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Non-parametric goodness-of-fit testing procedures based on kernel Steindiscrepancies (KSD) are promising approaches to validate general unnormaliseddistributions in various scenarios. Existing works focused on studying kernelchoices to boost test performances. However, the choices of (non-unique) Steinoperators also have considerable effect on the test performances. Inspired bythe standardisation technique that was originally developed to better deriveapproximation properties for normal distributions, we present a unifyingframework, called standardisation-function kernel Stein discrepancy (Sf-KSD),to study different Stein operators in KSD-based tests for goodness-of-fit. Wederive explicitly how the proposed framework relates to existing KSD-basedtests and show that Sf-KSD can be used as a guide to develop novel kernel-basednon-parametric tests on complex data scenarios, e.g. truncated distributions orcompositional data. Experimental results demonstrate that the proposed testscontrol type-I error well and achieve higher test power than existingapproaches.

Standardisation-function Kernel Stein Discrepancy: A Unifying View on Kernel Stein Discrepancy Tests for Goodness-of-fit