Concomitant tail behaviour for extremes

The influence of bivariate extremal dependence on the limiting behaviour of the concomitant of the largest order statistic is examined. Our approach is to fix the marginal distributions and derive a general tail characterisation of the joint survivor function. From this, we identify the normalisation required to obtain the limiting distribution of the concomitant of the largest order statistic, obtain its tail form, and investigate the limiting probability that the vector of componentwise maxima occurs as an observation of the bivariate process. The results are illustrated for a range of extremal dependence forms.

[1]  J. Angus.  The Asymptotic Theory of Extreme Order Statistics , 1990 .

[2]  M. Ivette Gomes.  Concomitants in a Multidimensional Extreme Model , 1984 .

[3]  L. Haan,et al.  Large quantile estimation in a multivariate setting , 1995 .

[4]  H. A. David.  Concomitants of Extreme Order Statistics , 1994 .

[5]  H. A. David,et al.  Distribution and Expected Value of the Rank of a Concomitant of an Order Statistic , 1977 .

[6]  J. D. T. Oliveira,et al.  The Asymptotic Theory of Extreme Order Statistics , 1979 .

[7]  Shie-Shien Yang,et al.  Concomitants of order statistics , 1976 .

[8]  K. Weintraub.  Sample and Ergodic Properties of Some Min-Stable Processes , 1991 .

[9]  Richard L. Smith.  Approximations in Extreme Value Theory. , 1987 .

[10]  Sidney I. Resnick,et al.  Estimating the limit distribution of multivariate extremes , 1993 .

[11]  A. Ledford,et al.  Statistics for near independence in multivariate extreme values , 1996 .

[12]  S. N. Joshi,et al.  Joint distribution of maxima of concomitants of subsets of order statistics , 1995 .

[13]  Janos Galambos,et al.  The asymptotic theory of concomitants of order statistics , 1974 .

[14]  H. Ruben.  An asymptotic expansion for the multivariate normal distribution and Mills' ratio , 1964 .

[15]  Jonathan A. Tawn,et al.  Modelling Dependence within Joint Tail Regions , 1997 .

[16]  Francis Galton,et al.  Natural inheritance. By Francis Galton. , 1889 .

[17]  H. Joe.  Multivariate extreme value distributions , 1997 .

[18]  H. A. David,et al.  On the dependence structure of order statistics and concomitants of order statistics , 1990 .

[19]  M. Fréchet.  Sur les tableaux de correlation dont les marges sont donnees , 1951 .

[20]  Jonathan A. Tawn,et al.  Bivariate extreme value theory: Models and estimation , 1988 .

[21]  H. A. David,et al.  Distribution of the Maximum of Concomitants of Selected Order Statistics , 1994 .

[22]  Jonathan A. Tawn,et al.  Statistical Methods for Multivariate Extremes: An Application to Structural Design , 1994 .