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Portrait of Krzysztof Podgórski. Photo.

Krzysztof Podgórski

Professor, Head of the Department of Statistics

Portrait of Krzysztof Podgórski. Photo.

Third cumulant for multivariate aggregate claim models

Author

  • Nicola Loperfido
  • Stepan Mazur
  • Krzysztof Podgórski

Summary, in English

The third 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. Two important special cases are considered. In the first 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. It is discussed how these results that deal only with dependence in the claim sizes can be used to obtain a formula for the third cumulant for more complex aggregate models of multivariate claims in which the dependence is also in the aggregating variables.

Department/s

  • Department of Statistics

Publishing year

2018-02-07

Language

English

Pages

109-128

Publication/Series

Scandinavian Actuarial Journal

Volume

2018

Issue

2

Document type

Journal article

Publisher

Taylor & Francis

Topic

  • Probability Theory and Statistics

Keywords

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

Status

Published

ISBN/ISSN/Other

  • ISSN: 0346-1238