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The two-sample problem in $$\mathbb R^m$$ and measure-valued martingales. (English) Zbl 1373.62191
de Gunst, Mathisca (ed.) et al., State of the art in probability and statistics. Festschrift for Willem R. van Zwet. Papers from the symposium, Leiden, Netherlands, March 23–26, 1999. Beachwood, OH: IMS, Institute of Mathematical Statistics (ISBN 0-940600-50-1). IMS Lect. Notes, Monogr. Ser. 36, 434-463 (2001).
Summary: The so-called two-sample problem is one of the classical problems in mathematical statistics. It is well-known that in dimension one the two-sample Smirnov test possesses two basic properties: it is distribution free under the null hypothesis and it is sensitive to ‘all’ alternatives. In the multidimensional case, i.e. when the observations in the two samples are random vectors in $$\mathbb R^{m}$$, $$m \geq 2$$, the Smirnov test loses its first basic property. In correspondence with the above, we define a solution of the two-sample problem to be a ‘natural’ stochastic process, based on the two samples, which is $$(\alpha)$$ asymptotically distribution free under the null hypothesis, and which is, intuitively speaking, $$(\beta)$$ as sensitive as possible to all alternatives. Despite the fact that the two-sample problem has a long and very diverse history, starting with some famous papers in the thirties, the problem is essentially still open for samples in $$\mathbb R^{m}$$, $$m\geq 2$$. In this paper we present an approach based on measure-valued martingales and we will show that the stochastic process obtained with this approach is a solution to the two-sample problem, i.e. it has both the properties $$(\alpha)$$ and $$(\beta)$$, for any $$m\in\mathbb N$$.
For the entire collection see [Zbl 0972.00075].

MSC:
 62G10 Nonparametric hypothesis testing 60G48 Generalizations of martingales
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