Minimax nonparametric multi-sample test under smoothing. (English) Zbl 07928652
Summary: We consider the problem of comparing probability densities among multiple groups. To this end, we develop a new probabilistic tensor product smoothing spline framework to model the joint density of two variables. Under such a framework, the probability density comparison is equivalent to testing the presence/absence of interactions, for which we propose a penalized likelihood ratio test. Here we show that the test statistic is asymptotically chi-squared distributed under the null hypothesis. Furthermore, we derive a sharp minimax testing rate based on the Bernstein width for nonparametric multi-sample tests, and show that our proposed test statistic is minimax optimal. In addition, we develop a data-adaptive tuning criterion for choosing the penalty parameter. The results of simulations and real applications demonstrate that the proposed test outperforms conventional approaches under various scenarios.
MSC:
| 62-XX | Statistics |
Keywords:
minimax optimality; multi-sample test; nonparametric test; penalized likelihood ratio test; smoothing splines; Wilks’ phenomenonReferences:
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| [50] | E-mail: xinxing@vt.edu Zuofeng Shang Department of Mathematical Sciences, New Jersey Institute of Technology, Newark, NJ 07102, USA. E-mail: zshang@njit.edu Pang Du Department of Statistics, Virginia Tech, Blacksburg, VA 24061, USA. |
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