Frequency


Frequency distribution is defined at the Line level, as an "accident" or "occurrence" frequency.  Before simuation, frequencies from different Lines are correlated to form second-level distributions. The simulator use copula or IC method to do the correlation.

Frequency distribution is defined as "Annual Frequency". Simulator would then convert the annual frequency to "monthly frequency". Poisson or Negative Binomial distributions are the two discrete distributions applied in the simulation. Frequency conversion utilizes Exposure, Trend, and SeasonalityNote, descriptions below only use Exposure "pro rata" for illustration purpose only.

1. Poisson univariate distribution: 

For a given line, the modeler inputs “lambda”, the mean of the Poisson distribution, which is called an “annual frequency”.  The modeler also specifies a vector < e1, e2, … e12 >  of relative exposures by calendar month within the year.  The model then determines the number of claims for each month j as a random number from the Poisson(lambdaj) distribution, where

 

 

lambdaj=  lambda * (ej / Sum(e))  for j = 1,2, 3 … 12.    

 

 

2. Negative Binomial univariate distribution:

For a given line, the modeler inputs the parameters “size” and “prob” for the “annual frequency”.  These are the same parameters as language R uses in its parameterization of the Negative Binomial (“Nbinom”) distribution.  For this discussion we will use the variable r to mean size and p to mean prob – these are variable names commonly used to describe the Nbinom.

If r is an integer, a random variable N with the nbinom(r, p) distribution represents the number of failures until the rth success is achieved in a serious of independent Bernoulli trials with probability of success p.  We can derive the density as

 

            pk = Pr[N=k] =                                               for k = 0,1,2…                                 (*)

 

The right-hand-side of (*) represents a density function even if r is not an integer.  Moreover, the documentation for the language R states that the size parameter for the nbinom distribution can be any positive real number. 

 

 

Incidentally the mean of N is r (1-p)/p ,and its variance is r (1-p)/p^2 

 

 

These properties allow us to convert “annual frequency” parameters to monthly frequency distributions in a manner analogous to the Poisson distribution as follows:

Let r and p be the respective “size” and “prob” input parameters for “annual frequency” and let

< e1, e2, … e12 >  be relative exposures by calendar month within the year.  The model then determines the number of claims Nj for each month j as a random number from the nbinom distribution:

 

 

 Nj = nbinom (size = 12 * ej / Sum(e) ,  prob = p)  for j =1, 2, 3, …12.

 

3. Test of Negative Binomial distribution in R:

To use R to generate negative binomial densities, we first calculated in Excel the various probabilities P[N=k] for the range:

p = 0.5 and 0.8;  r = (0.5, 1.0, 1.5, 2.0, 2.5, 3.0);  k = (0, 1, 2, … 9).

Then run the R session whose output is shown below.  The function dnbinom produces the probabilities corresponding to k = the values in vector x for the negative binomial distribution with parameters p and r.  This tested the dnbinom function for the following ranges:

p = 0.8;  r = (1, 2, 1.5, 0.5);  k = x = (0, 1, 2, … 8).

The output from the session follows:

> p <- 0.8

> x <- c(0,1,2,3,4,5,6,7,8)

> r <- 1

>y<-dnbinom(x,size=r,prob=p)

> y

[1] 8.000e-01 1.600e-01 3.200e-02 6.400e-03 1.280e-03 2.560e-04 5.120e-05

[8] 1.024e-05 2.048e-06

> r <- 2

> y <- dnbinom(x,r,p)

> y

[1] 6.40000e-01 2.56000e-01 7.68000e-02 2.04800e-02 5.12000e-03 1.22880e-03 2.86720e-04 6.55360e-05 1.47456e-05

> r <- 1.5

> y <- dnbinom(x,r,p)

> y

[1] 7.155418e-01 2.146625e-01 5.366563e-02 1.252198e-02 2.817446e-03 6.198380e-04 1.342982e-04 2.877819e-05 6.115366e-06

> r<- 0.5

> y <- dnbinom(x,r,p)

> y

[1] 8.944272e-01 8.944272e-02 1.341641e-02 2.236068e-03 3.913119e-04 7.043614e-05 1.291329e-05 2.398183e-06 4.496593e-07

These values agree with the values in the Excel spreadsheet.