Hostname: page-component-cd9895bd7-dzt6s Total loading time: 0 Render date: 2024-12-29T12:42:30.520Z Has data issue: false hasContentIssue false
  • English
  • Français

Bounds and asymptotic expansions for the distributionof the Maximum of a smooth stationary Gaussian process

Published online by Cambridge University Press:  15 August 2002

Jean-Marc Azaïs ,
Christine Cierco-Ayrolles  and
Alain Croquette
Jean-Marc Azaïs
Affiliation:
Laboratoire de Statistique et Probabilités, UMR C55830 du CNRS, Université Paul Sabatier, 118 route de Narbonne, 31062 Toulouse Cedex 4, France.
Christine Cierco-Ayrolles
Affiliation:
Laboratoire de Statistique et Probabilités, UMR C55830 du CNRS, Université Paul Sabatier, 118 route de Narbonne, 31062 Toulouse Cedex 4, France. Institut National de la Recherche Agronomique, Unité de Biométrie et Intelligence Artificielle, BP. 27, Chemin de Borde-Rouge, 31326 Castanet-Tolosan Cedex, France.
Alain Croquette
Affiliation:
Laboratoire de Statistique et Probabilités, UMR C55830 du CNRS, Université Paul Sabatier, 118 route de Narbonne, 31062 Toulouse Cedex 4, France.
Get access

Abstract

This paper uses the Rice method [18] to give bounds tothe distribution of the maximum of a smooth stationary Gaussianprocess. We give simpler expressions of the first two terms ofthe Rice series [3,13] for the distribution of the maximum. Our main contribution is a simpler form of the second factorial momentof the number of upcrossings which is in some sense a generalizationof Steinberg et al.'s formula ([7] p. 212). Then, we present a numerical application and asymptotic expansionsthat give a new interpretation of a result by Piterbarg [15].

Keywords

Asymptotic expansionsextreme valuesstationary Gaussian processRice seriesupcrossings.
Type
Research Article
Information
ESAIM: Probability and Statistics , Volume 3 , 1999 , pp. 107 - 129
Copyright
© EDP Sciences, SMAI, 1999

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

M. Abramowitz and I.A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs and Mathematical Tables. Dover, New York (1972).
R.J. Adler, An Introduction to Continuity, Extrema and Related Topics for General Gaussian Processes, IMS, Hayward, Ca (1990).
J.-M. Azaïs and M. Wschebor, Une formule pour calculer la distribution du maximum d'un processus stochastique. C.R. Acad. Sci. Paris Ser. I Math. 324 (1997) 225-230.
J-M. Azaïs and M. Wschebor, The Distribution of the Maximum of a Stochastic Process and the Rice Method, submitted.
C. Cierco, Problèmes statistiques liés à la détection et à la localisation d'un gène à effet quantitatif. PHD dissertation. University of Toulouse, France (1996).
C. Cierco and J.-M. Azaïs, Testing for Quantitative Gene Detection in Dense Map, submitted.
H. Cramér and M.R. Leadbetter, Stationary and Related Stochastic Processes, J. Wiley & Sons, New-York (1967).
Dacunha-Castelle, D. and Gassiat, E., Testing in locally conic models, and application to mixture models. ESAIM: Probab. Statist. 1 (1997) 285-317. CrossRef
Davies, R.B., Hypothesis testing when a nuisance parameter is present only under the alternative. Biometrika 64 (1977) 247-254. CrossRef
J. Ghosh and P. Sen, On the asymptotic performance of the log-likelihood ratio statistic for the mixture model and related results, in Proc. of the Berkeley conference in honor of Jerzy Neyman and Jack Kiefer, Le Cam L.M. and Olshen R.A., Eds. (1985).
Kratz, M.F. and Rootzén, H., On the rate of convergence for extreme of mean square differentiable stationary normal processes. J. Appl. Prob. 34 (1997) 908-923. CrossRef
M.R. Leadbetter, G. Lindgren and H. Rootzén, Extremes and Related Properties of Random Sequences and Processes. Springer-Verlag, New-York (1983).
Miroshin, R.N., Rice series in the theory of random functions. Vestnik Leningrad Univ. Math. 1 (1974) 143-155.
M.B. Monagan, et al. Maple V Programming guide. Springer (1998).
Piterbarg, V.I., Comparison of distribution functions of maxima of Gaussian processes. Theory Probab. Appl. 26 (1981) 687-705. CrossRef
Piterbarg, V.I., Large deviations of random processes close to gaussian ones. Theory Probab. Appl. 27 (1982) 504-524. CrossRef
V.I. Piterbarg, Asymptotic Methods in the Theory of Gaussian Processes and Fields. American Mathematical Society. Providence, Rhode Island (1996).
Rice, S.O., Mathematical Analysis of Random Noise. Bell System Tech. J. 23 (1944) 282-332; 24 (1945) 45-156. CrossRef
SPLUS, Statistical Sciences, S-PLUS Programmer's Manual, Version 3.2, Seattle: StatSci, a division of MathSoft, Inc. (1993).
Sun, J., Significance levels in exploratory projection pursuit. Biometrika 78 (1991) 759-769. CrossRef
M. Wschebor, Surfaces aléatoires. Mesure géometrique des ensembles de niveau. Springer-Verlag, New-York, Lecture Notes in Mathematics 1147 (1985).