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

2025/10/26 by Brueck, Florian, Reimoser, Veronika, Baier, Fabian
#FOS: Mathematics #Statistics Theory (math.ST)

paper · doi:10.48550/arxiv.2510.22792

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 χ²-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.

Citations

• Distribution free MMD tests for model selection with estimated parameters
• Stein's Method Meets Computational Statistics: A Review of Some Recent Developments
• Minimum Stein Discrepancy Estimators
• A Kernelized Stein Discrepancy for Goodness-of-fit Tests and Model Evaluation
• A Kernel Test of Goodness of Fit
• Measuring Sample Quality with Stein's Method
• A Wild Bootstrap for Degenerate Kernel Tests
• Dependent wild bootstrap for degenerateU- andV-statistics

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