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Third cumulant for multivariate aggregate claim models

Loperfido, Nicola; Mazur, Stepan LU and Podgórski, Krzysztof LU (2017) In Scandinavian Actuarial Journal
Abstract

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... (More)

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.

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Please use this url to cite or link to this publication:
author
organization
publishing date
type
Contribution to journal
publication status
epub
subject
keywords
Laplace motion, multivariate aggregate claim, skew-normal, Third cumulant
in
Scandinavian Actuarial Journal
pages
20 pages
publisher
Taylor & Francis
external identifiers
  • scopus:85016308730
ISSN
0346-1238
DOI
10.1080/03461238.2017.1306795
language
English
LU publication?
yes
id
2562b23f-6c27-4fbe-8a31-193d10538871
date added to LUP
2017-04-12 14:18:41
date last changed
2017-08-24 14:06:55
@article{2562b23f-6c27-4fbe-8a31-193d10538871,
  abstract     = {<p>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.</p>},
  author       = {Loperfido, Nicola and Mazur, Stepan and Podgórski, Krzysztof},
  issn         = {0346-1238},
  keyword      = {Laplace motion,multivariate aggregate claim,skew-normal,Third cumulant},
  language     = {eng},
  month        = {03},
  pages        = {20},
  publisher    = {Taylor & Francis},
  series       = {Scandinavian Actuarial Journal},
  title        = {Third cumulant for multivariate aggregate claim models},
  url          = {http://dx.doi.org/10.1080/03461238.2017.1306795},
  year         = {2017},
}