1. Significant models of claim number Introduction
Before we meet with complex risk model it can be useful to meet the most significant models of claim number.
In this chapter we will sign with η the number of claim during one pre-defined period (typically one year). The risk can be a contract, a business line or a product. Of course η is a non-negative variate with integer value. We will consider the most useful models.
Insurance mathematics VIII. lecture

2. Significant models of claim number (a,b,0) class of distributions I.
The most useable models are the so-called (a,b,0) class of distributions.
Definition: η is (a,b,0) class of distribution if
𝑃 η=𝑛 =(𝑎+ 𝑏 𝑛 )∙𝑃 η=𝑛−1 ; n=1,2,…
As we will see later in this case the total amount of claims can be calculated exactly and in the most practical cases the real number of claims can be reached with this class of distribution.
We have known from earlier this class as it follows from the next statement.
Insurance mathematics VIII. lecture

3. Significant models of claim number (a,b,0) class of distributions II.
Statement:
η has (a,b,0) class of distribution if and only if either is true in the next 4 cases:
(i) 𝑃 η=𝑛 = 1, 𝑖𝑓 𝑛=0 0, 𝑖𝑓 𝑛>0
(ii) 𝑃 η=𝑛 = λ 𝑛 𝑛! ∙ 𝑒 −λ ;λ>0
(iii) 𝑃 η=𝑛 = 𝛼+𝑛−1 𝑛 ∙ 𝑝 𝑛 ∙ 1−𝑝 𝛼 ; 𝛼>0;0<𝑝<1
(iv) 𝑃 η=𝑛 = 𝑁 𝑛 ∙ 𝑝 𝑛 ∙ 1−𝑝 𝑁−𝑛 ; 0<𝑝<1;𝑁=1,2,…
Insurance mathematics VIII. lecture

4. Significant models of claim number (a,b,0) class of distributions III.
Proof:
If a+b<0 then 𝑃 η=0 and 𝑃 η=1 would have different signs, which is impossible (because of 𝑃 η=1 =(𝑎+𝑏)∙𝑃 η=0 ).
If a+b=0 then 𝑃 η=1 = 𝑎+𝑏 ∙𝑃 η=0 =0 and it follows
𝑃 η=𝑛 =0;𝑛≥1. It means that 𝑃 η=0 has to be equal 1 (because η is distribution). We get (i) case.
η is a distribution, it means:
1= 𝑛=0 ∞ 𝑃 η=𝑛 =𝑃 η=0 ∙ 𝑛=0 ∞ 𝑏 𝑛 𝑛! = 𝑃 η=0 ∙ 𝑒 𝑏 𝑃 η=0 = 𝑒 −𝑏
η is Poisson with b parameter, (ii) case.
If a+b>0 and if a=0 then
𝑃 η=𝑛 = 𝑏 𝑛 ∙𝑃 η=𝑛−1 = 𝑏 𝑛 ∙ 𝑏 𝑛−1 ∙𝑃 η=𝑛−2 =…= 𝑏 𝑛 𝑛! ∙𝑃(η=0)
Insurance mathematics VIII. lecture

5. Significant models of claim number (a,b,0) class of distributions IV.
Proof (continued):
If a+b>0 and if a>0 then we sign 𝛼=1+ 𝑏 𝑎 . With this notation we get
𝑃 η=𝑛 =(𝑎+ 𝑏 𝑛 )∙𝑃 η=𝑛−1 = 1+ 𝑏 𝑎 +𝑛−1 𝑛 ∙𝑎∙𝑃 η=𝑛−1 =
= 𝛼+𝑛−1 𝑛 ∙𝑎∙𝑃 η=𝑛−1 =…=𝑃(η=0)∙ 𝛼+𝑛−1 𝑛 ∙ 𝑎 𝑛
In order that 𝑛=1 ∞ 𝑃 η=𝑛 <1 we must have a<1. Then we get the negative binomial distribution with p=a, (iii) case. You might have noticed:
𝑃 η=0 = (1−𝑎) 𝛼
Insurance mathematics VIII. lecture

6. Significant models of claim number (a,b,0) class of distributions V.
Proof (continued):
If a+b>0 and if a<0 then b>-a. It means that there are such K integer for which
𝑎+ 𝑏 𝑛 <0 ∀𝑛>𝐾
This is just only if can be true (because of avoiding negative probability) if ƎK for which
𝑎+ 𝑏 𝐾 =0
Then let 𝑁=𝐾−1 and 𝑝=− 𝑎 1−𝑎 , we will get the (iv) case.
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7. Significant models of claim number (a,b,0) class of distributions VI.
For these types the Poisson distribution the most applied distribution. The binomial distribution can be used naturally if related to one contract it can be just 0 or 1 claim. Regarding negative binomial distribution we can consider the next example:
We assume that distribution of η with fixed ν is Poisson distribution with ν∙𝑡
parameter. It means that the conditional probability-generating function will be as next:
𝐺 η ν 𝑧 =𝐸 𝑧 η ν = 𝑒 ν∙𝑡∙(𝑧−1)
From these equation we will get probability-generating function of η as follows (based on law of total expectation):
𝐺 η 𝑧 =𝐸(𝐸 𝑧 η ν )= 𝐸(𝑒 𝜈∙𝑡∙ 𝑧−1 )= 𝐿 ν (𝑡∙(1−𝑧))
If ν is Gamma distribution with (λ,α) parameters then its Laplace-transform is as next:
𝐿 ν 𝑠 = ( λ λ+𝑠 ) α
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8. Significant models of claim number (a,b,0) class of distributions VII.
Then we get:
𝐺 η 𝑧 = ( λ λ+𝑡∙(1−𝑧) ) α = (1− 𝑡 λ ∙(𝑧−1)) −α
which is probability-generating function of negative binomial distribution with (𝛼, 𝑡 𝑡+λ ) parameters.
It means that the negative binomial distribution is mixed Poisson distribution (the mixer variate is Gamma distribution).
Memo:
η has mixed Poisson distribution if ∃𝜏 mixer variate for which the conditional distribution of η with 𝜏 condition is Poisson distribution with 𝜏 parameter.
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9. Insurance mathematics VIII. lecture
Significant models of claim number (a,b,0) class of distributions VIII.
Definition:
η is compound Poisson distribution, if η= 𝑀 1 + 𝑀 2 +…+ 𝑀 𝑁
where N is Poisson distribution, 𝑀 1 , 𝑀 2 ,…, 𝑀 𝑁 are independent, identic distribution.
Statement:
Negative binomial distribution can be compound Poisson distribution.
Insurance mathematics VIII. lecture

10. Significant models of claim number (a,b,0) class of distributions IX.
Proof:
Let 𝑀 𝑗 the number of claim related to one claim event. We suppose that 𝑀 𝑗 is logarithmic distribution. It means that its probability-generating function will be as follows:
𝐺 𝑀 𝑗 𝑧 = ln⁡(1−𝑞∙𝑧) ln⁡(1−𝑞)
If the number of claims (N) is Poisson distribution then the total claim number, η= 𝑀 1 + 𝑀 2 +…+ 𝑀 𝑁 will be compound Poisson distribution. Probability-generating distribution of η will be the next:
𝐺 η 𝑧 =E 𝑧 𝑛 =𝐸 𝐸 𝑧 𝑛 𝑁 =𝐸 (𝐺 𝑀 𝑗 𝑧 ) 𝑁 = 𝐺 𝑁 𝐺 𝑀 𝑗 𝑧 =
=exp(λ∙ 𝐺 𝑀 𝑗 𝑧 −1 = exp λ∙ ln 1−𝑞∙𝑧 ln 1−𝑞 −1 =( 1−𝑞∙𝑧 1−𝑞 ) λ ln⁡(1−𝑞)
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11. Significant models of claim number (a,b,0) class of distributions X.
Proof (continued):
𝐺 η 𝑧 = (1− 𝑞 1−𝑞 𝑧−1 ) −(− λ ln⁡(1−𝑞) )
It means that η is negative binomial distribution with (− λ ln 1−𝑞 ,𝑞) parameters.
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12. Significant models of claim number Other models I.
The other models which can be used as claim number can come from the next families:
- generalizing (a,b,0) class of distributions;
- compound distributions;
- mixed distributions.
Definition: η is (a,b) class of distribution if
𝑃 η=𝑛 =(𝑎+ 𝑏 𝑛 )∙𝑃 η=𝑛−1 ; n=2,3,…
It seems that there is just a small difference between (a,b,0) and (a,b) classes (we do not require the equation in case of n=1), the real difference is significant.
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13. Significant models of claim number Other models II.
Definition: η is (a,b,1) class of distribution if
𝑃 η=𝑛 =(𝑎+ 𝑏 𝑛 )∙𝑃 η=𝑛−1 ; n=2,3,… and 𝑃 η=0 =0
It is affirmable if η is (a,b) distribution then ∃𝛼>0 and τ (a,b,0) or (a,b,1) distribution for which:
𝑃 η=0 =1−𝛼+𝛼∙𝑃 τ=0
𝑃 η=𝑛 =𝛼∙𝑃 τ=𝑛 ;n>0
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14. Significant models of claim number Other models III.
Compound distributions:
- compound Poisson
- compound geometric
If η is compound Poisson then its probability-generating function is as next:
𝐺 η 𝑧 = exp λ∙ 𝐺 𝑀 𝑧 −1
where 𝐺 𝑀 𝑧 is a probability-generating function of non-negative variate (with integer value). The two most important version will be the next:
Definition:
η is Poisson-binomial distribution if its probability-generating function is the following:
𝐺 η 𝑧 = exp λ∙ (1+𝑝∙ 𝑧−1 ) 𝑚 −1
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15. Significant models of claim number Other models IV.
Definition:
η is Neyman’s type A distribution if its probability-generating function is the following:
𝐺 η 𝑧 = exp λ 1 ∙ exp λ 2 ∙ 𝑧−1 −1
We will consider the two most important compound geometric distributions also as follows.
η is geometric-geometric distribution if its probability-generating function is the following:
𝐺 η 𝑧 = (1− 𝛽 1 ∙ 1 1− 𝛽 2 ∙ 𝑧−1 −1 ) −1
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16. Significant models of claim number Other models V.
Definition:
η is geometric-Poisson distribution if its probability-generating function is the following:
𝐺 η 𝑧 = (1−𝛽∙ exp λ∙ 𝑧−1 −1 ) −1
Mixed distributions
Because of law of total expectation and law of total variance:
E η =𝐸 𝐸 η τ and
𝐷 2 η = 𝐷 2 𝐸 η τ +𝐸 𝐷 2 η τ
From these equation we will get the following in case of mixed Poisson distribution:
E η =λ∙E(τ)
𝐷 2 η = λ 2 ∙𝐷 2 τ +λ∙𝐸 τ
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17. Significant models of claim number Other models VI.
Definition:
η distribution is indefinitely divisible if ∀𝑛∈𝑁 η is a convolution of n identic distribution.
Statement:
We assume that η distribution is such mixed Poisson distribution for which the mixer distribution is indefinitely divisible. Then η is compound Poisson distribution.
Insurance mathematics VIII. lecture

18. Significant models of claim number Other models VII.
Proof:
We sign L the Laplace-transform of mixer distribution. Because of indefinitely divisible property 𝐿 𝑛 = 𝑛 𝐿 is Laplace-transform. We know from earlier that the probability-generating function of mixed Poisson distribution is as follows:
𝐺 η 𝑧 =𝐿 λ∙ 1−𝑧 = ( 𝐿 𝑛 λ∙ 1−𝑧 ) 𝑛 = 𝐺 𝑛 (𝑧) 𝑛
Because of 𝐺 𝑛 (𝑧) is a probability-generating function of a mixed Poisson distribution and the above formula can be written for each n, η is indefinitely divisible. And if η is indefinitely divisible, nonnegative with integer value then η is compound Poisson distribution.
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19. Insurance mathematics VIII. lecture
Significant models of claim number Identification of distribution of claim number I.
We assume that a database exists from which there is clear which contract has how many claims event. (In the practice there is not so easy because the starting date and the duration would be different. We assume that the collection of data and the systematization happened.)
When we would like to identify the distribution of claim number at first we can try (a,b,0) class of distribution. What we know:
Poisson, binomial or negative binomial;
if we sign 𝑝 𝑛 =𝑃 η=𝑛 then let
𝑞 𝑛 = 𝑛+1 ∙ 𝑝 𝑛+1 𝑝 𝑛 =𝑎+𝑏+𝑛∙𝑎
If the distribution is Poisson then 𝐷 2 η =𝐸 η and 𝑎=0, it means 𝑞 𝑛 =𝑐𝑠𝑡.
If the distribution is binomial then 𝐷 2 η <𝐸 η and 𝑎<0, it means 𝑞 𝑛
is decreasing linear.
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20. Insurance mathematics VIII. lecture
Significant models of claim number Identification of distribution of claim number II.
If the distribution is negative binomial then 𝐷 2 η >𝐸 η and 𝑎>0, it means 𝑞 𝑛 is increasing linear.
Because of above remarks our process will be as next:
We sign 𝑈 𝑖 the number of such contracts which have exactly i claims. Let K is the number of contracts, and the maximum number of claim is j. Then the r-th empiric momentum will be as follows:
𝑀 𝑟 = 1 𝐾 𝑖=0 𝑗 𝑖 𝑟 ∙ 𝑈 𝑖
The empiric variance will be the next:
𝑆 2 = 𝑀 2 − 𝑀 1 2
Let 𝑇 𝑛 =(𝑛+1)∙ 𝑈 𝑛+1 𝑈 𝑛 .
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21. Insurance mathematics VIII. lecture
Significant models of claim number Identification of distribution of claim number III.
If 𝑆 2 ≈ 𝑀 1 and 𝑇 𝑛 ≈𝑐𝑠𝑡 then the distribution of claim number will be approximated with Poisson distribution.
If 𝑆 2 < 𝑀 1 and 𝑇 𝑛 is decreasing then the distribution of claim number will be approximated with binomial distribution.
If 𝑆 2 > 𝑀 1 and 𝑇 𝑛 is increasing then the distribution of claim number will be approximated with negative binomial distribution.
But there is insecure that the distribution of claim number comes from (a,b,0)-class of distribution. For example if we find that 𝑇 𝑛 is changing faster than linear then it can be useful to calculate third empiric momentum. Let
κ= 𝑀 3 −3 𝑀 2 𝑀 1 +2 𝑀 1 3
If η is negative binomial distribution then its third momentum is
3 𝐷 2 η −2𝐸 η + 2( 𝐷 2 η −𝐸 η ) 2 𝐸 η
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22. Insurance mathematics VIII. lecture
Significant models of claim number Identification of distribution of claim number IV.
It means that we have to compare κ with 3 𝑆 2 −2 𝑀 1 + 2( 𝑆 2 − 𝑀 1 ) 2 𝑀 1 . If there are similar we can accept negative binomial distribution. If κ is much more then we are using Neyman’s type A distribution or Poisson-geometric distribution. If κ is much less then we can try divers mixed distribution. Neyman’s type A distribution or Poisson-geometric distribution.
After determination of distribution type we can calculate estimation of parameters.
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23. Significant models of claim number Estimating the parameters I.
Example (the method of moments):
Let η 1 , η 2 ,…, η 𝐾 independent NB(r,q) data sample. It means that
𝑃 η 𝑖 =𝑛+𝑟 = 𝑛+𝑟−1 𝑟−1 ∙ 1−𝑞 𝑟 ∙ 𝑞 𝑛 ;𝑛=0,1,…
𝐸 η = 𝑟 1−𝑞 ; 𝐷 2 η = 𝑟𝑞 (1−𝑞) 2
From above equations we will get the following equation system:
𝑀 1 = 𝑟 1−𝑞 ; 𝑆 2 = 𝑟𝑞 (1−𝑞) 2
Insurance mathematics VIII. lecture

24. Significant models of claim number Estimating the parameters II.
Example continued:
From the solution of the equation system we will get the estimation of parameters:
𝑞 = 𝑆 2 𝑆 2 + 𝑀 1 ; 𝑟 = 𝑀 1 (1−𝑞)= 𝑀 1 𝑆 2 𝑆 2 + 𝑀 1
We can use maximum likelihood estimation also. The likelihood function will be as next:
𝐿 𝑟,𝑞 = 𝑃 𝑟,𝑞 η 1 = 𝑛 1 , η 2 = 𝑛 2 ,… η 𝐾 = 𝑛 𝐾 = 𝑖=1 𝐾 Γ(𝑟+ 𝑛 𝑖 ) Γ(𝑟)∙ 𝑛 𝑖 ! ∙ 1−𝑞 𝑟 ∙ 𝑞 𝑛 𝑖
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25. Significant models of claim number Estimating the parameters III.
Example continued:
The log likelihood function will be as follows:
𝑙 𝑟,𝑞 =𝑙𝑛𝐿 𝑟,𝑞 =𝑟𝐾𝑙𝑛 1−𝑞 +𝑙𝑛𝑞∙ 𝑖=1 𝐾 𝑛 𝑖 + 𝑖=1 𝐾 𝑚=0 𝑛 𝑖 −1 ln 𝑟+𝑚 − 𝑖=1 𝐾 𝑙𝑛 𝑛 𝑖 !
After derivation we will get the likelihood equations:
𝜕𝑙 𝜕𝑞 =−𝑟𝐾 1 1−𝑞 + 1 𝑞 ∙ 𝑖=1 𝐾 𝑛 𝑖 =0
𝜕𝑙 𝜕𝑟 =𝐾𝑙𝑛 1−𝑞 + 𝑖=1 𝐾 𝑚=0 𝑛 𝑖 −1 1 𝑟+𝑚 =0
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26. Significant models of claim number Estimating the parameters IV.
Example continued:
From this equation system we will get the next:
𝑟 1−𝑞 = 𝑀 1
ln 1−𝑞 = 1 𝐾 𝑖=1 𝐾 𝑚=0 𝑛 𝑖 −1 1 𝑟+𝑚
It can be approved that if 𝑆 2 >𝑀 1 then there exists unique solution of above equation system.
Insurance mathematics VIII. lecture

27. Significant models of claim number Remarks
It can be a lot of changes related to property of claim number that’s why these process has to be repeated year by year. Some possibilities regarding changes as follows:
There is a trend in claim frequency (for example mortality changes, motorization changes, etc.)
Periodicity with short period (for example yearly climate changes).
Periodicity with long period (because of changes of financial environment).
Simple occasional fluctuation.
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