Composite Goodness‐of‐Fit Test With the Kernel Stein Discrepancy and a Bootstrap for Degenerate U‐Statistics With Estimated Parameters

ABSTRACT This paper formally derives the asymptotic distribution of a goodness‐of‐fit test based on the Kernel Stein Discrepancy introduced in 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.