# Frequency Distributions

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.

## Interpretation of the Coefficient of Variation of the Mixing Distribution

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.

## Gamma Mixing

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 $$M_G(\zeta) = (1-\theta\zeta)^{-a}$$.

## Shifted Mixing (General)

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

## Delaporte Mixing (Shifted Gamma)

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

Other related frequency distributions include

• Poisson Inverse Gaussian Distribution
• Bernoulli Distribution
• Binomial Distribution
• Fixed Distribution

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