Polya distribution

Download a pdf copy of this help file here

Format: VosePolya(a, b, U)

Uses

There are several types of distribution in the literature that have been given the Polya name. We employ the name for a distribution that is very common in the insurance field.

A standard initial assumption of the frequency distribution of the number of claims is Poisson:

Number of claims = Poisson(l)

where l is the expected number of claims during the period of interest. The Poisson distribution has a mean and variance equal to l and one often sees historic claim frequencies with a variance greater than the mean so that the Poisson model underestimates the level of randomness of claim numbers. A standard method to incorporate greater variance is to assume that l is itself a random variable (and the claim frequency distribution is then called a mixed Poisson model). Because of its flexibility in shape, and ease of mathematics, a Gamma(a,b) distribution is most commonly used to describe the random variation of l between periods, so:

Claims = Poisson(Gamma(a,b)) (1)

This is the Polya (a,b) distribution.

Comments

For a process where the number of events occur randomly in a unit of time according to a Polya(α, β) distribution, the time between successive events follows a Pareto2(1/ β, α) distribution.

This is equivalent to how in a Poisson process, where the number of events occur randomly in a unit of time according to a Poisson(λ) distribution, the time between successive events follows an Exponential(1/ λ) distribution.

Relationship to the Negative Binomial

If a is an integer, we have:

Claims = Poisson(Gamma(a,b)) = NegBin(a,1/(1+b)) (2)

so one can say that the Negative Binomial distribution is a special case of the Polya .

Another parameterisation of the Polya

Another common actuarial parameterization of the Polya distribution comes from rewriting Equation 1 as follows:

Claims = Poisson(l * Gamma(h,1/h)) (3)

The Gamma(h,1/h) has mean = 1, so the Gamma distribution in Equation 3 is adding random variation about the expected rate l . Using this parameterization with our Polya distribution you would write:

Claims = Polya(h,l/h)

Other common Poisson mixture models

Two other variations of Equation 1 are of interest:

a) Claims = Poisson(Exponential(b))

which is a special case of Equation 1 with a = 1. From Equation 2 and recognizing that NegBin(1,p) = Geometric(p) this simplifies to:

Claims = Geometric(1/(1+b))

b) Claims = Poisson(l + Gamma(a,b))

which is introduced to give more flexibility, e.g. to try to match variance and skewness of claim frequencies as well as the mean rate. The result is a Delaporte distribution:

Poisson(l + Gamma(a,b)) = Delaporte(a,b,l)

We can split this equation up:

Poisson(l + Gamma(a,b)) = Poisson(l) + Poisson(Gamma(a,b))

= Poisson(l) + Polya(a,b)

So the Delaporte distribution models a claims frequency (or any other random countable integer) as having two components: a Poisson variable with a fixed expected rate; and an independent Poisson variable with an expected rate that is itself a random variable.

Use in microbiology

If the number of clumps in a sample are Poisson(λ) distributed and the number of organisms in a random cluster follows a Logarithmic(θ) distribution, then the total number of organisms in the sample follows a Pólya distribution.

Zero-modified version

When modeling or analyzing counting data, it is often desirable to modify probability of zero of the discrete distribution we use, to more accurately model the probability of "no event occurring". We can make two types of modifications to our distribution for this:

Zero-inflated model - we increase the probability of zero. (equations)
Zero-truncated model - we entirely remove the probability of zero events occurring. (equations)

VoseFunctions for this distribution

VosePolya generates values from this distribution or calculates a percentile

VosePolyaObject constructs a distribution object for this distribution

VosePolyaProb returns the probability mass or cumulative distribution function for this distribution

VosePolyaProb10 returns the log10 of the probability mass or cumulative distribution function

VosePolyaFit generates values from this distribution fitted to data, or calculates a percentile from the fitted distribution

VosePolyaFitObject constructs a distribution object of this distribution fitted to data

VosePolyaFitP returns the parameters of this distribution fitted to data