Compound mixed Poisson distribution

Let the random sum $Y=X_1+X_2+ \cdots +Y_N$ be the aggregate claims generated in a fixed period by an independent group of insureds. When the number of claims $N$ follows a Poisson distribution, the sum $Y$ is said to have a compound Poisson distribution. When the number of claims $N$ has a mixed Poisson distribution, the sum $Y$ is said to have a compound mixed Poisson distribution. A mixed Poisson distribution is a Poisson random variable $N$ such that the Poisson parameter $\Lambda$ is uncertain. In other words, $N$ is a mixture of a family of Poisson distributions $N(\Lambda)$ and the random variable $\Lambda$ specifies the mixing weights. In this post, we present several basic properties of compound mixed Poisson distributions. In a previous post (Compound negative binomial distribution), we showed that the compound negative binomial distribution is an example of a compound mixed Poisson distribution (with gamma mixing weights).

In terms of notation, we have:

• $Y=X_1+X_2+ \cdots +Y_N$,
• $N \sim$ Poisson$(\Lambda)$,
• $\Lambda \sim$ some unspecified distribution.

The following presents basic proeprties of the compound mixed Poisson $Y$ in terms of the mixing weights $\Lambda$ and the claim amount random variable $X$.

Mean and Variance

$\displaystyle E[Y]=E[\Lambda] E[X]$

$\displaystyle Var[Y]=E[\Lambda] E[X^2]+Var[\Lambda] E[X]^2$

Moment Generating Function

$\displaystyle M_Y(t)=M_{\Lambda}[M_X(t)-1]$

Cumulant Generating Function

$\displaystyle \Psi_Y(t)=ln M_{\Lambda}[M_X(t)-1]=\Psi_{\Lambda}[M_X(t)-1]$

Measure of Skewness
$\displaystyle E[(Y-\mu_Y)^3]=\Psi_Y^{(3)}(0)$

$\displaystyle =\Psi_{\Lambda}^{(3)}(0) E[X]^3 + 3 \Psi_{\Lambda}^{(2)}(0) E[X] E[X^2]+\Psi_{\Lambda}^{(1)}(0) E[X^3]$

$\displaystyle =\gamma_{\Lambda} Var[\Lambda]^{\frac{3}{2}} E[X]^3 + 3 Var[\Lambda] E[X] E[X^2]+E[\Lambda] E[X^3]$

Measure of skewness: $\displaystyle \gamma_Y=\frac{E[(Y-\mu_Y)^3]}{(Var[Y])^{\frac{3}{2}}}$

Previous Posts on Compound Distributions