Abstract
Statistical depth functions provide measures of the outlyingness, or centrality, of the elements of a space with respect to a distribution. It is a nonparametric concept applicable to spaces of any dimension, for instance, multivariate and functional. Liu and Singh (1993) presented a multivariate two-sample test based on depth-ranks. We dedicate this paper to improving the power of the associated test statistic and incorporating its applicability to functional data. In doing so, we obtain a more natural test statistic that is symmetric in both samples. We derive the null asymptotic of the proposed test statistic, also proving the validity of the testing procedure for functional data. Finally, the finite sample performance of the test for functional data is illustrated by means of a simulation study and a real data analysis on annual temperature curves of ocean drifters is executed.
Funding Statement
This work was supported by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) - 314838170, GRK 2297 MathCoRe, as well as KI 1443/6-1. Additionally, it was supported by grant PID2022-139237NB-I00 funded by MCIN/AEI/10.13039/501100011033 and “ERDF A way of making Europe”.
Acknowledgments
We would like to thank Adam Sykulski (Imperial College London) for bringing the ocean drifter data to our attention. Felix Gnettner wants to thank Norbert Gaffke (Otto-von-Guericke-Universität Magdeburg) for a helpful discussion on the proof of Theorem 2.6 and Łukasz Smaga (Uniwersytet im. Adama Mickiewicza, Poznań) for sending some -code on his tests for functional data.
Citation
Felix Gnettner. Claudia Kirch. Alicia Nieto-Reyes. "Symmetrisation of a class of two-sample tests by mutually considering depth ranks including functional spaces." Electron. J. Statist. 18 (2) 3021 - 3106, 2024. https://doi.org/10.1214/24-EJS2250
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