Xu, Wenkai; (2021) Advances in Non-parametric Hypothesis Testing with Kernels. Doctoral thesis (Ph.D), UCL (University College London).

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Abstract

Non-parametric statistical hypothesis testing procedures aim to distinguish the null hypothesis against the alternative with minimal assumptions on the model distributions. In recent years, the maximum mean discrepancy (MMD) has been developed as a measure to compare two distributions, which is applicable to two-sample problems and independence tests. With the aid of reproducing kernel Hilbert spaces (RKHS) that are rich-enough, MMD enjoys desirable statistical properties including characteristics, consistency, and maximal test power. Moreover, MMD receives empirical successes in complex tasks such as training and comparing generative models. Stein’s method also provides an elegant probabilistic tool to compare unnormalised distributions, which commonly appear in practical machine learning tasks. Combined with rich-enough RKHS, the kernel Stein discrepancy (KSD) has been developed as a proper discrepancy measure between distributions, which can be used to tackle one-sample problems (or goodness-of-fit tests). The existing development of KSD applies to a limited choice of domains, such as Euclidean space or finite discrete sets, and requires complete data observations, while the current MMD constructions are limited by the choice of simple kernels where the power of the tests suffer, e.g. high-dimensional image data. The main focus of this thesis is on the further advancement of kernel-based statistics for hypothesis testings. Firstly, Stein operators are developed that are compatible with broader data domains to perform the corresponding goodness-of-fit tests. Goodness-of-fit tests for general unnormalised densities on Riemannian manifolds, which are of the non-Euclidean topology, have been developed. In addition, novel non-parametric goodness-of-fit tests for data with censoring are studied. Then the tests for data observations with left truncation are studied, e.g. times of entering the hospital always happen before death time in the hospital, and we say the death time is truncated by the entering time. We test the notion of independence beyond truncation by proposing a kernelised measure for quasi-independence. Finally, we study the deep kernel architectures to improve the two-sample testing performances.

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