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A general framework for the analysis of kernel-based tests
Statistics Theory · Submitted 31 Aug 2022
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim Therefore, our framework leads to a much simpler and clean analysis of kernel tests, only requiring mild regularity conditions.
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The abstract, sentence by sentence
Kernel-based tests provide a simple yet effective framework that use the theory of reproducing kernel Hilbert spaces to design non-parametric testing procedures. In this paper we propose new theoretical tools that can be used to study the asymptotic behaviour of kernel-based tests in several data scenarios, and in many different testing problems. Unlike current approaches, our methods avoid using lengthy $U$ and $V$ statistics expansions and limit theorems, that commonly appear in the literature, and works directly with random functionals on Hilbert spaces. Therefore, our framework leads to a much simpler and clean analysis of kernel tests, only requiring mild regularity conditions. Furthermore, we show that, in general, our analysis cannot be improved by proving that the regularity conditions required by our methods are both sufficient and necessary. To illustrate the effectiveness of our approach we present a new kernel-test for the conditional independence testing problem, as well as new analyses for already known kernel-based tests.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | Kernel-based tests provide a simple yet effective framework that use the theory of reproducing kernel Hilbert spaces to design non-parametric testing procedures. | background 1.00 | 0.05 | 0.04 | asserted 1.00 |
| s1 | In this paper we propose new theoretical tools that can be used to study the asymptotic behaviour of kernel-based tests in several data scenarios, and in many different testing problems. | contribution 1.00 | 0.93 | 0.20 | asserted 0.87 |
| s2 | Unlike current approaches, our methods avoid using lengthy $U$ and $V$ statistics expansions and limit theorems, that commonly appear in the literature, and works directly with random functionals on Hilbert spaces. | method 0.72 | 0.84 | 0.12 | asserted 1.00 |
| s3 | Therefore, our framework leads to a much simpler and clean analysis of kernel tests, only requiring mild regularity conditions. | interpretation 0.55 | 0.87 | 0.36 | asserted 1.00 |
| s4 | Furthermore, we show that, in general, our analysis cannot be improved by proving that the regularity conditions required by our methods are both sufficient and necessary. | result 0.70 | 0.83 | 0.18 | asserted 1.00 |
| s5 | To illustrate the effectiveness of our approach we present a new kernel-test for the conditional independence testing problem, as well as new analyses for already known kernel-based tests. | contribution 0.98 | 0.94 | 0.10 | asserted 1.00 |
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The paper
Original abstract
Kernel-based tests provide a simple yet effective framework that use the theory of reproducing kernel Hilbert spaces to design non-parametric testing procedures. In this paper we propose new theoretical tools that can be used to study the asymptotic behaviour of kernel-based tests in several data scenarios, and in many different testing problems. Unlike current approaches, our methods avoid using lengthy $U$ and $V$ statistics expansions and limit theorems, that commonly appear in the literature, and works directly with random functionals on Hilbert spaces. Therefore, our framework leads to a much simpler and clean analysis of kernel tests, only requiring mild regularity conditions. Furthermore, we show that, in general, our analysis cannot be improved by proving that the regularity conditions required by our methods are both sufficient and necessary. To illustrate the effectiveness of our approach we present a new kernel-test for the conditional independence testing problem, as well as new analyses for already known kernel-based tests.
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How the literature uses it
1 arXiv paper cite this in 1 sentence
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Builds on 1 paper
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The third application is a novel test for conditional independent testing that kernelises the very recently proposed Weighted Generalised Covariance Measure [ref] which is a weighted generalisation of the introduced Generalised Covariance Measure [ref] .
The Weighted Generalised Covariance Measure · Introduction · 0.95
Uses the method or tool of 1 paper
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Our first experiment considers $d = 1$, and in such case we consider the weight function $ω(z)=sign(z)$ (as it was used in the experiments of [ref] ).
The Weighted Generalised Covariance Measure · Experiments · 0.92
Compares with 1 paper
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The above conditions are slightly stronger than the corresponding conditions of [ref] , but they allow us to avoid splitting the data as in [ref] (e.g. use half of the data to estimate $f$ and $g$, and the other half in the testing procedure).
The Weighted Generalised Covariance Measure · A test based on the Kernelised GCM · 0.99
Cites as background 6 papers
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Other testing problems where kernel methods have been applied are: conditional independence [ref] , composite goodness-of-fit [ref] , and several testing problems in survival analysis [ref] .
Kernel-based Conditional Independence Test and Application in Causal Discovery · Introduction · 0.95
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Other testing problems where kernel methods have been applied are: conditional independence [ref] , composite goodness-of-fit [ref] , and several testing problems in survival analysis [ref] .
Composite Goodness-of-fit Tests with Kernels · Introduction · 0.95
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Other testing problems where kernel methods have been applied are: conditional independence [ref] , composite goodness-of-fit [ref] , and several testing problems in survival analysis [ref] .
A Multiple kernel testing procedure for non-proportional hazards in factorial designs · Introduction · 0.95
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