Abstract
Bivariate extreme random pairs, with extreme margins, have for the distribution function, in the case of maxima, or for the survival function, in the case of minima, a dependence function. For the cases of Gumbel margins for maxima or of the exponential margins for minima an index of dependence as well the correlation coefficient are obtained; analogous results could be obtained for other margins and also study the non-parametric correlation coefficients.
Parametric models either for the cases where a density in IR2 does exist (differentiable models) or does not (non-differentiable models) are considered and some statistical procedures are proposed.
Finally a reference to the intrinsic estimation of the dependence function (i.e., verifying some conditions and, thus, with restrictions on its coefficients) is made and some difficulties of the usual technique are referred.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
B.V. Finkelshteyn (1953) - Limiting distribution of extremes of a variational series of a two dimensional random variable, Dokl. Ak. Nauk. S.S.S.R„ vol. 91 (in russian), 209–211.
R.A. Fisher and L.H.C. Tippet (1928) - Limiting forms of frequency distribution of largest and smallest members of a sample, Proc. Cambr. Phil. Soc., vol. 24, pt. 2, 180–190.
M. Fréchet (1927) - Sur la loi de probabilité de l’écart maximum, Ann. Soc. Polon. Math. (Krakov), vol. 6, 93–116.
J. Galambos (1968) - The Asymptotic Theory of Extreme Order Statistics, J. Wiley, N. Y.
J. Geffroy (1958–59) - Contributions à la théorie des valeurs extrêmes, Thése de doctorat, Publ. Inst. Stat. Paris, vol 7/8, 37–185.
B.V. Gnedenko (1943) - Sur la distribution limite du terme maximum d’une série aléatoire, An. Math., vol. 44, 423–433.
E.J. Gumbel (1935) - Les valeurs extrêmes des distributions statistiques, Ann. Inst. H.Poincaré, vol. V, 115–158.
E.J. Gumbel (1958) Statistics of Extremes, Columbia University Press.
E.J. Gumbel (1961) - Multivariate extremal distributions, Bull. Int. Stat. Inst., 33e sess., 2e livr., Paris, 191–193.
E.J. Gumbel (1961’) - Bivariate logistic distribution, J. Amer Stat. Assoc., vol. 56, # 307, 194–816.
E. J. Gumbel and Neil Goldstein (1964) - Analysis of empirical bivariate Extremal distributions, J. Amer. Stat. Assoc., vol. 59, 794–816.
K. Joag - Dev (1983) - Independence via uncorrelatedness under certain dependence conditions, Ann. Prob., vol. 11, 1037–1041.
Albert W. Marshall and Ingram Olkin (1967) - A multivariate exponential distribution, J. Amer. Stat Assoc., vol. 62, 30–44.
Albert W. Marshall and Ingram Olkin (1983) - Domains of attraction of multivariate extreme value distributions, Ann. Prob., vol. 11, 168–177.
R. von Mises (1935) - La distribution de la plus grande de n valeurs, Rev. Math. Union Interbalkanique, vol. I, 141–160.
J. Pickands III (1981) - Multivariate extreme value theory, Bull. Int. Stat. Inst., 49th session I.S.I., Buenos Aires, 859–878.
M. Sibuya (1960) - Bivariate extremal statistics I Ann. Inst. Stat. Math. vol. XI, 195–210.
J.A. Tawn (1987) - Bivariate extreme value theory - models and estimation, Techn. Rep. n0 57, Dep. of Math., University of Surrey.
J. Tiago de Oliveira (1958) - Extremal distributions, Rev. Fac. Ciências Lisboa 2 ser., A, Mat., vol. VIII, 299–310.
J. Tiago de Oliveira (1962–63) - Structure theory of bivariate extremes; extensions, Estudos Mat.. e Econom., vol. VII, 165–195.
J. Tiago de Oliveira (1965) - Statistical decision for bivariate extremes, Portug. Math. vol. 24, 145–154.
J. Tiago de Oliveira (1968) - Extremal processes; definition and properties, Publ. Inst. Stat. Univ. Paris, vol. XVII, 25–36.
J. Tiago de Oliveira (1970) - Biextremal distributions: statistical decision, Trab. Estad. y Inv. Oper., vol. XXI, Madrid, 107–117.
J. Tiago de Oliveira (1971) - A new model of bivariate extremes: statistical decision, Studi di Probabilitè. Statistica e Ricerca Operativa in Onore di Giuseppe Pompilj, Tip. Oderisi, Gubbio, Italia, 1–13.
J. Tiago de Oliveira (1974) - Regression in the non-differentiable bivariate models J. Amer. Stat. Assoc., vol. 69, 816–818.
J. Tiago de Oliveira (1975) - Bivariate and multivariate extreme distributions Statistical Distributions in Scientific Work, G.P. Patil, S. Kotz and J.K. Ord. eds., vol. 1, D. Reidell Publ. Co, 355–361.
J. Tiago de Oliveira (1975’) - Statistical decision for extremes, Trab. Estad. y Inv.Oper., vol. XXVI, Madrid, 453–471.
J. Tiago de Oliveira (1980) - Bivariate extremes: foundations and statistics, Proc. Int. Symp. Multiv. Analysis, P.R. Krishnaiah, ed., North Holland, 349–366.
J. Tiago de Oliveira (1982) - Decision and modelling for extremes, Some Recent Advances in Statistics, J. Tiago de Oliveira and B. Epstein eds., Academic Press, London, 101–110.
J. Tiago de Oliveira (1984) - Bivariate models for extremes; statistical decision, Statistical Extremes and Applications, J. Tiago de Oliveira ed., D. Reidel and Co, 131–153.
J. Tiago de Oliveira (1985) - Statistical Choice of non-separate models, Trab. Estad. y Inv. Oper. (Madrid), vol. 36, 136–152.
J. Tiago de Oliveira (1987) - Comparaison entre les modèles bivariées logistique et naturel pour les maxima et extensions, C.R. Acad. Sc. Paris, 1.305, ser. I, 481–484.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1989 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Tiago de Oliveira, J. (1989). Statistical Decision for Bivariate Extremes. In: Hüsler, J., Reiss, RD. (eds) Extreme Value Theory. Lecture Notes in Statistics, vol 51. Springer, New York, NY. https://doi.org/10.1007/978-1-4612-3634-4_21
Download citation
DOI: https://doi.org/10.1007/978-1-4612-3634-4_21
Publisher Name: Springer, New York, NY
Print ISBN: 978-0-387-96954-1
Online ISBN: 978-1-4612-3634-4
eBook Packages: Springer Book Archive