Abstract
Some asymptotic results are proved for the distribution of the maximum of a centered Gaussian random field with unit variance on a compact subset S of ℝN. They are obtained by a Rice method and the evaluation of some moments of the number of local maxima of the Gaussian field above an high level inside S and on the border ∂ S. Depending on the geometry of the border we give up to N+1 terms of the expansion sometimes with exponentially small remainder. Application to waves maximum is shown.
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References
Adler, R.J., The Geometry of Random Fields, Wiley, New York, 1981.
Adler, R.J., An Introduction to Continuity, Extrema and Related Topics for General Gaussian Processes, IMS, Hayward, CA, 1990.
Adler, R.J., “On excursion sets, tube formulae, and maxima of random fields,” Ann. of Appl. Probab. 10, 1–74, (2000).
Azaïs, J.M. and DELMAS, C., “Asymptotic expansions for the distribution of the maximum of Gaussian random fields,” Prépublication du Laboratoire de Statistique et Probabilités, Université Toulouse III. 2001.
Azaïs, J.M. and Wschebor, M., “Une formule pour calculer la distribution du maximum d'un processus stochastique,” C. R. Acad. Sci. Paris Sér. I Math. 324, 225–230, (1997).
Azaïs, J.M. and Wschebor, M., “The distribution of the maximum of a Gaussian process: Rice method revisited,” In: In and Out of Equilibrium: Probability with a Physical Flavour (V. Sidoravicius ed.) Progress in probability, Birhauser (2002).
Belyaev, Yu. K. and Piterbarg, V. I., “The asymptotic formula for the mean number of A-points of excursions of Gaussian fields above high levels (in Russian),” In: Bursts of Random Fields, Moscow University Press, Moscow, 62–89, (1972a).
Belyaev, Yu. K. and Piterbarg, V. I., “Asymptotics of the average number of A-points of overshoot of a Gaussian field beyond a high level,” Dockl. Akad. Nauk SSSR 203, 309–313, (1972b).
Brillinger, D. R., “On the number of solutions of systems of random equations,” Ann. of Math. Statist. 43, 534–540, (1972).
Cramer, H. and Leadbetter, M. R., Stationary and Related Stochastic Processes, Wiley, New York, 1967.
Dacunha-Castelle, D. and Gassiat, E., “Testing in locally conic model and application to mixture models,” ESAIM: Probability and Statistics 1, 285–317, (1997).
Dacunha-Castelle, D. and Gassiat, E., “Testing the order of a model using locally conic parametrisation: population mixtures and stationary ARMA processes,” Ann. of Statist. 27(4), 1178–1209, (1999).
Davies, R.B., “Hypothesis testing when a nuisance parameter is present only under the alternative,” Biometrika 64, 247–254, (1977).
Delmas, C., “An asymptotic expansion for the distribution of the maximum of a class of Gaussian fields,” C. R. Acad. Sci. Paris Sér. I Math. 327, 393–397, (1998).
Delmas, C., “Distribution du maximum d'un champ aléatoire et applications statistiques,” Doctoral thesis, University of Toulouse III, 2001.
Hasofer, A. M., “The Mean Number of Maxima Above High Levels in Gaussian Random Fields,” J. Appl. Probab. 13, 377–379, (1976).
Kac, M., “On the average number of real roots of a random algebraic equation,” Bull. Amer. Math. Soc. 49, 314–320, (1943).
Konakov, V. and Mammen, E., “The shape of kernel density estimates in higher dimensions,” Mathematical Methods of Statistics 6(4), 440–464, (1997).
Konakov, V. and Piterbarg, V. I., “High level excursions of Gaussian fields and the weakly optimal choice of the smoothing parameter I,” Mathematical Methods of Statistics 4(4), 421–434, (1995).
Konakov, V. and Piterbarg, V. I., “High level excursions of Gaussian fields and the weakly optimal choice of the smoothing parameter II,” Mathematical Methods of Statistics 6(1), 112–124, (1997).
Mikhaleva, T. L. and Piterbarg V. I., “On the distribution of the maximum of a Gaussian field with constant variance on a smooth manifold,” Theor. Prob. Appl. 41, 367–379, (1996).
Park, M. G. and Sun, J., “Tests in projection pursuit regression,” Journal of Statistical Planning and Inference 75, 65–90, (1998).
Piterbarg, V. I., “Comparison of Distribution Functions of Maxima of Gaussian Processes,” Th. Prob. Appl. 26, 687–705, (1981).
Piterbarg, V. I., Asymptotic Methods in the Theory of Gaussian Processes and Fields. American Mathematical Society. Providence, Rhode Island, (1996a).
Piterbarg, V. I., “Rice's method for large excursions of Gaussian random fields,” Technical Report No. 478, University of North Carolina. Translation of Rice's method for Gaussian random fields. Fundamental and Applied Mathematics 2, 187–204 (in Russian), (1996b).
Piterbarg, V. I. and Tyurin, Y. N., “Testing for homogeneity of two multivariate samples: a Gaussian field on a sphere,” Mathematical Methods of Statistics 2, 147–164, (1993).
Rice, S. O., “Mathematical analysis of random noise,” Bell System Tech. J. 23, 282–332; 24, 45–156, (1944–1955).
Siegmund, D. O. and Worsley, K. J., “Testing for a signal with unknown location and scale in a stationary Gaussian random field,” Ann. of Statist. 23, 608–639.
Sun, J., “Significance levels in exploratory projection pursuit,” Biometrika 78, 759–769, (1991).
Sun, J., “Tail probabilities of the maxima of Gaussian random fields,” Ann. Probab. 21, 34–71, (1993).
Takemura, A. and Kuriki, S., “Maximum of Gaussian field on piecewise smooth domain: Equivalence of tube method and Euler Characteristic method,” Preprint, 1999.
Talagrand, M., “Majorising measures: the general chaining,” Ann. of Probab. 24, 1049–1103, 1996.
Worsley, K. J., “Local maxima and the expected Euler characteristic of excursion sets of χ 2, F and t fields” Adv. Appl. Probab. 26, 13–42, (1994).
Worsley, K. J., “Estimating the number of peaks in a random field using the Hadwiger characteristic of excursion sets, with applications to medical images,” Ann. of Statist. 23, 640–669, (1995a).
Worsley, K. J., “Boundary corrections for the expected Euler characteristic of excursion sets of random fields, with an application to astrophysics,” Adv. Appl. Probab. 27, 943–959, (1995b).
Worsley, K. J., “The geometry of random images,” Chance 9(1), 27–40, (1997).
Worsley, K. J., Evans, A. C., Marret, S., and Neelin, P., “A three-dimensional statistical analysis for CBF activation studies in human brain,” Journal of Cerebral Blood Flow and Metabolism 12, 900–918, (1992).
Wschebor, M., “Surfaces aléatoires. Mesure géométrique des ensembles de niveau,” Lecture Notes in Mathematics 1147, Springer-Verlag, 1985.
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AZAÏS, JM., Delmas, C. Asymptotic Expansions for the Distribution of the Maximum of Gaussian Random Fields. Extremes 5, 181–212 (2002). https://doi.org/10.1023/A:1022123321967
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DOI: https://doi.org/10.1023/A:1022123321967