Delaporte distribution | Vose Software

Delaporte distribution

Format: Delaporte(a, b, l)

Uses

A very common starting point for modeling the numbers of events that occur randomly distributed in time and/or space (e.g. the number of claims that will be received by an insurance company) is the Poisson distribution:

Number of events = Poisson(l)

where l is the expected number of events during the period of interest. The Poisson distribution has a mean and variance equal to l and one often sees historic data (e.g. frequency of insurance claims) with a variance greater than the mean so that the Poisson model underestimates the level of randomness. A standard method to incorporate greater variance is to assume that l is itself a random variable (and the resultant frequency distribution is called a mixed Poisson model). A Gamma(a,b) distribution is most commonly used to describe the random variation of l between periods, so:

Number of events = Poisson(Gamma(a, b))                     (1)

This is the same as the Polya(a, b) distribution.

Alternatively, one might consider that some part of the Poisson intensity is constant and has an additional component that is random, following a Gamma distribution:

Number of events = Poisson(l + Gamma(a, b))               (2)

This is thesame as the Delaporte distribution, i.e:

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)

Special cases of the Delaporte distribution

Delaporte(l,a,0) = Poisson(l)

Delaporte(0,a,b) = Polya(a,b)

Delaporte(0,1,b) = Geometric(1/(1+b))

Generating values from the Delaporte distribution

The cumulative distribution function for the Delaporte is a double summation equation (see below), which makes using the inversion method for simulating from this distribution computationally burdensome, particularly when the mean = (a*b+l) is large. The ModelRisk function VoseDelaporte(a,b,l,U) therefore uses two approaches to generate values:

1. If the optional U parameter is omitted, the function samples from a Gamma and Poisson distribution using the identity Delaporte(a,b,l) = Poisson(l+Gamma(a,b))

2. If the U parameter is included, the function performs the double summation to construct the cumulative distribution function and applies the inversion method.

Thus, unless you wish to control the sampling of the Delaporte distribution we recommend omitting the U parameter to increase simulation speed.

Zero-modified versions

When modeling or analyzing counting data, it is often desirable to modify the 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.

• Zero-truncated model - we entirely remove the probability of zero events occurring.

ModelRisk functions added to Microsoft Excel for the Delaporte distribution

VoseDelaporte generates random values from this distribution for Monte Carlo simulation, or calculates a percentile if used with a U parameter.

VoseDelaporteObject constructs a distribution object for this distribution.

VoseDelaporteProb returns the probability mass or cumulative distribution function for this distribution.

VoseDelaporteProb10 returns the log10 of the probability mass or cumulative distribution function.

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

VoseDelaporteFitObject constructs a distribution object of this distribution fitted to data.

VoseDelaporteFitP returns the parameters of this distribution fitted to data.

ModelRisk functions added to Microsoft Excel for the Zero-Inflated Delaporte distribution

VoseZIDelaporte generates random values from this distribution for Monte Carlo simulation, or calculates a percentile if used with a U parameter.

VoseZIDelaporteObject constructs a distribution object for this distribution.

VoseZIDelaporteProb returns the probability mass or cumulative distribution function for this distribution.

VoseZIDelaporteProb10 returns the log10 of the probability mass or cumulative distribution function.

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

VoseZIDelaporteFitObject constructs a distribution object of this distribution fitted to data.

VoseZIDelaporteFitP returns the parameters of this distribution fitted to data.

ModelRisk functions added to Microsoft Excel for the Zero-Truncated Delaporte distribution

VoseZTDelaporte generates random values from this distribution for Monte Carlo simulation, or calculates a percentile if used with a U parameter.

VoseZTDelaporteObject constructs a distribution object for this distribution.

VoseZTDelaporteProb returns the probability mass or cumulative distribution function for this distribution.

VoseZTDelaporteProb10 returns the log10 of the probability mass or cumulative distribution function.

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

VoseZTDelaporteFitObject constructs a distribution object of this distribution fitted to data.

VoseZTDelaporteFitP returns the parameters of this distribution fitted to data.