Semiparametric KSD test: unifying score and distance-based approaches for goodness-of-fit testing

TL;DR

This paper introduces a new semiparametric goodness-of-fit test called SKSD, unifying score-based and distance-based approaches, which is computationally efficient, versatile, and achieves high power for general models.

Contribution

It proposes the SKSD test, a novel nonparametric score-based method that unifies classical distance-based tests and offers computational efficiency and broad applicability.

Findings

SKSD test is computationally efficient and universally consistent.

It attains Pitman efficiency and high power comparable to specialized normality tests.

The method works well with models having intractable likelihoods but tractable scores.

Abstract

Goodness-of-fit (GoF) tests are fundamental for assessing model adequacy. Score-based tests are appealing because they require fitting the model only once under the null. However, extending them to powerful nonparametric alternatives is difficult due to the lack of suitable score functions. Through a class of exponentially tilted models, we show that the resulting score-based GoF tests are equivalent to the tests based on integral probability metrics (IPMs) indexed by a function class. When the class is rich, the test is universally consistent. This simple yet insightful perspective enables reinterpretation of classical distance-based testing procedures-including those based on Kolmogorov-Smirnov distance, Wasserstein-1 distance, and maximum mean discrepancy-as arising from score-based constructions. Building on this insight, we propose a new nonparametric score-based GoF test through a special class of IPM induced by kernelized Stein's function class, called semiparametric kernelized Stein discrepancy (SKSD) test. Compared with other nonparametric score-based tests, the SKSD test is computationally efficient and accommodates general nuisance-parameter estimators, supported by a generic parametric bootstrap procedure. The SKSD test is universally consistent and attains Pitman efficiency. Moreover, SKSD test provides simple GoF tests for models with intractable likelihoods but tractable scores with the help of Stein's identity and we use two popular models, kernel exponential family and conditional Gaussian models, to illustrate the power of our method. Our method achieves power comparable to task-specific normality tests such as Anderson-Darling and Lilliefors, despite being designed for general nonparametric alternatives.