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Third Cumulant for Multivariate Aggregate Claim Models

Loperfido, Nicola; Mazur, Stepan LU and Podgorski, Krzysztof LU (2015) In Working Papers in Statistics
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... (More)
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. (Less)
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author
organization
publishing date
type
Working Paper
publication status
published
subject
keywords
Third cumulant, multivariate aggregate claim, skew-normal, Laplace motion
in
Working Papers in Statistics
issue
13
pages
30 pages
publisher
Department of Statistics, Lund university
language
English
LU publication?
yes
id
2eb47fca-f6d5-45cd-bdb5-af331591ce6c (old id 8310038)
date added to LUP
2015-12-09 16:36:51
date last changed
2016-04-16 10:03:20
@misc{2eb47fca-f6d5-45cd-bdb5-af331591ce6c,
  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.},
  author       = {Loperfido, Nicola and Mazur, Stepan and Podgorski, Krzysztof},
  keyword      = {Third cumulant,multivariate aggregate claim,skew-normal,Laplace motion},
  language     = {eng},
  number       = {13},
  pages        = {30},
  publisher    = {ARRAY(0x9581680)},
  series       = {Working Papers in Statistics},
  title        = {Third Cumulant for Multivariate Aggregate Claim Models},
  year         = {2015},
}