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Resampling-free Inference for Time Series via RKHS Embedding

Deep Ghoshal, Xiaofeng Shao

TL;DR

The paper addresses nonparametric inference for infinite-dimensional time-series parameters (e.g., goodness-of-fit, change points, and independence) by eliminating resampling. It develops a kernel-based framework using RKHS mean embeddings, sample splitting, projection, and self-normalization to obtain pivotal null distributions, applicable to vector, functional, and object-valued data. The authors derive FCLTs in RKHS and establish limiting distributions for GOF ($U$), change-point ($G$ and $G^*$), and independence tests (HSIC-type), with power analyses under local alternatives. Empirically, the SS-SN tests exhibit excellent size accuracy, competitive power, and substantial computational efficiency relative to bootstrap-based competitors, highlighting practical advantages for large-scale and complex time-series settings.

Abstract

In this article, we study nonparametric inference problems in the context of multivariate or functional time series, including testing for goodness-of-fit, the presence of a change point in the marginal distribution, and the independence of two time series, among others. Most methodologies available in the existing literature address these problems by employing a bandwidth-dependent bootstrap or subsampling approach, which can be computationally expensive and/or sensitive to the choice of bandwidth. To address these limitations, we propose a novel class of kernel-based tests by embedding the data into a reproducing kernel Hilbert space, and construct test statistics using sample splitting, projection, and self-normalization (SN) techniques. Through a new conditioning technique, we demonstrate that our test statistics have pivotal limiting null distributions under absolute regularity and mild moment assumptions. We also analyze the limiting power of our tests under local alternatives. Finally, we showcase the superior size accuracy and computational efficiency of our methods as compared to some existing ones.

Resampling-free Inference for Time Series via RKHS Embedding

TL;DR

The paper addresses nonparametric inference for infinite-dimensional time-series parameters (e.g., goodness-of-fit, change points, and independence) by eliminating resampling. It develops a kernel-based framework using RKHS mean embeddings, sample splitting, projection, and self-normalization to obtain pivotal null distributions, applicable to vector, functional, and object-valued data. The authors derive FCLTs in RKHS and establish limiting distributions for GOF (), change-point ( and ), and independence tests (HSIC-type), with power analyses under local alternatives. Empirically, the SS-SN tests exhibit excellent size accuracy, competitive power, and substantial computational efficiency relative to bootstrap-based competitors, highlighting practical advantages for large-scale and complex time-series settings.

Abstract

In this article, we study nonparametric inference problems in the context of multivariate or functional time series, including testing for goodness-of-fit, the presence of a change point in the marginal distribution, and the independence of two time series, among others. Most methodologies available in the existing literature address these problems by employing a bandwidth-dependent bootstrap or subsampling approach, which can be computationally expensive and/or sensitive to the choice of bandwidth. To address these limitations, we propose a novel class of kernel-based tests by embedding the data into a reproducing kernel Hilbert space, and construct test statistics using sample splitting, projection, and self-normalization (SN) techniques. Through a new conditioning technique, we demonstrate that our test statistics have pivotal limiting null distributions under absolute regularity and mild moment assumptions. We also analyze the limiting power of our tests under local alternatives. Finally, we showcase the superior size accuracy and computational efficiency of our methods as compared to some existing ones.
Paper Structure (24 sections, 18 theorems, 110 equations, 4 figures, 4 tables)

This paper contains 24 sections, 18 theorems, 110 equations, 4 figures, 4 tables.

Table of Contents

  1. Introduction
  2. Methodology
  3. Review of RKHS
  4. Goodness-of-Fit Testing
  5. Change Point Testing
  6. Testing Independence between Two Time Series
  7. Asymptotic Theory
  8. Goodness-of-Fit Testing
  9. Change Point Testing
  10. Testing Independence between Two Time Series
  11. Numerical Experiments
  12. Goodness-of-Fit Testing
  13. Testing for a Change Point in Euclidean Data
  14. Testing for a Change Point in Functional Time Series
  15. Testing for Independence between Two Functional Time Series
  16. ...and 9 more sections

Key Result

Lemma 2.1

If $K(.,.)$ is measurable and $\mathbb{E}_{X \sim Q}\sqrt{K(X,X)}$ is finite, then mean embedding of $Q$ into $\mathcal{H}(K)$ exists.

Figures (4)

  • Figure 1: Size-adjusted power in goodness-of-fit testing
  • Figure 2: Size-adjusted power in change point testing in Euclidean data
  • Figure 3: Size-adjusted power in change point testing in functional data
  • Figure 4: Size-adjusted power in testing independence between two functional series

Theorems & Definitions (46)

  • Lemma 2.1: JMLR:v13:gretton12a
  • Lemma 2.2: sriperumbudur2010hilbert
  • Definition 2.3
  • Remark 1
  • Remark 2
  • Example
  • Remark 3
  • Remark 4
  • Proposition 1
  • Remark 5
  • ...and 36 more