# Mixed Poisson distributions for modeling the number...

## Mixed Poisson distributions for modeling the number of insured events

In practice, the Poisson distribution parameter λ is often unstable for the following reasons:

• the difference in the parameters of the Poisson distribution among different insurers when modeling the number of cases in individual models;

• the parameter difference for different years in a portfolio with the same risks in the case of a collective model (weather conditions, economic conditions, etc.) (for example, when car insurance against an accident, the intensity depends on the number of days with bad weather and is not constant).

In this case, there arises the problem of entering an additional random variable , which is responsible for changing the parameter λ and reflecting the heterogeneity of the portfolio (in the first case) or serving to model annually changing external influences in a homogeneous portfolio of the collective model (in the second case).

independent identically distributed random variables that characterize the individuality of the insured in the first case and the "quality of the year" in the second. The distribution is called the mixing distribution and acts as a measure of the heterogeneity of the portfolio.

Jean Lemaire , Thomas Mack ( Thomas Mack ) propose to take into account the heterogeneity of policyholders using a certain function called the structural function and (X), which leads to the so-called mixed (composite, complex) Poisson distribution .

So, suppose that the number of insured events on the account of each insured person has a Poisson distribution

And each insured is characterized by its value λ, which allows you to take into account the heterogeneity of risks.

The discrete random variable K has a mixed Poisson distribution law , if it takes the values ​​ with the parameter-function with probabilities:

(3.19)

where is the distribution of the random variable (structural function).

In practice, a certain assumption is made about the form of the mixing distribution, i.e. distribution of a random variable

Various functions can be selected as the structural or mixing function . The most common as a mixing distribution and lead to adequate results:

is the gamma distribution;

is the inverse Gaussian distribution.

Mixed Poisson/Gamma distribution

The gamma distribution with the parameters a and b is often used to simulate the Poisson parameter in actuarial calculations:

where is the Euler gamma function, ,

- a natural number.

Its numerical characteristics:

It is the gamma distribution that well describes the situation when the values ​​of λ fluctuate around a certain value, while both very small and very large λ values ​​are possible, but unlikely.

The distribution of the number of insured events in the portfolio ( p k, } then is reduced to the following form:

If a - is an integer, then, given that :

Thus, we came to the negative binomial model of the form (3.16):

with parameters and numerical characteristics:

Calculating the probabilities of a negative binomial distribution does not require a gamma function value table. Sequential use of the property allows us to go to the recurrence formula:

(3.20)

with an initial value

(3.21)

Estimates of the parameters of the distribution of the sample using the method of moments are carried out by the formulas:

(3.22)

EXAMPLE 3.7

It is known that:

1) the number of insured events K has a Poisson distribution with mean A

2) A has a gamma distribution with average 1 and variance 2. Determine the probability that K = 1 (in the contract there will be 1 insurance event).

Variants of answers: a) 0.19; b) 0.24; c) 0.31; d) 0.34; e) 0.37. Solution

By the condition of the problem, it is clear that the random variable K has a mixed Poisson/gamma distribution, which is reduced to a negative binomial distribution of the form (3.16):

with parameters:

where a and b are the parameters of the gamma distribution that are related to its mathematical expectation and variance by formulas of the form:

By the condition

Hence the gamma distribution parameters:

Then

Now, using the formula of negative binomial distribution, we can find the required probability:

Therefore, the correct answer is a).

EXAMPLE 3.8

Let's explore a portfolio consisting of n = 2512 contracts for auto hull insurance. During the year, suits were received for t = 888 contracts in connection with insured events. The number of insured events kk, occurred under one contract, varied in the portfolio studied from 0 to 10 (see table).

It is necessary to check whether the mixed Poisson/gamma distribution (negative binomial) is suitable for modeling the distribution of the number of insured events in a given auto-hull portfolio.

Solution

We will find selective estimates of the distribution parameters of the number of insured events in one contract (mean value):

and selective variance

The lookup table looks like this:

Calculation of sample estimates of the distribution parameters of the number of insured events in one contract

 k 0 1624 0 0 0 1 490 490 1 490 2 208 416 4 832 3 98 294 9 882 4 48 192 6 768 5 23 115 25 575 6 10 60 36 360 7 5 35 49 245 8 3 24 64 192 9 2 18 81 162 10 1 10 100 100 n 2512

So, based on the results of calculating the estimates of the distribution parameters of the random variable K - the number of insurance cases in one contract, we get:

- sample mean:

- in one contract for a portfolio, an average of 0.658 insured events occur during the year;

- selective variance:

Checking this empirical distribution on the law of Poisson gave a serious discrepancy between empirical and theoretical frequencies and = 7,815. Thus, in the example considered, the Poisson distribution can not serve as an adequate model, the empirical distribution has a long right tail - up to 10 insurance cases, it is necessary to try mixed Poisson distributions. So, we will calculate a mixed Poisson/gamma distribution.

Considering the sampling characteristics found, the mean and sample variance, calculate the estimates of the gamma distribution parameters by (3.22):

Using the recurrence formula (3.20) and the formula for calculating the probability of zero payments (3.21), we calculate the theoretical frequencies, the results are presented in the table.

Calculate the distribution of the number of claims in the portfolio of auto hull contracts using a negative binomial distribution (mixed Poisson/gamma distribution)

 The number of insured events in the contract k Empirical Theoretical (negative by bin.) Frequencies t to Probabilities p kT frequencies t kT 0 1624 0.6434 1616 1 490 0.1992 501 2 208 0.0836 210 3 98 0.0382 96 4 48 0.0181 46 5 23 0.0088 22 6 10 0.0043 11 7 5 0.0022 5 8 3 0.0011 3 9 2 0.0005 1 10 1 0.0003 1 Total 2512 1,000 2512

Further, in accordance with the requirements of Pearson's agreement criterion for combining intervals with small theoretical frequencies less than 5, the observations were grouped together, and the value% 211a6L was calculated.

Comparison of x? and 6n = 0.806 and x? pound (a = 0.05, v = 9 - 2 - 1 = 6) = 12.592 showed that x, laft; l & lt; Therefore, the test hypothesis is not rejected, i.e. The negative binomial model is considered adequate and fairly accurately reflects the distribution of the number of claims received. A negative binomial model adopted at this level of reliability can be used to perform actuarial calculations.

The results obtained and the consistency of the empirical and negative binomial distributions are graphically illustrated by the graph (Figure 3.5).

In the works of Jean Lemer, Thomas Mack, IA Kornilov and other scientists, the successful application of a negative binomial distribution in fitting the number of insurance cases for heterogeneous portfolios is noted. Let's give an example (short results -

Selective and theoretical negative binomial distribution of the number of insured events and calculation of Pearson's agreement criteria

 k 0 1624 1616 0.0378 1 490 501 0.2190 2 208 210 0.0198 3 98 96 0.0481 4 48 46 0.1333 5 23 22 0.0367 6 10 11 0.0736 7 5 5 0.0339 & gt; 8 6 5 0.2 The sum of x2abl 0.8062

theta calculations) even more heterogeneous portfolio with a much longer right tail.

Fig. 3.5. Modeling the number of claims by type of auto hull insurance using a negative binomial distribution

EXAMPLE 3.9

For the insurance portfolio of voluntary medical insurance, the distribution of the number of insurance cases in which was given as an example in Fig. 3.4, in the number n = 44 114 contracts, selective estimates of the model parameters were calculated using the torque method using the formulas (3.22):

The results of calculating the theoretical frequencies of the mixed Poisson/gamma distribution and their comparison with the empirical frequencies observed in the portfolio are shown in the table and in Fig. 3.6.

In the model under consideration, (a = 0,05; v = 14) = 23,685.

Therefore, the hypothesis being tested is not rejected at the significance level a = 0.05, and the negative binomial model is considered adequate to approximate the distribution of settled losses in modeling the number of insurance payments in the portfolio risks under medical insurance contracts (LCA).

Table and Fig. 3.6 clearly demonstrate the excellent consistency of the random number of the insured event with a theoretical mixed Poisson/gamma distribution.

Mixed Poisson/inverse Gaussian distribution

Now we will study another variant of the mixed Poisson distribution, which is often advanced as an alternative to the negative binomial distribution in actuarial calculations-the Poisson/inverse Gaussian distribution.

So, another common version of the mixed Poisson law :

Empirical ( t to ) and theoretical (t kT ) < strong> frequencies of the mixed Poisson/gamma distribution and verification of their agreement by the Pearson criterion

 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 & gt; 16 27,981 7719 3652 1908 1148 612 423 256 162 106 66 34 20 9 10 4 4 27 680 7810 3682 1970 1117 654 392 238 146 91 57 36 22 14 9 6 8 0.53 1.06 0.24 1.95 0.86 2.70 2.45 1.36 1.75 2.47 1.42 0.11 0.18 1.79 0.11 0.67 2.00

Fig. 3.6. Empirical and theoretical frequencies (according to the negative binomial distribution) of the LCA portfolio

is the use for the value as a structural or mixing function of the probability density of the inverse Gaussian distribution with the parameters g and h.

Then we get a mixed Poisson distribution, which is called a Poisson inverse Gaussian distribution with numerical characteristics:

Selective estimates for the distribution parameters g and h using the method of moments are:

(3.23)

The probabilities of a Poisson/inverse Gaussian distribution can be calculated recurrently:

(3.24)

Then theoretical frequencies are calculated and the adequacy of the model to the portfolio under investigation is checked with the help of the consent criterion.

EXAMPLE 3.10

Using the data of Example 3.8, we find sample estimates for the parameters of the inverse Gaussian distribution g and h using the moment method (3.23): g = 0.658; h = 1,126.

We calculate the probabilities of the inverse Gaussian distribution by recurrence formulas (3.24). The results are given in the table.

Calculation of the distribution model of the number of claims in the portfolio of auto hull contracts using a mixed Poisson/inverse Gaussian distribution

 The number of insured events in the contract k Empirical Theoretical frequencies m k the probability of p kT frequencies t kt 0 1624 0.6252 1571 1 490 0.2282 573 2 208 0.0812 204 3 98 0.0332 84 4 48 0.0153 39 5 23 0.0076 19 6 10 0.0040 10 7 5 0.0022 6 8 3 0.0013 3 9 2 0.0007 2 10 1 0.0004 1 Total 2512 1,000 2512

In the future, by analogy with the previous distributions, observations with theoretical frequencies less than 5 were combined into groups, and the value of y, is determined.

Calculation of Pearson's agreement criteria for a mixed Poisson/inverse Gaussian distribution of the number of insured events

 0 1624 1571 1.7880331 1 490 573 12,02268761 2 208 204 0,078431373 3 98 84 2,333333333 4 48 39 2,076923077 5 23 19 0.842105263 6 10 10 0 7 5 6 0.166666666 & gt; 8 6 6 0 The sum of μ2 aba 19,308

Comparison of у211ал = 19,308 and χ2κρΗΤ (a = 0,05, v = 9- 2 1 = 6) = 12,592 has shown that χ2π; ι6ι & gt; χ2 um, so the test hypothesis is rejected, i.e. The inverse Gaussian distribution is recognized as an inadequate model and does not accurately reflect the distribution of the number of claims received. Although the value of the statistics of the agreement criterion x2uab, = 19.308 shows that it is only slightly higher than the critical value, and the distribution under study can be put in second place in the approximation of the studied distribution after the negative binomial model.

EXAMPLE 3.11

In another researched portfolio of auto hull insurance, the best was just a mixed Poisson/inverse Gaussian distribution.

Briefly give only the main results obtained. The empirical distribution obtained using the SUMMARY TABLE in Excel, is presented in the table.

Based on the selective characteristics obtained from the portfolio, estimates of the parameters (3.23) were found:

Further, according to the formula (3.24), theoretical probabilities are calculated, and then - frequencies.

Calculate the distribution of the number of claims in the portfolio of auto hull contracts for a mixed Poisson/inverse Gaussian distribution

 The number of insured events in the contract k Empirical Theoretical frequencies t to the probability of p kT frequencies m kT 0 8045 0.5467 7996 1 3807 0.2677 3916 2 1614 0.1066 1559 3 654 0.0434 635 4 264 0.0188 274 5 124 0.0086 125 6 58 0.0041 60 7 29 0.0020 30 8 12 0.0010 15 9 10 0.00053 8 10 5 0.00028 4 11 3 0.00015 2 12 2 0.00008 1 n 14,627 1,000 14 626

Next, as was done in the previous examples, we check the model's correspondence with the help of the Pearson criterion. We perform the grouping and combine the 10th, 11th and 12th intervals so that the value of the theoretical frequency is a total greater than or equal to 5. The results of testing the hypothesis that the theoretical distribution is consistent with the empirical distribution.

Selective and theoretical inverse Gaussian distribution of the number of insured events and calculation of Pearson's agreement criteria

 0 8045 7997 0.288 1 3807 3916 3,046 2 1614 1559 1,922 3 654 635 0.596 4 264 274 0.389 5 124 126 0.032 6 58 60 .056 7 29 30 0.010 8 12 15 0.590 9 10 8 0.655 & gt; 10 10 7 1.286 8.87

where 11 is the number of intervals (after combining the right tail with small frequencies); 2 - the number of parameters evaluated by the sample.

As the criterion of consent has shown, the hypothesis that the distribution of the number of insurance cases is subject to the inverse Gaussian distribution is not rejected. Consequently, this distribution adequately reflects the empirical distribution, which is confirmed by the graph (Figure 3.7).

Model "good/bad risks Lemer

Jean Lemer, a professor at the University of Pennsylvania, a well-known Belgian actuary, one of the largest researchers in European automobile insurance, proposed another model for calculating the number of suits that relates to the mixed Poisson distributions - model> good/bad risks " . In this model, designed to approximate the number of claims in the portfolio of auto insurance contracts, it is assumed that there are two categories of drivers with different levels accidents - good (for the simulation of which a Poisson distribution with parameter λ1 is introduced) and bad

Fig. 3.7. Simulation of the number of claims by type of auto hull insurance using a mixed Poisson/inverse Gaussian distribution

Drivers (characterized by the value of the parameter λ2). The mixing distribution A here is a two-point discrete distribution. Estimates of theoretical probabilities are calculated using the formula

(3.25)

where

Estimates of the distribution parameters by the method of moments are calculated by the following formulas:

(3.26)

(3.27)

where - initial selective moments of the 1st, 2nd and 3rd order of the random variable K - the number of insured events , which came in a year in one contract.

Next, theoretical frequencies are calculated and, with the help of the consent criterion, checked for consistency with empirical ones.

EXAMPLE 3.12

Let's analyze the real portfolio of insurance contracts for OSAGO one of the major Moscow insurance companies.

Having grouped all contracts according to the number of insured events that occurred in a year, we get the following empirical distribution of the number of claims:

Selective distribution of the number of received claims for K under OSAGO insurance contracts

 Number of claims k 0 1 2 3 4 5 Number of contracts t to 105 925 4940 554 73 7 1

For our portfolio of 111,500 contracts, the following estimates of the distribution parameters according to (3.26) and (3.27) are calculated:

The calculated sample estimates of the distribution parameters of the model "good/bad risks"

 a b C 0.0565 0.0713 0.1069 0.0565 0.0148 0.0060 A In 1 a g 0.4426 0.0102 0.0244 0.4182 0.9185 0.0815

The results indicate that the share of "good" of drivers in the portfolio is about 91.85% and on average they fall into 0.02 accidents per year. The remaining 8.15% - bad drivers, they have an accident rate of 0.42 accidents per year.

For further analysis, calculate the theoretical probabilities and frequencies by substituting the found parameters into the law of the Lemaire distribution (3.25):

The table shows the theoretical probabilities and frequencies calculated from the resulting formula of the approximating law: It should be noted that the proportion of good drivers are much more than "bad", which makes the choice of the Lemer model quite

 The number of insurance cases in the contract k Empirical Theoretical frequencies m k the probabilities p kt frequencies m kT 0 105 925 0.9500 105 925 1 4940 0,0443 4941 2 554 0.0050 553 3 73 0.0007 73 4 7 0.0001 8 5 1 0.0000 1 Total 111,500 1,000 111,500

adequate. In addition, the probability of an accident as a whole for a portfolio is quite small.

As you can conclude from the table and Fig. 3.8, the Lemer model very accurately reflects the distribution of insurance cases in one contract, since the discrepancy in theoretical and empirical frequencies is minimal.

Let's test this argument with Pearson's criterion. So, (recall that the sample was evaluated by three distribution parameters, unlike all previous models). The hypothesis is not rejected, but the model of "good and bad risks" Lemer is recognized as adequate for approximating the distribution of the number of insurance cases -

Fig. 3 .8. Simulation of the number of claims for the type of insurance of OSAGO with the help of the Lemer model "good/bad risks"

teas in the portfolio of OSAGO. Such a rare agreement between empirical and theoretical frequencies reminds that the model was developed by Lemaire specifically for the insurance of motor vehicle owners' liability, and the insurance portfolio of the compulsory civil liability insurance of a United States insurance company was also suitable for this distribution.

The presented model is "good/bad risks can be used to approximate the distribution of the number of settled losses in many other types of insurance, where the policyholders can also be divided into groups with different probability of occurrence of the insured event.

EXAMPLE 3.13

According to examples 3.8 and 3.10, we calculate the Lemer model.

For the portfolio of insurance contracts for motor hull insurance, the following estimates of distribution parameters are calculated.

 a b C 0.658 1.833 7,4108 0.658 1,175 3,2269 A In 3,3078 1.0028 0.3376 2,9702 0.8781 0.1219

The obtained estimates can be interpreted as follows: 87.81% (β) of the "good" quotes fell into the portfolio under consideration; drivers whose frequency of insurance cases is 0.33 accidents per year (λ,), and 12.19% of "bad" drivers who make an average of 2.97 accidents a year.

Thus, the probabilities of the mixed Poisson distribution of the "good risks/bad risks" model approximating the distribution of the number of claims in the portfolio under investigation will be calculated using the following formula:

The results of the calculations are given in the table.

Now, using the χ2 consensus criterion, let's test the hypothesis of the adequacy of the constructed model.

X2nail = 37.882 & gt; χ2κρΜΤ (a = 0.05, ν = 8-3-1 = 4) = 9.488 (note - here we selected three parameters - α , λ i, λ2), therefore, the hypothesis is rejected. Model good risks/bad risks is recognized as inadequate for the distribution of settled losses.

Calculate the distribution of the number of claims in the portfolio of auto hull contracts using the mixed Poisson distribution of the "good risks/bad risks" model. (according to J. Lemaire)

 The number of insured events in the contract k Empirical frequencies t to Theoretical (ridiculous Poisson, chorus/bad risks) Probabilities Pb frequencies t kT 0 1624 0.6328 1590 1 490 0.2301 578 2 208 0.0633 159 3 98 0.0313 79 4 48 0.0206 52 5 23 0.0121 30 6 10 0.0060 15 7 5 0.0025 6 8 3 0.0009 2 9 2 0.0003 1 10 1 0.00009 0 Total 2512 1,000 2512

Selective and theoretical mixed Poisson distribution of the model "good/bad risks" number of insured events and calculation of Pearson's agreement criteria

 k 0 1624 1590 0.749 1 490 578 13,401 2 208 159 15,125 3 98 79 4,751 4 48 52 0.275 5 23 30 1.761 6 10 15 1.654 & gt; 7 11 9 0.168 The sum of x2in6l 37,882

The results of applying the Pearson consensus criterion to all the distributions studied (based on the results of the calculations of Examples 3.8, 3.10, 3.10, 3.13) are summarized in the summary table.

The results of calculating Pearson's agreement criteria for all considered distributions that simulate the number of insured events in one auto-hull insurance contract

 Distribution type Poisson Mixed Poisson Gamma The inverse Gaussian The Lemer Model 829,541 0.806 19,308 37,882 7,815 12.592 12.592 9,488 The hypothesis is rejected The hypothesis is not rejected The hypothesis is rejected The hypothesis is rejected

In addition, we give a graph (see Figure 3.9), which shows all the results obtained and what theoretical distribution as simulates all the frequencies of the empirical.

Statistical study and modeling of the number of claims in the portfolio of insurance contracts considered allowed us to conclude that the most suitable model in the study of insurance contracts is the negative binomial distribution, which is a mixed Poisson/gamma distribution.

There are works in the actuarial literature in which other mixed Poisson models were also considered.

Willmot ( Willmot ) received a simple recurrence formula that works for a wide range of continuous mixing distributions. As mixing distributions, he used a beta distribution, a uniform distribution, a Pareto distribution, and a generalized Pareto distribution. In addition, he considered a negative binomial distribution, a Poisson beta distribution, and generalized Gaussian models.

Albrecht ( Albrecht ) considered mixtures of a Poisson distribution with distributions such as the Pearson family, Weibull distribution, Pareto distribution, Bessel distribution, truncated normal distribution, chi-square, etc. He also

Fig. 3.9. Modeling the number of auto hull insurance claims using Poisson and three mixed Poisson distributions

voiced considerations in favor of discrete mixtures of Poisson distributions.

Delaporte ( Delaporte ) first introduced, and Ruohonen ( Ruohonen ), Willmotte and Sundt ( Sundt ) continued the study of the mixed Poisson distribution with mixing distribution, which is a gamma distribution with a three-parameter shift. With this approach, the process of occurrence of insured events consists of two independent components, a Poisson process that reflects the overall contribution of all risks, and a negative binomial process that is responsible for the individual contribution of a particular risk.

The three-parameter mixed Poisson Pangeer model ( Panjer ) is a Pascalian generalization of the Poisson distribution and contains, as special cases, a negative binomial distribution, Poya distribution, a Neumann type A and a Poisson inverse Gaussian distribution.

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