Collective model

A collective model is a pair of two random variables with great application in actuarial science .

definition

Let be a random variable with and a sequence of real random variables, then the pair is called a collective model . ${\displaystyle N}$ ${\displaystyle P(N\in \mathbb {N} _{0})=1}$ ${\displaystyle (X_{n})_{n\in \mathbb {N} }}$ ${\displaystyle \langle N,(X_{n})_{n\in \mathbb {N} _{0}}\rangle }$

interpretation

One possible interpretation is very important in non-life actuarial mathematics when considering a homogeneous set of risks. This is interpreted as the number of all damages that have occurred in a period and as the amount of damage caused by the -th damage. ${\displaystyle N}$ ${\displaystyle X_{i}}$ ${\displaystyle i}$

However, caution is advised when using it in practice, since all random variables are assumed to be independently distributed, which does not always have to be the case in practice.

The collective model is a generalization of the individual model.

Furthermore, it makes sense in actuarial mathematics to define a total loss : ${\displaystyle S:\Omega \rightarrow \mathbb {R} }$

${\displaystyle S:=\sum _{n=0}^{\infty }\chi _{\{\omega \in \Omega |N(\omega )=n\}}\sum _{i=1}^{n}X_{i}}$

${\displaystyle S}$itself is then again a random variable, which is described by the collective model on which it is based. One is then often interested in certain properties of such as the expected value or the variance . ${\displaystyle S}$

The total damage can be calculated recursively using the Panjer algorithm . ${\displaystyle S}$

literature

• Schmidt, Klaus D .: Actuarial Mathematics , Springer Dordrecht Heidelberg London New York 2009, ISBN 978-3-642-01175-7

See also

• Panjer algorithm