A random variable \(N\) is **\(G\)-mixed Poisson** if \(N\mid G\) has a Poisson \(nG\) distribution for some fixed non-negative \(n\) and a non-negative mixing distribution \(G\) with \(\text{E}(G)=1\). Let \(\text{Var}(G)=c\) (Glenn Meyers calls \(c\) the contagion) and let \(\text{E}(G^3)=g\).

The MGF of a \(G\)-mixed Poisson is \[\label{mgfi} M_N(\zeta)=\text{E}(e^{\zeta N})=\text{E}(\text{E}(e^{\zeta N} \mid G))=\text{E}(e^{n G(e^\zeta-1)})=M_G(n(e^\zeta-1)) \] since \(M_G(\zeta):=\text{E}(e^{\zeta G})\) and the MGF of a Poisson with mean \(n\) is \(\exp(n(e^\zeta-1))\). Thus \[ \text{E}(N)=M_N'(0)=n M_G'(0)=n, \] because \(\text{E}(G)=M_G'(0)=1\). Similarly \[ \text{E}(N^2)=M_N''(0)=n^2M_G''(0)+n M_G'(0)=n^2(1+c)+n \] and so \[ \text{Var}(N)=n(1+cn). \] Finally \[\begin{align*} \text{E}(N^3) &= M_N'''(0) =n^3M_G'''(0)+3n^2M_G''(0)+n M_G'(0) \\ &= gn^3 + 3n^2(1+c) + n \end{align*}\] and therefore the central moment \[ \text{E}(N-\text{E}(N))^3 = n^3(g -3c -1) + 3cn^2 + n. \]

We can also assume \(G\) has mean \(n\) and work directly with \(G\) rather than \(nG\), \(\text{E}(G)=1\). We will call both forms mixing distributions.

Per Actuarial Geometry, if \(\nu\) is the CV of \(G\) then the \(\nu\) equals the asymptotic coefficient of variation for any \(G\)-mixed compound Poisson distribution whose variance exists. The variance will exist iff the variance of the severity term exists.

A negative binomial is a gamma-mixed Poisson: if \(N \mid G\) is distributed as a Poisson with mean \(G\), and \(G\) has a gamma distribution, then the unconditional distribution of \(N\) is a negative binomial. A gamma distribution has a shape parameter \(a\) and a scale parameter \(\theta\) so that the density is proportional to \(x^{a-1}e^{x/\theta}\), \(\text{E}(G)=a\theta\) and \(\text{Var}(G)=a\theta^2\).

Let \(c=\text{Var}(G)=\nu^2\), so \(\nu\) is the coefficient of variation of the mixing distribution. Then

- \(a\theta=1\) and \(a\theta^2=c\)
- \(\theta=c=\nu^2\), \(a=1/c\)

The non-central moments of the gamma distribution are \(\text{E}(G^r)=\theta^r\Gamma(a+r)/\Gamma(a)\). Therefore \(Var(G) = a\theta^2\) and \(E(G-E(G))^3 = 2a\theta^3\). The skewness of \(G\) is \(\gamma = 2/\sqrt(a) = 2\nu\).

Applying the general formula for the third central moment of \(N\) we get an expression for the skewness \[ \text{skew}(N) = \frac{n^3(\gamma -3c -1) + n^2(3c+2) + n}{(n(1+cn))^{3/2}}. \]

The corresponding MGF of the gamma is \(M_G(\zeta) = (1-\theta\zeta)^{-a}\).

We can adjust the skewness of mixing with shifting. In addition to a target CV \(\nu\) assume a proportion \(f\) of claims are sure to occur. Use a mixing distribution \(G=f+G'\) such that

- \(E(G)= f + E(G') = 1\) and
- \(CV(G) = SD(G') = \nu\).

As \(f\) increases from 0 to 1 the skewness of \(G\) will increase. Delaporte first introduced this idea.

Since \(skew(G)=skew(G')\) we have \(g=\mathsf{E}(G^3)=\nu^3 skew(G')+3c+1\).

Inputs are target CV \(\nu\) and proportion of certain claims \(f\), \(0\leq f \leq 1\). Find parameters \(f\), \(a\) and \(\theta\) for a shifted gamma \(G=f+G'\) with \(E(G')=1-f\) and \(SD(G')=\nu\) as

- \(f\) is input
- mean \(a\theta=1-s\) and \(CV=\nu=\sqrt{a}\theta\) so \(a=(1-f)^2/\nu^2=(1-f)^2/c\) and \(\theta=(1-f)/a\)

The skewness of \(G\) equals the skewness of \(G'\) equals \(2/\sqrt{a}= 2\nu/(1-f)\), which is then greater than the skewness \(2\nu\) when \(f=0\). The third non-central moment \(g=2\nu^4/(1-f)+3c+1\)

Other related frequency distributions include

- Poisson Inverse Gaussian Distribution
- Bernoulli Distribution
- Binomial Distribution
- Fixed Distribution

posted 2018-09-15 | tags: probability, distribution, frequency, mixed

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