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The Supremum of Chi-Square Processes

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Abstract

We describe a lower bound for the critical value of the supremum of a Chi-Square process. This bound can be approximated using an RQMC simulation. We compare numerically this bound with the upper bound given by Davies, only suitable for a regular Chi-Square process. In a second part, we focus on a non regular Chi-Square process: the Ornstein–Uhlenbeck Chi-Square process. Recently, Rabier et al. (2009) have shown that this process has an application in genetics: it is the limiting process of the likelihood ratio test process related to the test of a gene on an interval representing a chromosome. Using results from Delong (Commun Stat Theory Method A10(20):2197–2213, 1981), we propose a theoretical formula for the supremum of such a process and we compare it in particular with our simulated lower bound.

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Author notes

  1. Charles-Elie Rabier

    Present address: Statistics Department, Medical Science Center, University of Wisconsin-Madison, 1300 University Avenue, Madison, WI, 53706-1532, USA

Authors and Affiliations

  1. Institut de Mathématiques de Toulouse, Université de Toulouse, U.P.S., Toulouse, France

    Charles-Elie Rabier

  2. INRA UR631, Station d’Amélioration Génétique des Animaux, Auzeville, France

    Charles-Elie Rabier

  3. Department of Mathematics, Washington State University, Pullman, WA, 99164-3113, USA

    Alan Genz

Authors
  1. Charles-Elie Rabier
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  2. Alan Genz
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Correspondence to Charles-Elie Rabier.

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Rabier, CE., Genz, A. The Supremum of Chi-Square Processes. Methodol Comput Appl Probab 16, 715–729 (2014). https://doi.org/10.1007/s11009-013-9331-1

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  • DOI: https://doi.org/10.1007/s11009-013-9331-1

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