Frequency Distributions

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 \(\mathsf{E}(G)=1\). Let \(\mathsf{Var}(G)=c\) (Glenn Meyers calls \(c\) the contagion) and let \(\mathsf{E}(G^3)=g\).

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

We can also assume \(G\) has mean \(n\) and work directly with \(G\) rather than \(nG\), \(\mathsf{E}(G)=1\). 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 \(x^{a-1}e^{x/\theta}\), \(\mathsf{E}(G)=a\theta\) and \(\mathsf{Var}(G)=a\theta^2\).

Let \(c=\mathsf{Var}(G)=\nu^2\), 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 \(\mathsf{E}(G^r)=\theta^r\Gamma(a+r)/\Gamma(a)\). 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 \[ \mathsf{skew}(N) = \frac{n^3(\gamma -3c -1) + n^2(3c+2) + n}{(n(1+cn))^{3/2}}. \]

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

• \(\mathsf E(G)= f + \mathsf E(G') = 1\) and
• \(\mathsf{CV}(G) = \mathsf{SD}(G') = \nu\).

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

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

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 \(\mathsf E(G')=1-f\) and \(\mathsf{SD}(G')=\nu\) 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