This paper unifies and extends results on a class of multivariate Extreme Value (EV) models studied by Hougaard, Crowder, and Tawn. In these models both unconditional and conditional distributions are EV, and all lower-dimensional marginals and maxima belong to the class. This leads to substantial economies of understanding, analysis and prediction. One interpretation of the models is as size mixtures of EV distributions, where the mixing is by positive stable distributions. A second interpretation is as exponential-stable location mixtures (for Gumbel) or as power-stable scale mixtures (for non-Gumbel EV distributions). A third interpretation is through a Peaks over Thresholds model with a positive stable intensity. The mixing variables are used as a modeling tool and for better understanding and model checking. We study extreme value analogues of components of variance models, and new time series, spatial, and continuous parameter models for extreme values. The results are applied to data from a pitting corrosion investigation.
Multivariate models for extreme value data are attracting substantial interest, see e.g. Kotz and Nadarajah (2000) and Fougères (2004). However, with the exception of Smith (2004) and Heffernan and Tawn (2004), few applications involving 1 more than two or three dimensions have been reported. One main application area is environmental extremes. Dependence between extreme wind speeds and rain fall can be important for reservoir safety (Anderson and Nadarajah (1993), Ledford and Tawn (1996)), high mean water levels occurring together with extreme waves may cause flooding (Bruun and Tawn (1998), de Haan and de Ronde (1998)), and simultaneous high water levels at different spatial locations pose risks for large floods (Coles and Tawn (1991)). Another set of applications is in economics where multivariate extreme value theory has been used to model the risk that extreme fluctuations of several exchange rates or of prices of several assets, such as stocks, occur together (Mikosch (2004), Smith (2004), Stȃricȃ (1999)). A third use, perhaps somewhat unlikely, is in the theory of rational choice (McFadden (1978)). Below we will also consider a fourth problem, analysis of pitting corrosion measurements (Kowaka (1994), Scarf and Laycock (1994)).
The papers cited above all use multivariate Extreme Value (EV) distributions.
The rationale is the “extreme value argument”: maxima of many individually small variables often have approximately a (univariate or multivariate as the case may be) extreme value distribution. However in “random effects” situations this argument becomes less clear. Suppose e.g. a number of groups each has its own i.i.d variation but in addition each group is affected by some overall random effect. Then, is it the unconditional distributions which belong to the extreme value family, or is it the conditional distribution, given the value of the random effect? In many situations the extreme value argument seems equally compelling for unconditional and conditional distributions. So, should one use an EV model for the conditional distribution; or is it perhaps the unconditional distributions which are extreme value?
In the present paper this problem is overcome by using models where both conditional and unconditional distributions are EV. The models have the further attractive properties that all lower-dimensional marginals belong to the same class of models, and that maxima of all kinds, e.g. over a number of “groups” with differing numbers of elements, also have distributions which belong to the class.
The models are obtained by mixing EV distributions over a positive stable distribution. They were first noted by Watson and Smith (1985) and, in a survival analysis context, apparently independently introduced by Hougaard (1986) and Crowder (1989). Further interesting applications of such models were made in Crowder (1998). The most general versions of these distributions were called the asymmetric logistic distribution and the nested logistic distribution by Tawn (1990) and McFadden (1978) and were further studied in Coles and Tawn (1991). Crowder (1985) and Crowder and Kimber (1997) contain some related material. However, we believe that the full potential of these models is still far from being realized. In this paper we have attempted to take three more steps towards making them more widely useful.
The first step is to revisit the papers of Hougaard, Crowder and Tawn, to collect and solidify the results in these papers. We concentrated on two parts: the physical motivation for the models, and a clear mathematical formulation of the general results. The second step is to use the stable mixing variables not just as a “trick” to obtain multivariate distributions, but as a modeling tool. Insights obtained from taking the mixing variable seriously are new model checking tools, and better understanding of identifiability of parameters and of the model in general.
The final important step is the realization that through suitable choices of the mixing variables it is possible to obtain new natural time series models, spatial models, and continuous parameter models for extreme value data. This provides classes of models for extreme value data which go beyond dimensions two and three.
It is not immediately obvious from the forms of the asymmetric logistic distribution and the nested logistic distribution how to simulate values from them, see e.g. Kotz and Nadarajah (1999, Section 3.7). However the representation as stable mixtures makes simulation straightforward. According to it, one can first simulate the stable variables, using e.g. the method of Chambers et al. (1976), and then simulate independent variables from the conditional distribution given the stable variables, cf. Stephenson (2003). This adds to the usefulness of the models.
Our results can be presented in two closely related ways: as mixture models for Gumbel distributions, and as mixture models for the general family of EV distributions.
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