Download a complete copy of this risk
analysis resource for free here.
|
Format: VoseDelaporte(a, b, l, U)
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:
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:
Events = Poisson(Gamma(a, b)) (1)
This is 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:
Events = Poisson(l + Gamma(a, b)) (2)
This is 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))
The cumulative distribution function for the Delaporte is a double summation equation, which makes using the inversion method for simulating from this distribution computationally large, particularly when the mean = (a*b+l) is large. The ModelRisk function VoseDelaporte(a,b,l,U) therefore uses two approaches to generate values:
If the optional U parameter is omitted, the function samples from a Gamma and Poisson distribution using the identity Delaporte(a,b,l) = Poisson(Gamma(l+a,b))
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.
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)
See also: Zero-modified counting distributions
VoseDelaporte generates values from this distribution or calculates a percentile
VoseDelaporteObject constructs a distribution object for this distribution
VoseDelaporteProb returns the probability density or cumulative distribution function for this distribution
VoseDelaporteProb10 returns the log10 of the probability density 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
*The Delaporte distribution was first studied in Delaporte (1959).
