Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Composite goodness-of-fit test with the Kernel Stein Discrepancy and a bootstrap for degenerate U-statistics with estimated parameters

Florian Brueck, Veronika Reimoser, Fabian Baier

TL;DR

This work addresses the problem of composite goodness-of-fit testing when the null encompasses a parametric family of distributions, using Kernel Stein Discrepancy (KSD) as the testing statistic. It derives the full asymptotic distribution of the composite KSD estimator under parameter estimation, revealing a limit that combines a weighted sum of chi-square terms with a disturbance from estimating the parameter, and provides a joint limit involving the estimator and score derivatives. To enable valid inference, the authors develop a general bootstrap CLT for degenerate, parameter-dependent U-statistics, showing that naive bootstrap procedures fail and introducing a corrected bootstrap term that delivers the correct limiting law; this framework is then specialized to the KSD. Simulations demonstrate that the proposed bootstrap-based KSD test maintains level under the null and exhibits higher power than competing MMD-based tests and the prior wild-bootstrap KSD approach, particularly in moderate to high dimensions. Overall, the paper offers both a rigorous theoretical foundation and a practical resampling strategy for composite goodness-of-fit testing with KSD in the presence of estimated parameters, with broader relevance to degenerate U-statistics beyond the KSD case.

Abstract

This paper formally derives the asymptotic distribution of a goodness-of-fit test based on the Kernel Stein Discrepancy introduced in (Oscar Key et al., "Composite Goodness-of-fit Tests with Kernels", Journal of Machine Learning Research 26.51 (2025), pp. 1-60). The test enables the simultaneous estimation of the optimal parameter within a parametric family of candidate models. Its asymptotic distribution is shown to be a weighted sum of infinitely many $χ^2$-distributed random variables plus an additional disturbance term, which is due to the parameter estimation. Further, we provide a general framework to bootstrap degenerate parameter-dependent $U$-statistics and use it to derive a new Kernel Stein Discrepancy composite goodness-of-fit test.

Composite goodness-of-fit test with the Kernel Stein Discrepancy and a bootstrap for degenerate U-statistics with estimated parameters

TL;DR

This work addresses the problem of composite goodness-of-fit testing when the null encompasses a parametric family of distributions, using Kernel Stein Discrepancy (KSD) as the testing statistic. It derives the full asymptotic distribution of the composite KSD estimator under parameter estimation, revealing a limit that combines a weighted sum of chi-square terms with a disturbance from estimating the parameter, and provides a joint limit involving the estimator and score derivatives. To enable valid inference, the authors develop a general bootstrap CLT for degenerate, parameter-dependent U-statistics, showing that naive bootstrap procedures fail and introducing a corrected bootstrap term that delivers the correct limiting law; this framework is then specialized to the KSD. Simulations demonstrate that the proposed bootstrap-based KSD test maintains level under the null and exhibits higher power than competing MMD-based tests and the prior wild-bootstrap KSD approach, particularly in moderate to high dimensions. Overall, the paper offers both a rigorous theoretical foundation and a practical resampling strategy for composite goodness-of-fit testing with KSD in the presence of estimated parameters, with broader relevance to degenerate U-statistics beyond the KSD case.

Abstract

This paper formally derives the asymptotic distribution of a goodness-of-fit test based on the Kernel Stein Discrepancy introduced in (Oscar Key et al., "Composite Goodness-of-fit Tests with Kernels", Journal of Machine Learning Research 26.51 (2025), pp. 1-60). The test enables the simultaneous estimation of the optimal parameter within a parametric family of candidate models. Its asymptotic distribution is shown to be a weighted sum of infinitely many -distributed random variables plus an additional disturbance term, which is due to the parameter estimation. Further, we provide a general framework to bootstrap degenerate parameter-dependent -statistics and use it to derive a new Kernel Stein Discrepancy composite goodness-of-fit test.
Paper Structure (20 sections, 8 theorems, 44 equations, 4 figures, 1 algorithm)

This paper contains 20 sections, 8 theorems, 44 equations, 4 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. The KSD
  3. Asymptotic distribution of the composite KSD estimator
  4. Assumptions
  5. Asymptotic distribution under $H_0^C$ and $H_1^C$
  6. A bootstrap CLT for degenerate $U$-statistics with estimated parameters
  7. Bootstrapping the KSD
  8. Simulations
  9. Monte Carlo study under the null hypothesis
  10. Monte Carlo study under the alternative hypothesis
  11. Proofs
  12. Technical results
  13. Proof of Theorem \ref{['thm.: Convergence of KSD under H0']}
  14. Proof of Theorem \ref{['thm.: KSD Consistency under H_1^C']}
  15. Proof of Lemma \ref{['lem:jointconvbootstrapUstat']}
  16. ...and 5 more sections

Key Result

Theorem 1

Under Assumptions KSD assumpt. 0-KSD assumpt. 6 and under $H_0^C$, we have where $H^*:= \left(\frac{\partial^2}{\partial \theta_i \partial \theta_j} \mathop{\mathrm{KSD}}\nolimits^2(P_{\theta_0} , Q) \right)_{i,j\in[p]}$.

Figures (4)

  • Figure 1: Simulation under the null hypothesis for dimension $d=1$ (left) and dimension $d=4$ (right) with $c=0.2$. KSD denotes the test proposed in this paper whereas Wild denotes the KSD test from Key_2023.
  • Figure 2: Empirical rejection probabilities under the alternative for dimensions $d\in\{1,2,4\}$, $\mu=1$ (left) and $\mu=2$ (right) with $c=1$. KSD$_U$ denotes the test proposed in this paper, KSD$_V$ (resp. MMD) denotes the wild bootstrapped KSD (resp. MMD) tests from Key_2023 and BFM denotes the MMD test from brueckminfermanian2024
  • Figure 3: Simulation under the null hypothesis for dimension $d=2$.
  • Figure 4: Simulation under the alternative for dimensions $d\in\{1,2,4\}$ and $\mu=4$.

Theorems & Definitions (14)

  • Definition 1
  • Definition 2
  • Theorem 1: Convergence under Null Hypothesis
  • Theorem 2: Consistency under Alternative Hypothesis
  • Lemma 1
  • Proposition 1
  • Theorem 3
  • Remark 1
  • Corollary 1
  • Remark 2: Wild bootstrap for the KSD
  • ...and 4 more