Statistics & Probability Letters 17 (1993) 361-368 North-Holland 5 August 1993 Multivariate distributions with generalized Pareto conditionals Barry C. Arnold zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA University of California, Riverside, USA Enrique Castillo and Jose Maria Sarabia Universidad de Cantabria, Santander, Spain Received July 1992 Revised December 1992 Abstract: Two classes of k-dimensional distributions with generalized Pareto conditionals are characterized. This subsumes and extends earlier work on distributions with Pareto conditionals. zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFE 1. introduction The selection of models of multidimensional socio-economic variables is a nontrivial exercise. One popular approach is to take logarithms of the variables and assume a multivariate normal model. The whole armory of multivariate normal methodology becomes immediately available. In many situations however such assumptions of log-normal marginal distributions are clearly not based on theoretical insights, they are mathematical conveniences bolstered by a vaguely formulated belief that, thanks in some way to the central limit theorem, they will provide at least approximately appropriate analyses. Equally strong and sometimes stronger arguments can be marshalled in favor of models with Pareto Marginal distributions. The first paper discussing multivariate distributions with Pareto marginals was Mardia (1962). A survey of models with Pareto marginals may be found in Arnold (1983, Chapters 2 and 3). Arnold, Castillo and Sarabia (1992) call into question the process of marginal specification of multivariate distributions, preferring instead conditional specification. If we focus on a particular multivariate economic variable, for example income from several sources, it is not at all clear that we will be able to confidently visualize marginal features of the distribution. Rather we might be able to speculate that for given levels of income from sources 2, 3,. . . , k the income from source 1 will have a Pareto-like distribution with parameters perhaps depending on the levels of income from the other sources. Similarly income from source j, given the levels of income from other sources might have a Pareto-like distribution. In Arnold, Castillo and Sarabia (1992) an array of conditionally specified models was presented and analysed. Special emphasis was put on the relatively tractable conditionals in exponential families case. In that monograph and earlier in Arnold (1987) some attention was paid to Pareto conditionals models. Correspondence to: Barry C. Arnold, Department of Statistics, University of California, Riverside, CA 92521-0138, USA. 0167-7152/93/$06.00 0 1993 - Elsevier Science Publishers B.V. All rights reserved 361