The distribution of maxima of approximately Gaussian random fields
Yuval Nardi, David O. Siegmund, and Benjamin Yakir
Source: Ann. Statist. Volume 36, Number 3
(2008), 1375-1403.
Abstract
Motivated by the problem of testing for the existence of a signal of known parametric structure and unknown “location” (as explained below) against a noisy background, we obtain for the maximum of a centered, smooth random field an approximation for the tail of the distribution. For the motivating class of problems this gives approximately the significance level of the maximum score test. The method is based on an application of a likelihood-ratio-identity followed by approximations of local fields. Numerical examples illustrate the accuracy of the approximations.
First Page:
Show
Hide
Full-text: Open access
Links and Identifiers
Permanent link to this document: http://projecteuclid.org/euclid.aos/1211819568
Digital Object Identifier: doi:10.1214/07-AOS511
Mathematical Reviews number (MathSciNet): MR2418661
Zentralblatt MATH identifier: 1148.60029
References
[1] Adler, R. J. (2000). On excursion sets, tube formulas and maxima of random fields. Ann. Appl. Probab. 10 1–74.
Mathematical Reviews (MathSciNet): MR1765203
Project Euclid: euclid.aoap/1019737664
[2] Adler, R. J. (1981). The Geometry of Random Fields. Wiley, Chichester.
Mathematical Reviews (MathSciNet): MR611857
Zentralblatt MATH: 0478.60059
[3] Azaïs, J.-M. and Wschebor, M. (2005). On the distribution of the maximum of a Gaussian field with d parameters. Ann. Appl. Probab. 15 254–278.
[4] Azaïs, J.-M., Bardet, J.-M. and Wschebor, M. (2002). On the tails of the distribution of the maximum of a smooth stationary Gaussian process. ESAIM Probab. Statist. 6 177–184 (electronic).
[5] Barndorff-Nielsen, O. E. and Cox, D. R. (1989). Asymptotic Techniques for Use in Statistics. Chapman and Hall, London.
Mathematical Reviews (MathSciNet): MR1010226
Zentralblatt MATH: 0672.62024
[6] Bhattacharya, R. N. and Rao, R. R. (1976). Normal Approximation and Asymptotic Expansions. Wiley, New York.
Mathematical Reviews (MathSciNet): MR436272
Zentralblatt MATH: 0331.41023
[7] Kuriki, S. and Takemura, A. (1999). Distribution of the maximum of Gaussian random field: Tube method and Euler characteristic method. Proc. Inst. Statist. Math. 47 201–221.
Mathematical Reviews (MathSciNet): MR1720026
[8] Luenberger, D. G. (2003). Linear and Nonlinear Programming, 2nd ed. Kluwer Academic, Boston.
[9] Nardi, Y. (2006). Gaussian, and asymptotically Gaussian, random fields and their maxima. Ph.D. dissertation, Hebrew Univ. of Jerusalem.
[10] Peng, J. and Siegmund, D. O. (2003). The admixture model in linkage analysis. J. Statist. Plann. Inference 130 317–324.
Mathematical Reviews (MathSciNet): MR2128010
Digital Object Identifier: doi:10.1016/j.jspi.2003.07.022
Zentralblatt MATH: 1085.62127
[11] Rabinowitz, D. (1994). Detecting clusters in disease incidence. In Change-Point Problems (H.-G. Müller, E. Carlstein and D. Siegmund, eds.) 255–275. IMS, Hayward, CA.
[12] Rabinowitz, D. and Siegmund, D. (1997). The approximate distribution of the maximum of a smoothed Poisson random field. Statist. Sinica 7 167–180.
Mathematical Reviews (MathSciNet): MR1441152
[13] Shafie, K., Sigal, S., Siegmund, D. O. and Worsley, K. J. (2003). Rotation space ranmdom fields with an application to fMRI data. Ann. Statist. 31 1732–1771.
Mathematical Reviews (MathSciNet): MR2036389
Digital Object Identifier: doi:10.1214/aos/1074290326
Project Euclid: euclid.aos/1074290326
Zentralblatt MATH: 1043.92019
[14] Siegmund, D. O. and Yakir, B. (2000). Tail probabilities for the null distribution of scanning statistics. Bernoulli 6 191–213.
Mathematical Reviews (MathSciNet): MR1748719
Digital Object Identifier: doi:10.2307/3318574
Project Euclid: euclid.bj/1081788026
Zentralblatt MATH: 0976.62048
[15] Siegmund, D. O. and Yakir, B. (2003). Significance level in interval mapping. In Development of Modern Statistics and Related Topics. Ser. Biostat. 1 10–19. World Scientific, River Edge, NJ.
Mathematical Reviews (MathSciNet): MR2039775
Zentralblatt MATH: 1081.62104
[16] Siegmund, D. O. and Worsley, K. J. (1995). Testing for a signal with unknown location and scale in a stationary Gaussian random field. Ann. Statist. 23 608–639.
Mathematical Reviews (MathSciNet): MR1332585
Digital Object Identifier: doi:10.1214/aos/1176324539
Project Euclid: euclid.aos/1176324539
Zentralblatt MATH: 0898.62119
[17] Sun, J. (1993). Tail probabilities of the maxima of Gaussian random fields. Ann. Probab. 21 34–71.
Mathematical Reviews (MathSciNet): MR1207215
Digital Object Identifier: doi:10.1214/aop/1176989393
Project Euclid: euclid.aop/1176989393
Zentralblatt MATH: 0772.60038
[18] Taylor, J., Takemura, A. and Adler, R. J. (2005). Validity of the expected Euler characteristic heuristic. Ann. Probab. 33 1362–1396.
Mathematical Reviews (MathSciNet): MR2150192
Digital Object Identifier: doi:10.1214/009117905000000099
Project Euclid: euclid.aop/1120224584
Zentralblatt MATH: 1083.60031
[19] Worsley, K. J., Evans, A. C., Marrett, S. and Neelin, P. (1992). A three dimensional statistical analysis for CBF activation studies in human brain. J. Cerebral Blood Flow and Metabolism 12 900–918.
[20] Yakir, B. and Pollak, M. (1998). A new representation for the renewal-theoretic constant appearing in asymptotic approximations of large deviations. Ann. Appl. Probab. 8 749–774.
Mathematical Reviews (MathSciNet): MR1627779
Digital Object Identifier: doi:10.1214/aoap/1028903449
Project Euclid: euclid.aoap/1028903449
Zentralblatt MATH: 0937.60082
