2.4.5. The Frequency Clause

The exposure and severity clauses determine the expected claim count. The frequency clause specifies the other particulars of the claim count distribution. As with severity, the syntax is different for non-parametric and parametric distributions.

2.4.5.1. Non-Parametric Frequency Distributions

An exposure clause:

dfreq [outcomes] <[probabilities]>

directly specifies the frequency distribution. The outcomes and probabilities are specified as in Non-Parametric Severity Distributions. There is no need for a frequency clause at the end.

Examples.

agg A dfreq [1 2 3] [.5 3/8 1/8] sev lognorm 50 cv 1.75
agg A dfreq [1 2 3] [.5 3/8 1/8] dsev [1:11]

The first specifies a frequency distribution with outcomes 1, 2, or 3 occurring with probabilities 0.5, 0.375, and 0.125 respectively. Probabilities can be entered as decimals or fractions. The second combines a non-parametric frequency and severity.

2.4.5.2. Parametric Frequency Distributions

The following parametric frequency distributions are supported. Remember that the exposure clause determines the expected claim count.

  • poisson, no additional parameters required.

  • geometric, no additional parameters required.

  • fixed, no additional parameters required, expected claim count must be an integer.

  • bernoulli, no additional parameters required; expected claim count must be \(\le 1\).

  • binomial SHAPE, the shape parameter sets \(p\) and \(n=\mathsf{E}[N]/p\).

  • neyman SHAPE (or neymana or neymanA), the Neyman A Poisson-compound Poisson. The shape variable gives the average number of claimants per claim. See JKK and Consul and Shenton [1973].

  • pascal SHAPE1 SHAPE2 (the generalized Poisson-Pascal, see REF), where SHAPE1 gives the cv and SHAPE2 the number of claims per occurrence.

Example.

agg A 100 claims sev lognorm 50 cv 0.75 poisson
agg A 100 claims sev lognorm 50 cv 0.75 mixed gamma 0.2

specifies a Poisson frequency. and negative binomial frequency respectively. For the latter, frequency CV equals (1 + .2**2 * 100) ** .5 / 10 = 0.22361.

2.4.5.3. Mixed-Poisson Frequency Distributions

A \(G\)-mixed Poisson frequency (see Mixed Frequency Distributions), where \(G\) has expectation 1, can be specified using the mixed keyword, followed by the name and shape parameters of the mixing distribution:

mixed DIST_NAME SHAPE1 <SHAPE2>

SHAPE1 specifies cv of the mixing distribution. The following mixing distributions are supported:

  • gamma SHAPE1 is a gamma-Poisson, i.e., negative binomial. Since the mix mean (shape times scale) equals one \(\alpha\beta=1\) and hence the mix variance equals \(c:=\alpha=(cv)^{-2}\), which is sometimes called the contagion. The negative binomial variance equals \(n(1+cn)\).

  • delaporte SHAPE1 SHAPE2, a shifted gamma and the second parameter equals the proportion of certain claims (which determines a minimum claim count).

  • ig SHAPE1 the inverse Gaussian distribution

  • sig SHAPE1 SHAPE2 the shifted inverse Gaussian, parameter 2 as for Delaporte.

  • beta SHAPE1 a beta-Poisson with mean 1 and cv SHAPE1. Use with caution.

  • sichel SHAPE1 SHAPE2 is Sichel’s (generalized inverse Gaussian) distribution with SHAPE2 equal to \(\lambda\).

    • sichel.gamma SHAPE1 is the same as Delaporte

    • sichel.ig SHAPE1 is the same as a shifted inverse Gaussian.

Example.

agg A 100 claims sev lognorm 50 cv 0.75 mixed gamma 0.2

specifies a negative binomial (gamma-mixed Poisson) frequency respectively. The variance equals \(100\times (1 + 0.2^2 \times 100)\) and the CV equals (1 + .2**2 * 100) ** .5 / 10 = 0.22361.

Warning

Fixed frequency will accept non-integer input, but will not return a distribution (it will have negative probabilities). Be careful!

2.4.5.4. Zero Modification and Zero Truncation

Todo

Not yet implemented.