Third Cumulant for Multivariate Aggregate Claim Models

Authors

  • Nicola Loperfido Dipartimento di Economia, Societa e Politica, Universita degli Studi di Urbino "Carlo Bo", Via Sa 42, 61029 Urbino (PU), Italy
  • Stepan Mazur Department of Statistics Lund University School of Economics and Management Box 743, SE-22007 Lund, Sweden
  • Krzysztof Podgórski Department of Statistics Lund University School of Economics and Management Box 743, SE-22007 Lund, Sweden

Keywords:

Third cumulant, multivariate aggregate claim, skew-normal, Laplace motion

Abstract

The third moment cumulant for the aggregated multivariate claims is considered. A formula is presented for the general case when the aggregating variable is independent of the multivariate claims. It is discussed how this result can be used to obtain a formula for the third cumulant for a classical model of  multivariate claims. Two important special cases are considered. In the rst one, multivariate skewed normal claims are considered and aggregated by a Poisson variable. The second case is dealing with multivariate asymmetric generalized Laplace and aggregation is made by a negative binomial variable. Due to the invariance property the latter case can be derived directly leading to the identity involving the cumulant of the claims and the aggregated claims. There is a well established relation between asymmetric Laplace motion and negative binomial process that corresponds to the invariance principle of the aggregating claims for the generalized asymmetric Laplace distribution. We explore this relation and provide multivariate continuous time version of the results.

 

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Section

Working Papers in Statistics