ON A MULTIVARIATE PARETO DISTRIBUTION 1 Alexandru V. Asimit School of Mathematics, University of Manchester, Manchester M13 9PL, United Kingdom. E-mail: [email protected] Edward Furman2; 3 Department of Mathematics and Statistics, York University, Toronto, Ontario M3J 1P3, Canada. E-mail: [email protected] Raluca Vernic Faculty of Mathematics and Informatics, Ovidius University of Constanta, Constanta, Romania. E-mail: [email protected] Abstract. A multivariate distribution possessing arbitrarily parameter- ized Pareto margins is formulated and studied. The distribution is believed to allow for an adequate modeling of dependent heavy tailed risks with a non-zero probability of simultaneous loss. Numerous links to certain nowa- days existing probabilistic models, as well as seemingly useful character- istic results are proved. Expressions for, e.g., decumulative distribution functions, densities, (joint) moments and regressions are developed. An application to the classical pricing problem is considered, and some formu- las are derived using the recently introduced economic weighted premium calculation principles. Keywords and phrases: Multivariate Pareto distributions, characteriza- tions, mixtures, dependence, simultaneous loss, economic weighted pricing. Mathematics Subject Classi¯cation: IM01, IM10, IM54. 1We are indebted to Professors Gheoghita Zbaganu (University of Bucharest, Romania) and Ri·cardas Zitikis (University of Western Ontario, Canada) for many illuminating discussions, and we also sincerely thank the anonymous referee for helpful comments and suggestions. 2The ¯nancial support of Natural Sciences and Engineering Research Council (NSERC) of Canada is gratefully appreciated. 3Corresponding author. 2 1. Introduction Pareto probability laws were formulated by Pareto (1897) and initially dealt with the distribution of income over a population. They are nowadays popular in describing a variety of other phenomena such as city population sizes, occurrence of natural resources, stock price fluctuations, to name a few. In actuarial mathematics, the excess-of-loss random variable (rv) Y = X ¡d given that X > d, possesses the decumulative distribution function (ddf) F X;d(x) = P[X ¡ d > xj X > d]; (1.1) and it is of central importance in both life and non-life insurance contexts. Loosely speaking, ddf (1.1) on the one hand describes a risk inherent in a reinsurance treaty bearing a deductible d, and on the other hand it corresponds to the age-at-death of a new born child given survival to age d. Interestingly, it can be shown that for large d, appropriately chosen function σ(d) > 0 and a constant ® > 0, the approximation µ ¶ x ¡® F (x) ¼ 1 + ; x > 0 (1.2) X;d σ(d) holds for a large class of distribution functions (see, e.g., Balkema and de Haan, 1974; Pickands, 1975). Thus, the excess-of-loss rv can be generally approximated by a Pareto of the second kind. This, in turn, immediately hints at utilizing this probability law for modeling, e.g., catastrophe insurance treaties, which are known to be characterized by rather heavy tailed risks. In view of the high popularity of the univariate Pareto distributions and the aforemen- tioned implication of equation (1.2), it is quite natural to ask for a multivariate extension. Indeed, the concept of dependence, unfairly neglected in the `classical' actuarial science, has been receiving its merited attention in the recent years, thus leading to numerous papers touching on various aspects of the multivariate distribution theory with applica- tions to insurance (see, e.g., Vernic, 1997, 2000; Pfeifer and Ne·slehov¶a,2004; Bauerle and Grubel, 2005; Roger et al., 2005; Centeno, 2005; Furman, 2008; Furman and Landsman, 2008, 2009; Chiragiev and Landsman, 2009). Recently, probably due to the observation formulated by equation (1.2), the multivariate Pareto distribution of the second kind having the ddf à !¡® Xn x ¡ ¹ F (x ; : : : ; x ) = 1 + j j ; x > ¹ ; (1.3) 1 n σ j j j=1 j 3 where ¹j 2 (¡1; 1) := R, σj 2 (0; 1) := R+ and j = 1; 2; : : : ; n, has been employed to model insurance risks in the context of risk capital allocations and optimal reinsur- ance retentions by Chiragiev and Landsman (2007) (see, also, Vernic, 2009) and Cai and Tan (2007), respectively. In what follows, we denote the random vectors distributed ¤ multivariate Pareto of the second kind with ddf (1.3) as X v P an(II)(¹; σ; ®), where 0 n n 0 n n ¹ = (¹1; : : : ; ¹n) 2 (¡1; 1) := R and σ = (σ1; : : : ; σn) 2 (0; 1) := R+ are two constant vectors, and ® is a positive constant. Although the model seems to be quite tractable, it inconveniently results in common shape parameters for all univariate mar- gins Xj; j = 1; 2; : : : ; n, and thus imposes the somewhat restrictive correlation coe±cient given by 1 Corr[X ;X ] = ; for all j 6= l 2 f1; 2; : : : ; ng and ® > 2: j l ® Another well-known inconvenience associated with (1.3) is that it does not allow to model independent Pareto distributed risks. In the present paper we propose an alternative to (1.3) multivariate generalization of the univariate Pareto probability laws. The probabilistic model introduced herein is constructed using the multivariate reduction method (see, e.g., Furman and Landsman, 2009; and references therein). The motivation for, as well as the interpretation of the aforementioned construction stem from, e.g., the background economy and the common shock models (see, e.g., Tsanakas, 2008; Boucher et al., 2008). The resulting multivariate Pareto, denoted in the sequel as X v P an(II)(¹; σ; ®; ®), with ¹, σ and ® as before, 0 n and ® = (®1; : : : ; ®n) 2 R+, is marginally closed, allows for non-negative probabilities of simultaneous losses and possesses a more flexible than (1.3) dependence structure. In addition, setting ® ´ 0 yields a probabilistic model having independent Pareto distributed margins. The rest of the paper is organized as follows: In Section 2 the multivariate Pareto of interest is introduced, and its various properties are derived. In Section 3 the discussion is specialized to the bivariate case, for which the (joint) moments, the conditional dis- tributions along with the conditional moments are developed. Finally, in Section 4 it is shown that the regression function of the multivariate Pareto introduced herein is `sepa- rable', which is then employed to facilitate its applications to insurance pricing. Section 5 concludes the paper. 4 2. The multivariate Pareto of the second kind In the sequel, we ¯x the probability space (­; F; P), and we are interested in construct- 0 ing a random vector X = (X1;:::;Xn) , which is a map from the aformentioned space n n into the n-dimensional Borel space (R+; B ), such that the j-th coordinate, j = 1; : : : ; n, of X is a univariate Pareto of the second kind. In other words, Xj v P a(II)(¹j; σj; ®j), and µ ¶¡®j x ¡ ¹j F Xj (x) = 1 + ; x > ¹j; and σj; ®j > 0: σj 0 To this end, let Y = (Y0;Y1;:::;Yn) be an (n + 1) variate random vector possess- ing mutually independent univariate Pareto margins Y0 v P a(II)(0; 1; ®0) and Yj v n+1 n P a(II)(¹j; σj; ®j); j = 1; : : : ; n. Denote by T the functional map T : R+ ! R+. De¯nition 2.1. The random vector X = T (Y), such that Xj = min (σjY0 + ¹j;Yj) ; j = 1; : : : ; n; is said to follow the multivariate Pareto distribution of the second kind P an (II)(¹; σ; ®; ®0), n n with ¹ and σ, ® being constant vectors in R and R+, respectively, and ®0 > 0. Note 2.1. It is possible to formulate the de¯nition above for a slightly more general random vector Y by assuming Y0 v P a(II)(¹0; σ0; ®0), which is a non-standardized Pareto n+1 n of the second kind. In such a case, the map T : R+ ! R+ is such that µ ¶ σj Xj = min (Y0 ¡ ¹0) + ¹j;Yj ; j = 1; : : : ; n: σ0 However, as it is readily observed from Proposition 2.1, neither ¹0 nor σ0 influences the ddf of X, and they are hence set to ¹0 ´ 0 and σ0 ´ 1 without loss of generality. We note in passing that various particular cases of the just de¯ned multivariate Pareto have been studied. Namely, when ¹j = σj ´ const for j = 1; : : : ; n, the model is called the Marshall-Olkin type (see, Marshall and Olkin, 1967) multivariate Pareto in Kotz et al. (2000), and it has been explored by Muliere and Scarsini (1987) in the two dimensional, and by Hanagal (1996) in the multidimensional contexts. We further derive the multivariate ddf of the distribution de¯ned in De¯nition 2.1. To enhance the readability of the paper, we relegate the proof of the next proposition, as well as other technical proofs to the Appendix. 5 Proposition 2.1. Let X v P an(II)(¹; σ; ®; ®0). The ddf of X is then µ ¶ µ ¶ ¡®0 Yn ¡®j ¹ xj ¡ ¹j xj ¡ ¹j FX (x) = 1 + max 1 + ; xj > ¹j; j = 1; : : : ; n; (2.1) j=1;:::;n σ σ j j=1 j where ¹j are real, and σj; ®j; ®0 are positive constants. Note 2.2. It is straightforwardly observed that ddf (2.1) is invariant under linear trans- formations, and it thus belongs to a (multivariate) location-scale family of distributions. Consequently, in the sequel, we shall frequently use the so-called standardized form of the ddf, which certainly does not reduce the level of generality, but strongly contributes to the clarity of the results. More speci¯cally, we shall often consider the multivariate Pareto Z v P an(II)(0; 1; ®; ®0), with ¹ = 0 and σ = 1 being n-variate constant vectors of zeros and ones, respectively, and the ddf µ ¶ ¡®0 Yn ¹ ¡®j FZ (z1; : : : ; zn) = 1 + max zj (1 + zj) ; zj > 0; j = 1; : : : ; n: (2.2) j=1;:::;n j=1 It seems worth noticing that there exist alternative to De¯nition 2.1 ways to arrive at ddf (2.1).