×

zbMATH — the first resource for mathematics

A conditional approach for multivariate extreme values. (With discussion). (English) Zbl 1046.62051
Summary: Multivariate extreme value theory and methods concern the characterization, estimation and extrapolation of the joint tail of the distribution of a \(d\)-dimensional random variable. Existing approaches are based on limiting arguments in which all components of the variable become large at the same rate. This limit approach is inappropriate when the extreme values of all the variables are unlikely to occur together or when interest is in regions of the support of the joint distribution where only a subset of components is extreme. In practice this restricts existing methods to applications where \(d\) is typically 2 or 3.
Under an assumption about the asymptotic form of the joint distribution of a \(d\)-dimensional random variable conditional on its having an extreme component, we develop an entirely new semiparametric approach which overcomes these existing restrictions and can be applied to problems of any dimension. We demonstrate the performance of our approach and its advantages over existing methods by using theoretical examples and simulation studies. The approach is used to analyse air pollution data and reveals complex extremal dependence behaviour that is consistent with scientific understanding of the process. We find that the dependence structure exhibits marked seasonality, with extremal dependence between some pollutants being significantly greater than the dependence at non-extreme levels.

MSC:
62G32 Statistics of extreme values; tail inference
62H12 Estimation in multivariate analysis
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Arnold B., Conditional Specification of Statistical Models (1999) · Zbl 0932.62001
[2] Besag J., J. R. Statist. Soc. B 36 pp 192– (1974)
[3] Besag J., Statistician 24 pp 179– (1975)
[4] DOI: 10.1111/1467-9876.00177 · Zbl 0944.62110 · doi:10.1111/1467-9876.00177
[5] Bortot P., Biometrika 85 pp 851– (1998)
[6] Brimblecombe P., Air Pollution Reviews pp 1– (2001)
[7] Caperaa P., Biometrika 84 pp 567– (1997)
[8] DOI: 10.1023/A:1009963131610 · Zbl 0972.62030 · doi:10.1023/A:1009963131610
[9] Coles S. G., J. R. Statist. Soc. B 53 pp 377– (1991)
[10] Coles S. G., Appl. Statist. 43 pp 1– (1994)
[11] Colls J., Air Pollution, 2. ed. (2002)
[12] Crowder M. J., J. R. Statist. Soc. B 51 pp 93– (1989)
[13] DOI: 10.1111/1467-9868.00275 · Zbl 0976.62068 · doi:10.1111/1467-9868.00275
[14] Davison A. C., Bootstrap Methods and Their Application (1997) · Zbl 0886.62001
[15] Davison A. C., J. R. Statist. Soc. B 52 pp 393– (1990)
[16] Dekkers A. L. M., Ann. Statist. 17 pp 1833– (1989)
[17] Department of the Environment, Transport and the Regions, The Air Quality Strategy for England, Scotland, Wales and Northern Ireland (2000)
[18] G. Draisma (2000 ) Parametric and semi-parametric methods in extreme value theory .Thesis. Erasmus University Rotterdam, Rotterdam.
[19] G. Draisma, H. Drees, A. Ferreira, and L. Haan (2004 ) Bivariate tail estimation: dependence in asymptotic independence .Bernoulli, 10 , no. 2, in the press. · Zbl 1058.62043
[20] Embrechts P., Modelling Extremal Events (1997) · Zbl 0873.62116
[21] Gumbel E. J., J. Am. Statist. Ass. 55 pp 698– (1960)
[22] Haan L., Z. Wahrsch. Theor. 40 pp 317– (1977)
[23] Haan L., Stochast. Models 9 pp 275– (1993)
[24] DOI: 10.1023/A:1009909800311 · Zbl 0921.62144 · doi:10.1023/A:1009909800311
[25] Hall P., Bernoulli 6 pp 835– (2000)
[26] Hand D., Practical Longitudinal Data Analysis (1996) · Zbl 0885.62002
[27] DOI: 10.1023/A:1011459127975 · Zbl 0979.62040 · doi:10.1023/A:1011459127975
[28] Housley D., Air Pollution Reviews pp 247– (2001)
[29] Joe H., Can. J. Statist. 22 pp 47– (1994)
[30] Joe H., Multivariate Models and Dependence Concepts (1997) · Zbl 0990.62517
[31] Joe H., J. R. Statist. Soc. B 54 pp 171– (1992)
[32] Leadbetter M. R., Extremes and Related Properties of Random Sequences and Series (1983) · Zbl 0518.60021
[33] Ledford A. W., Biometrika 83 pp 169– (1996)
[34] DOI: 10.1111/1467-9868.00080 · doi:10.1111/1467-9868.00080
[35] DOI: 10.1239/aap/1035228000 · Zbl 0905.60034 · doi:10.1239/aap/1035228000
[36] Ledford A. W., J. R. Statist. Soc. B 65 pp 521– (2003)
[37] DOI: 10.1016/S0378-4266(99)00077-1 · doi:10.1016/S0378-4266(99)00077-1
[38] DOI: 10.1239/jap/1037816012 · Zbl 1090.90017 · doi:10.1239/jap/1037816012
[39] DOI: 10.1016/S0167-7152(98)00280-6 · Zbl 0958.62049 · doi:10.1016/S0167-7152(98)00280-6
[40] Photochemical Oxidants Review Group, Fourth Report of the Photochemical Oxidants Review Group (1997)
[41] Pickands J., Ann. Statist. 3 pp 119– (1975)
[42] Pickands J., Proc. 43rd Sess. Int. Statist. Inst. pp 859– (1981)
[43] DOI: 10.1093/rfs/hhg058 · doi:10.1093/rfs/hhg058
[44] Resnick S. I., Extreme Values, Regular Variation, and Point Processes (1987) · Zbl 0633.60001
[45] DOI: 10.1214/aoap/1019487509 · Zbl 1083.60521 · doi:10.1214/aoap/1019487509
[46] Schlather M., Biometrika 90 pp 139– (2003)
[47] Shi D., Technical Report 2074 (1992)
[48] A. K. Sinha (1997 ) Estimating failure probability when failure is rare: multidimensional case .Thesis. Erasmus University Rotterdam, Rotterdam.
[49] Smith R. L., Statist. Sci. 4 pp 367– (1989)
[50] Starica C., J. Emp. Finan. 6 pp 513– (2000)
[51] Tawn J. A., Biometrika 75 pp 397– (1988)
[52] Tawn J. A., Biometrika 77 pp 245– (1990)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.