dorsal/arxiv
View SchemaSecond-Order Asymptotics of Two-Sample Tests
| Authors | K V Harsha, Jithin Ravi, Tobias Koch |
|---|---|
| Categories | |
| ArXiv ID | 2601.09196vv1 |
| URL | https://arxiv.org/abs/2601.09196 |
| License | http://creativecommons.org/licenses/by/4.0/ |
Abstract
In two-sampling testing, one observes two independent sequences of independent and identically distributed random variables distributed according to the distributions $P_1$ and $P_2$ and wishes to decide whether $P_1=P_2$ (null hypothesis) or $P_1\neq P_2$ (alternative hypothesis). The Gutman test for this problem compares the empirical distributions of the observed sequences and decides on the null hypothesis if the Jensen-Shannon (JS) divergence between these empirical distributions is below a given threshold. This paper proposes a generalization of the Gutman test, termed \emph{divergence test}, which replaces the JS divergence by an arbitrary divergence. For this test, the exponential decay of the type-II error probability for a fixed type-I error probability is studied. First, it is shown that the divergence test achieves the optimal first-order exponent, irrespective of the choice of divergence. Second, it is demonstrated that the divergence test with an invariant divergence achieves the same second-order asymptotics as the Gutman test. In addition, it is shown that the Gutman test is the GLRT for the two-sample testing problem, and a connection between two-sample testing and robust goodness-of-fit testing is established.
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"abstract": "In two-sampling testing, one observes two independent sequences of independent and identically distributed random variables distributed according to the distributions $P_1$ and $P_2$ and wishes to decide whether $P_1=P_2$ (null hypothesis) or $P_1\\neq P_2$ (alternative hypothesis). The Gutman test for this problem compares the empirical distributions of the observed sequences and decides on the null hypothesis if the Jensen-Shannon (JS) divergence between these empirical distributions is below a given threshold. This paper proposes a generalization of the Gutman test, termed \\emph{divergence test}, which replaces the JS divergence by an arbitrary divergence. For this test, the exponential decay of the type-II error probability for a fixed type-I error probability is studied. First, it is shown that the divergence test achieves the optimal first-order exponent, irrespective of the choice of divergence. Second, it is demonstrated that the divergence test with an invariant divergence achieves the same second-order asymptotics as the Gutman test. In addition, it is shown that the Gutman test is the GLRT for the two-sample testing problem, and a connection between two-sample testing and robust goodness-of-fit testing is established.",
"arxiv_id": "2601.09196",
"authors": [
"K V Harsha",
"Jithin Ravi",
"Tobias Koch"
],
"categories": [
"cs.IT",
"math.IT"
],
"license": "http://creativecommons.org/licenses/by/4.0/",
"title": "Second-Order Asymptotics of Two-Sample Tests",
"url": "https://arxiv.org/abs/2601.09196",
"version": "v1"
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