, even further, to derive the corresponding rate of convergence. A small simulation study sheds light on the finite sample properties of this bootstrap proposal.
Paper Structure (6 sections, 6 theorems, 96 equations, 1 figure)

This paper contains 6 sections, 6 theorems, 96 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Set up and Assumptions
  3. Gaussian Approximation for Spectral Density Estimators
  4. Confidence Bands for the Spectral Density
  5. Simulations
  6. Auxiliary Lemmas and Proofs

Key Result

Theorem 1

Suppose that $\{X_t, t\in \mathbb Z\}$ fulfils Assumption 1 and 2. Let $\widehat{f}_T$ be a lag-window estimator of $f$ as given in (spec_dens), where the lag-window $w$ satisfies Assumption 3 and let $M_T\rightarrow \infty$ and $M_T\sim T^{a_s}$. Assume further that Let $\xi _k, k=1, \ldots , N_T$, be jointly normally distributed random variables with zero mean and covariance $E \xi_{k_1} \xi_{k

Figures (1)

  • Figure 1: Plot of $\widetilde{f}(\lambda_{j,T})$ (solid line) together with $90\%$ and $95\%$ averaged confidence bands (dotted and dashed lines, respectively), for time series of length $T=512$ and $b_T=10$ (top) and $T=1024$ and $b_T=11.5$ (bottom) stemming from Model II.

Theorems & Definitions (16)

  • Theorem 1
  • Remark 1
  • Remark 2
  • Theorem 2
  • Remark 3
  • Proposition 1
  • Theorem 3
  • Remark 4
  • Lemma 1
  • proof
  • ...and 6 more