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Higher-order approximation for uncertainty quantification in time series analysis

Annika Betken, Marie-Christine Düker

TL;DR

This work tackles the slow convergence of the empirical process for time series with strong temporal correlation by developing higher-order approximations based on Hermite expansions. It proves a two-parameter limit theorem for the higher-order term and derives Hermite-process limits for the dominating lower-order components, enabling confidence intervals for distributional functionals (e.g., marginal distribution and quantiles) that are robust to long-range dependence. The authors propose HOA-based confidence intervals that incorporate the higher-order corrections, leading to improved coverage and shorter interval lengths compared to classical asymptotic intervals, especially under high Hurst parameters. Numerical studies demonstrate the practical gains and discuss estimation challenges for the long-range variance and the Hurst parameter, highlighting the method’s potential for uncertainty quantification in change-point analysis and goodness-of-fit testing under long-range dependence.

Abstract

For time series with high temporal correlation, the empirical process converges rather slowly to its limiting distribution. Many statistics in change-point analysis, goodness-of-fit testing and uncertainty quantification admit a representation as functionals of the empirical process and therefore inherit its slow convergence. As a result, inference based on the asymptotic distribution of those quantities is significantly affected by relatively small sample sizes. We assess the quality of higher-order approximations of the empirical process by deriving the asymptotic distribution of the corresponding error terms. Based on the limiting distribution of the higher-order terms, we propose a novel approach to calculate confidence intervals for statistical quantities such as the median. In a simulation study, we compare coverage rates and lengths of these confidence intervals with those based on the asymptotic distribution of the empirical process and highlight some benefits of higher-order approximations of the empirical process.

Higher-order approximation for uncertainty quantification in time series analysis

TL;DR

This work tackles the slow convergence of the empirical process for time series with strong temporal correlation by developing higher-order approximations based on Hermite expansions. It proves a two-parameter limit theorem for the higher-order term and derives Hermite-process limits for the dominating lower-order components, enabling confidence intervals for distributional functionals (e.g., marginal distribution and quantiles) that are robust to long-range dependence. The authors propose HOA-based confidence intervals that incorporate the higher-order corrections, leading to improved coverage and shorter interval lengths compared to classical asymptotic intervals, especially under high Hurst parameters. Numerical studies demonstrate the practical gains and discuss estimation challenges for the long-range variance and the Hurst parameter, highlighting the method’s potential for uncertainty quantification in change-point analysis and goodness-of-fit testing under long-range dependence.

Abstract

For time series with high temporal correlation, the empirical process converges rather slowly to its limiting distribution. Many statistics in change-point analysis, goodness-of-fit testing and uncertainty quantification admit a representation as functionals of the empirical process and therefore inherit its slow convergence. As a result, inference based on the asymptotic distribution of those quantities is significantly affected by relatively small sample sizes. We assess the quality of higher-order approximations of the empirical process by deriving the asymptotic distribution of the corresponding error terms. Based on the limiting distribution of the higher-order terms, we propose a novel approach to calculate confidence intervals for statistical quantities such as the median. In a simulation study, we compare coverage rates and lengths of these confidence intervals with those based on the asymptotic distribution of the empirical process and highlight some benefits of higher-order approximations of the empirical process.
Paper Structure (25 sections, 9 theorems, 163 equations, 7 figures)

This paper contains 25 sections, 9 theorems, 163 equations, 7 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Setting
  4. Gaussian subordination
  5. Higher-order approximation
  6. Main result
  7. Statement
  8. Proof
  9. Confidence intervals
  10. Confidence intervals for the marginal distribution
  11. Confidence intervals for quantiles
  12. Discussion
  13. Numerical Studies
  14. Estimation of long-run variance and Hurst parameter
  15. Confidence intervals for the marginal distribution
  16. ...and 10 more sections

Key Result

Theorem 4.1

Suppose $X_{n}, n \in \mathbb N,$ satisfies Model model and $X_{n}$ has a strictly monotone, continuous distribution function $F$ and $\frac{1}{D} \notin \mathbb N$. Then, as $N \to \infty$, in $D([-\infty, \infty] \times [0,1])$, where $S(x,t)$ is a mean zero Gaussian process with cross-covariances

Figures (7)

  • Figure 1: The empirical distribution of the centered and standardized empirical distribution $F_N(x)$ evaluated at zero ($x=0$) under the assumption of Gaussian long-range dependent data with different Hurst parameters and for sample sizes $m = 100, 200, 1000$. The red line depicts the standard Gaussian density function.
  • Figure 2: Number of summands in the "lower-order term" given that the Hermite rank $r=1$.
  • Figure 3: The coverage rate and length of confidence intervals for the marginal distribution $F(x)$ evaluated at different $x$. The two displayed methods to calculate the confidence intervals are based on the asymptotic distribution (asymp) and our higher-order approximation (HOA). The simulations are based on 2000 repetitions for Gaussian time series of length $N=200$ (first row) and $N=1000$ (second row) with Hurst parameter $H=0.55$. The dashed gray line depicts the significance level of $95\%$.
  • Figure 4: The coverage rate and length of confidence intervals for the marginal distribution $F(x)$ evaluated at different $x$. The two displayed methods to calculate the confidence intervals are based on the asymptotic distribution (asymp) and our higher-order approximation (HOA). The simulations are based on 2000 repetitions for Gaussian time series of length $N=200$ (first row) and $N=1000$ (second row) with Hurst parameter $H=0.95$. The dashed gray line depicts the significance level of $95\%$.
  • Figure 5: Confidence intervals for the marginal distribution $F$. The two displayed methods to calculate the confidence intervals are based on the asymptotic distribution (asymp) and our higher-order approximation (HOA). The simulations are based on 1000 repetitions for Gaussian time series of length $N=100$ with Hurst parameters $H=0.6, H=0.75$ and $H=0.9$.
  • ...and 2 more figures

Theorems & Definitions (25)

  • Example 2.2: Definition 2.8.3 in PipirasTaqqu
  • Definition 2.3
  • Definition 2.4: Definition 5.2.1 in PipirasTaqqu
  • Theorem 4.1
  • Remark 4.2
  • Example 5.1
  • Example 5.2
  • proof : Proof of \ref{['eq:tightness_Mn_part2']}
  • Lemma B.1
  • proof
  • ...and 15 more