Open Access
October 2020 Asymptotic distribution and detection thresholds for two-sample tests based on geometric graphs
Bhaswar B. Bhattacharya
Ann. Statist. 48(5): 2879-2903 (October 2020). DOI: 10.1214/19-AOS1913


In this paper, we consider the problem of testing the equality of two multivariate distributions based on geometric graphs constructed using the interpoint distances between the observations. These include the tests based on the minimum spanning tree and the $K$-nearest neighbor (NN) graphs, among others. These tests are asymptotically distribution-free, universally consistent and computationally efficient, making them particularly useful in modern applications. However, very little is known about the power properties of these tests. In this paper, using the theory of stabilizing geometric graphs, we derive the asymptotic distribution of these tests under general alternatives, in the Poissonized setting. Using this, the detection threshold and the limiting local power of the test based on the $K$-NN graph are obtained, where interesting exponents depending on dimension emerge. This provides a way to compare and justify the performance of these tests in different examples.


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Bhaswar B. Bhattacharya. "Asymptotic distribution and detection thresholds for two-sample tests based on geometric graphs." Ann. Statist. 48 (5) 2879 - 2903, October 2020.


Received: 1 March 2019; Revised: 1 September 2019; Published: October 2020
First available in Project Euclid: 19 September 2020

MathSciNet: MR4152627
Digital Object Identifier: 10.1214/19-AOS1913

Primary: 60C05 , 60D05 , 60F05 , 62F07 , 62G10

Keywords: efficiency , geometric probability , local power , nearest-neighbor graphs , nonparametric hypothesis testing

Rights: Copyright © 2020 Institute of Mathematical Statistics

Vol.48 • No. 5 • October 2020
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