Maxima of moving sums in a Poisson random field
Chan Hock Peng
Source: Adv. in Appl. Probab. Volume 41, Number 3 (2009), 647-663.
Abstract
In this paper we examine the extremal tail probabilities of moving sums in a marked Poisson random field. These sums are computed by adding up the weighted occurrences of events lying within a scanning set of fixed shape and size. We also provide an alternative representation of the constants of the asymptotic formulae in terms of the occupation measure of the conditional local random field at zero, and extend these representations to the constants of asymptotic tail probabilities of Gaussian random fields.
Primary Subjects: 60F10
Secondary Subjects: 60G10, 60G55
Keywords: Change of measure; Gaussian process; large deviations; marked Poisson process; moving sums; random field; scan statistics
Full-text: Access denied (no subscription detected)
We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber.
If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text
Links and Identifiers
Permanent link to this document: http://projecteuclid.org/euclid.aap/1253281058
Digital Object Identifier: doi:10.1239/aap/1253281058
Zentralblatt MATH identifier:
05625062
References
Adler, R. J. (2000). On excursion sets, tube formulas and maxima of random fields. Ann. Appl. Prob. 10, 1--74.
Mathematical Reviews (MathSciNet):
MR1765203
Zentralblatt MATH:
01500285
Project Euclid: euclid.aoap/1019737664
Albin, J. M. P. (1990). On extremal theory for stationary processes. Ann. Prob. 18, 92--128.
Mathematical Reviews (MathSciNet):
MR1043939
Zentralblatt MATH:
0704.60029
Digital Object Identifier: doi:10.1214/aop/1176990940
Project Euclid: euclid.aop/1176990940
Alm, S. E. (1997). On the distribution of scan statistics of a two-dimensional Poisson process. Adv. Appl. Prob. 29, 1--18.
Mathematical Reviews (MathSciNet):
MR1432927
Zentralblatt MATH:
0883.60046
Digital Object Identifier: doi:10.2307/1427857
JSTOR: links.jstor.org
Anderson, N. H. and Titterington, D. M. (1997). Some methods for investigating spatial clustering with epidemiological applications. J. R. Statist. Soc. A 160, 87--105.
Berman, S. M. (1982). Sojourns and extremes of stationary processes. Ann. Prob. 10, 1--46.
Mathematical Reviews (MathSciNet):
MR637375
Zentralblatt MATH:
0498.60035
Digital Object Identifier: doi:10.1214/aop/1176993912
Project Euclid: euclid.aop/1176993912
Bickel, P. and Rosenblatt, M. (1973). Two-dimensional random field. In Multivariate Analysis, III, Academic Press, New York, pp. 3--15.
Mathematical Reviews (MathSciNet):
MR348832
Zentralblatt MATH:
0297.60020
Chan, H. P. and Lai, T. L. (2006). Maxima of asymptotically Gaussian random fields and moderate deviation approximations to boundary crossing probabilities of sums of random variables with multidimensional indices. Ann. Prob. 34, 80--121.
Mathematical Reviews (MathSciNet):
MR2206343
Zentralblatt MATH:
1106.60022
Digital Object Identifier: doi:10.1214/009117905000000378
Project Euclid: euclid.aop/1140191533
Chan, H. P. and Zhang, N. R. (2007). Scan statistics with weighted observations. J. Amer. Statist. Assoc. 102, 595--602.
Mathematical Reviews (MathSciNet):
MR2370856
Zentralblatt MATH:
05191529
Digital Object Identifier: doi:10.1198/016214506000001392
Cressie, N. (1993). Statistics for Spatial Data. John Wiley, New York.
Mathematical Reviews (MathSciNet):
MR1239641
Zentralblatt MATH:
0799.62002
Feller, W. (1971). An Introduction to Probability Theory and Its Applications, Vol. 2, 2nd edn. John Wiley, New York.
Mathematical Reviews (MathSciNet):
MR270403
Glaz, J., Naus, J. and Wallenstein, S. (2001). Scan Statistics. Springer, New York.
Mathematical Reviews (MathSciNet):
MR1869112
Hogan, M. and Siegmund, D. O. (1986). Large deviations for the maxima of some random fields. Adv. Appl. Math. 7, 2--22.
Mathematical Reviews (MathSciNet):
MR834217
Zentralblatt MATH:
0612.60029
Digital Object Identifier: doi:10.1016/0196-8858(86)90003-5
Kabluchko, Z. and Spodarev, E. (2009). Scan statistics of Lévy noises and marked empirical processes. Adv. Appl. Prob. 41, 13--37.
Mathematical Reviews (MathSciNet):
MR2514944
Digital Object Identifier: doi:10.1239/aap/1240319575
Loader, C. R. (1991). Large-deviation approximations to the distribution of scan statistics. Adv. Appl. Prob. 23, 751--771.
Mathematical Reviews (MathSciNet):
MR1133726
Zentralblatt MATH:
0741.60036
Digital Object Identifier: doi:10.2307/1427674
JSTOR: links.jstor.org
Naus, J. I. (1965). Clustering of random points in two dimensions. Biometrika 52, 263--267.
Mathematical Reviews (MathSciNet):
MR211433
Zentralblatt MATH:
0132.39702
Digital Object Identifier: doi:10.2307/2333829
JSTOR: links.jstor.org
Pickands, J., III (1969). Upcrossing probabilities for stationary Gaussian processes. Trans. Amer. Math. Soc. 145, 51--75.
Mathematical Reviews (MathSciNet):
MR250367
Zentralblatt MATH:
0206.18802
Digital Object Identifier: doi:10.2307/1995058
Piterbarg, V. (1996). Asymptotic Methods in the Theory of Gaussian Processes and Fields (Transl. Math. Monogr. 148), American Mathematical Society, Providence, RI.
Mathematical Reviews (MathSciNet):
MR1361884
Qualls, C. and Watanabe, H. (1973). Asymptotic properties of Gaussian random fields. Trans. Amer. Math. Soc. 177, 155--171.
Mathematical Reviews (MathSciNet):
MR322943
Zentralblatt MATH:
0274.60030
Digital Object Identifier: doi:10.2307/1996589
Shafie, K., Sigal, B., Siegmund, D. and Worsley, K. J. (2003). Rotation space random fields with an application to fMRI data. Ann. Statist. 31, 1732--1771.
Mathematical Reviews (MathSciNet):
MR2036389
Zentralblatt MATH:
1043.92019
Digital Object Identifier: doi:10.1214/aos/1074290326
Project Euclid: euclid.aos/1074290326
Siegmund, D. (1985). Sequential Analysis. Springer, New York.
Mathematical Reviews (MathSciNet):
MR799155
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
Zentralblatt MATH:
0898.62119
Digital Object Identifier: doi:10.1214/aos/1176324539
Project Euclid: euclid.aos/1176324539
Spivak, M. (1965). Calculus on Manifolds: A Modern Approach to Classical Theorems of Advanced Calculus. W. A. Benjamin, New York.
Mathematical Reviews (MathSciNet):
MR209411
Storey, J. D. and Siegmund, D. O. (2001). Approximate $p$-values for local sequence alignments: numerical studies. J. Comput. Biol. 8, 549--556.
Taylor, J. E. (2006). A Gaussian kinematic formula. Ann. Prob. 34, 122--158.
Mathematical Reviews (MathSciNet):
MR2206344
Zentralblatt MATH:
1094.60025
Digital Object Identifier: doi:10.1214/009117905000000594
Project Euclid: euclid.aop/1140191534
Taylor, J., Takemura, A. and Adler, R. J. (2005). Validity of the expected Euler characteristic heuristic. Ann. Prob. 33, 1362--1396.
Mathematical Reviews (MathSciNet):
MR2150192
Zentralblatt MATH:
1083.60031
Digital Object Identifier: doi:10.1214/009117905000000099
Project Euclid: euclid.aop/1120224584
Tu, I.-P. (2009). Asymptotic overshoots for arithmetic i.i.d. random variables. Statistica Sinica 19, 315--323.
Mathematical Reviews (MathSciNet):
MR2487892
Zentralblatt MATH:
1153.62063
Tu, I. and Siegmund, D. O. (1999). The maxima of a function of a Markov chain and application to linkage analysis. Adv. Appl. Prob. 31, 510--531.
Mathematical Reviews (MathSciNet):
MR1724565
Zentralblatt MATH:
0941.60088
Digital Object Identifier: doi:10.1239/aap/1029955145
Project Euclid: euclid.aap/1029955145
Woodroofe, M. (1979). Repeated likelihood ratio tests. Biometrika 66, 454--463.
Mathematical Reviews (MathSciNet):
MR556732
Zentralblatt MATH:
0418.62063
Digital Object Identifier: doi:10.1093/biomet/66.3.453
JSTOR: links.jstor.org
Advances in Applied Probability
