Bühlmann's Credibility Model with Claims of Negative Binomial and 2-Poisson Distribution CAUCHY –Jurnal Matematika Murni dan Aplikasi Volume 7(4) (2023), Pages 493-502 p-ISSN: 2086-0382; e-ISSN: 2477-3344 Submitted: June 10, 2022 Reviewed: February 22, 2023 Accepted: March 09, 2023 DOI: http://dx.doi.org/10.18860/ca.v7i4.16400 Bühlmann's Credibility Model with Claims of Negative Binomial and 2-Poisson Distribution Ikhsan Maulidi1, Taufiq Iskandar2, Nurbahari3, Alim Misbullah4, Vina Apriliani5* 1,2,3Department of Mathematics, Syiah Kuala University, Banda Aceh, Indonesia 4Department of Informatics, Syiah Kuala University, Banda Aceh, Indonesia 5Department of Mathematics Education, UIN Ar-Raniry, Banda Aceh, Indonesia 1,5Graduate School of Natural Science and Technology, Kanazawa University, Kanazawa, Japan Email: vina.apriliani@ar-raniry.ac.id ABSTRACT One technique for determining the premium is using the credibility theory. In this study, a credibility premium determination model was derived with the best accuracy approach in the form of Bühlmann’s credibility premium. The approach used was a parameteric approach where the claim data is assumed to have a Negative Binomial and 2-Poisson distribution. The Bühlmann's credibility premium formula is given explicitly for these two data distributions. The obtained model is also applied to the correct data following these distributions. From the simulation results, it is obtained that the premium values are very close in value so that both models can be applied to the data and have a high level of credibility because they have a high credibility factor value. The results of this study provide a basic contribution to the development of actuarial science, especially in the technique of determining insurance premiums. Copyright © 2023 by Authors, Published by CAUCHY Group. This is an open access article under the CC BY-SA License (https://creativecommons.org/licenses/by-sa/4.0/) Keywords: 2-Poisson distribution; Bühlmann's credibility; Negative Binomial distribution INTRODUCTION In everyday life, humans are very vulnerable to risk. For example, the risk of accidents, property loss, illness, and even loss of life. This risk causes humans to lose their assets. Therefore, insurance comes with the aim of minimizing loss if the risk does occur. Insurance is an agreement between the insured (policyholder) and the insurer which requires the policyholder to pay a premium as compensation for the insurance benefits to be provided by the insurer in the event of a risk of failure to the policyholder [1]. One of the problems of insurance companies is how to determine the premium of a product. One of the techniques used is credibility theory. This theory predicts the amount of premium rates in the future based on experience data in the past. There are two predictive models that can be formed, namely a model for the number of claims and a model for the amount of claims (claim severity) that occur. This of course will be related to the type of data distribution used. The statistical approach that can be used in modeling the data is a parametric approach and a nonparametric approach. In this approach, we used a http://dx.doi.org/10.18860/ca.v7i4.16400 mailto:vina.apriliani@ar-raniry.ac.id https://creativecommons.org/licenses/by-sa/4.0/ Bühlmann's Credibility Model with Claims of Negative Binomial and 2-Poisson Distribution Ikhsan Maulidi 494 parametric approach where the claim data is assumed to follow a certain distribution [2]. One type of credibility that is widely used is the best accuracy credibility which consists of the Bühlmann model and the Bühlmann-Straub model [3]. In the Bühlmann model, policyholders are assumed to be the same number between time periods, while the Bühlmann-Straub model is a general form of the Bühlmann model where the number of policyholders may differ between time periods. Several studies related to the determination of premiums with Bühlmann and Bühlmann-Straub credibility parametrically can be seen in the research conducted by [4]–[7]. The distributions that are commonly used to model many claims are the Negative Binomial distribution and the Poisson distribution [8]. According to [9], mixed distribution is a distribution that can be considered in modeling the data. This is because, data modeling becomes more accurate. One of the mixed distributions used in this study is the 2-Poisson distribution [10]. The use of 2-Poisson distribution has not been found in previous studies. By using the assumption of a Negative Binomial and 2-Poisson distribution on the data, the Bühlmann’s credibility model will be determined on the data that satisfies this distribution. The equation for determining Bühlmann's credibility parameter is given explicitly. In addition, the prediction results obtained through the application of the data are also compared. Applications on nonparametric data using R can be seen in [11]. METHODS Poisson Distribution The Poisson distribution is a distribution with one parameter (𝜆). The probability function for the Poisson distribution is 𝑝𝑘 = 𝑒−𝜆𝜆𝑘 𝑘! , (1) with 𝑘 = 0,1,2, …. The expected value and variance of the Poisson distribution are 𝐸(𝑋) = 𝜆, (2) 𝑉𝑎𝑟(𝑋) = 𝜆 (3) [12]. The other properties and the applications of this distribution can be seen in [13]. Gamma Distribution A random variable is said to have a Gamma distribution with parameters 𝛼 and 𝛽, if it has a probability density function 𝑔𝑋 (𝑥; 𝛼, 𝛽) = 𝑥𝛼−1𝑒 −𝑥 𝛽⁄ 𝛤(𝛼)𝛽𝛼 , 𝑥 ∈ ℝ+ (4) with 𝛼 > 0, 𝛽 > 0, Γ(𝛼) > 0, and Γ(𝛼) = ∫ 𝑦𝛼−1 ∞ 0 𝑒−𝑦 𝑑𝑦. The parameter 𝛼 is called the shape parameter associated with the Gamma distribution and the parameter 𝛽 is generally called the scale parameter because it multiplies the random variable with a Gamma distribution by a positive constant. The expected value and variance of the Gamma distribution are 𝐸(𝑋) = 𝛼 𝜏 , (5) 𝑉𝑎𝑟(𝑋) = 𝛼 𝜏2 . (6) The evidence can be found at [14]. Bühlmann's Credibility Model with Claims of Negative Binomial and 2-Poisson Distribution Ikhsan Maulidi 495 Negative Binomial Distribution A Negative Binomial distribution is formed by an experiment that satisfies the following conditions: 1. An experiment consists of a series of independent experiments. 2. Each experiment can only produce one of two possible outcomes, failure and success. 3. The experiment continues until a total number of 𝑥 successes. The Negative Binomial distribution can be formed from a mixed distribution of the Poisson distribution and the Gamma distribution. By using Equation (1) and (4), it can be shown that this distribution has two parameters (𝛼 and 𝜏). The probability density function is 𝑝𝑋 (𝑥) = ( 𝑥 + 𝛼 − 1 𝛼 − 1 ) (1 − 𝜏)𝑥 𝜏𝛼 , (7) with 𝑥 = 0,1,2, … [15]. 2-Poisson Distribution The 2-Poisson distribution is a distribution with three parameters (𝜆1, 𝜆2, and 𝑝). The probability function for the 2-Poisson distribution is 𝑝𝑋 (𝑥) = 𝑒−𝜆1 𝜆1 𝑥 𝑥! 𝑝 + (1 − 𝑝) 𝑒−𝜆2 𝜆2 𝑥 𝑥! , (8) with 𝑥 = 0,1,2, … [16]. Some of the applications of this distribution can be seen in [17]-[18]. Bühlmann’s Credibility Bühlmann's credibility is a credibility model with the best accuracy approach. In this model, the number of policyholders observed is assumed to be the same every year. Premiums are determined based on a linear model between past data and theoretical premiums. The parameters used in the Bühlmann's credibility model are as follows: 1. The average value of individual claims or premiums and the expected values 𝝁(𝜽) = 𝑬(𝑿|𝚯), (𝟗) 𝝁 = 𝑬(𝝁(𝜽)), (𝟏𝟎) 2. The variance of the hypothetical mean 𝒂 = 𝑽𝒂𝒓(𝝁(𝜽)) = 𝑽𝒂𝒓(𝚯), (𝟏𝟏) 3. Variance process and the expected value of variance 𝒗(𝜽) = 𝑽𝒂𝒓(𝑿|𝚯), (𝟏𝟐) 𝒗 = 𝑬(𝒗(𝜽)), (𝟏𝟑) 4. Credibility coefficient 𝑲 = 𝒗 𝒂 , (𝟏𝟒) 5. Credibility factor 𝒁 = 𝑵 𝑵 + 𝑲 , (𝟏𝟓) 6. Credibility premium 𝑃𝐶 = 𝑍�̅� + (1 − 𝑍) (16) [19]. Goodnes-of-Fit Test Suppose 𝑚 is the largest value in the distribution of the observed data and 𝑟 is the number of parameters to be estimated, then with several predicted parameters, statistics Bühlmann's Credibility Model with Claims of Negative Binomial and 2-Poisson Distribution Ikhsan Maulidi 496 𝝌𝟐 = ∑ (𝒏𝒙 − 𝒏𝒑𝑿(𝒙)) 𝟐 𝒏𝒑𝒙(𝒙) 𝒎 𝒙=𝟎 (𝟏𝟕) asymptotically spread 𝝌𝟐 with degrees of freedom 𝒎 − 𝒓. Hypothesis testing is one of the statistical tests carried out for testing the suitability of the parameter 𝛽𝑖 which is made with the following hypothesis: 𝐻0: 𝜃 = 𝛽𝑖 , (data has a distribution that matches the distribution of the test) 𝐻1: 𝜃 ≠ 𝛽𝑖 . (data does not have a distribution that matches the distribution of the test) By using the values of the calculated chi-square and the table chi-square, the following decision rules apply: If 𝜒𝑐𝑎𝑙𝑐𝑢𝑙𝑎𝑡𝑒𝑑 2 ≤ 𝜒𝑡𝑎𝑏𝑙𝑒 2 then the null hypothesis is accepted and if 𝜒𝑐𝑎𝑙𝑐𝑢𝑙𝑎𝑡𝑒𝑑 2 > 𝜒𝑡𝑎𝑏𝑙𝑒 2 then the null hypothesis is rejected, by setting the alpha value as well as the degrees of freedom of the Chi-Square distribution [20]. RESULTS AND DISCUSSION Estimating Bühlmann’s Credibility Parameters using Negative Binomial Distribution Bühlmann's credibility model with claims of Negative Binomial distribution can be derived by providing an estimate of the credibility parameter assuming the frequency of claims with a Negative Binomial distribution. We know that the Negative Binomial distribution is a mixed distribution of the Poisson distribution and the Gamma distribution. Suppose 𝑋|Θ ∽ Poisson(𝜃) and Θ ∽ Gamma(𝛼, 𝜏), it can be proven that 𝑋 has a Negative Binomial distribution with parameters 𝛼 and 𝜏 [21]. The following formula is given for the Bühlmann’s credibility parameters using this distribution assumption. Hypothetical Mean and the Expected Value for Negative Binomial Model The hypothetical mean for Negative Binomial model can be determined using Equation (9). Since 𝑋|Θ ∽ Poisson(𝜃) then 𝜇(θ) = 𝐸(𝑋|Θ = θ) = 𝜃. The expected value of the hypothetical mean or known as the individual premium (𝜇) can be determined by Equation (10) as follows: 𝜇 = 𝐸(𝜇(Θ)) = 𝐸(Θ). Since Θ has Gamma distribution (𝛼, 𝜏), then according to Equation (5), 𝜇 = 𝛼 𝜏 . (18) To determine the credibility coefficient, it is necessary to estimate the value of parameter 𝑎. The value of 𝑎 is the variance of the hypothetical mean. Using Equation (11), the formula for the variance value of the hypothetical mean can be determined as follows: 𝑎 = 𝑉𝑎𝑟(𝜇(𝜃)) = 𝑉𝑎𝑟(Θ) = 𝐸(Θ2) − (𝐸(Θ)) 2 . Since Θ has Gamma distribution, then according to Equation (6), 𝑎 = 𝛼 𝜏2 . (19) Variance Process and the Expected Value for Negative Binomial Model The variance process formula and the expected value can be determined using Equation (12) and (13). Since 𝑋|Θ ∽ Poisson(𝜃) then 𝑣(𝜃) = 𝑉𝑎𝑟(𝑋|Θ = θ) = 𝜃. The expected value of the variance process (𝑣) ishypothetical𝑣 = 𝐸(𝑣(Θ)) = 𝐸(Θ) 𝑣 = 𝛼 𝜏 . (20) Bühlmann’s Credibility Coefficient for Negative Binomial Model The credibility coefficient is the ratio between the expected value of variance process Bühlmann's Credibility Model with Claims of Negative Binomial and 2-Poisson Distribution Ikhsan Maulidi 497 and variance of the hypothetical mean. By using Equation (11), (19), and (20), it is obtained 𝐾 = 𝑣 𝑎 = 𝜏. (21) Bühlmann's Credibility Premium for Negative Binomial Model After obtaining the formula from the credibility parameter for the Negative Binomial model, it can be formulated a formula for determining the premium with the Negative Binomial model using Equation (15) and (16) as follows: 𝑃𝐶 = 𝑍�̅� + (1 − 𝑍) μ, (22) with 𝑍 = 𝑁 𝑁 + 𝐾 = 𝑁 𝑁 + 𝜏 , �̅� = ∑ 𝑥𝑖 (𝑛𝑥𝑖 ) 𝑁 𝑖=1 (𝑛𝑥𝑖 ) . �̅� is the average of the observed data while μ is the individual premium which can be determined using Equation (18). The 𝑍 variable is also called the Bühlmann’s credibility factor for the frequency of claims with a Negative Binomial distribution, where 𝐾 is a credibility coefficient that satisfies Equation (21). Estimating Bühlmann's Credibility Parameters using 2-Poisson Distribution The following formula is given for the Bühlmann’s credibility parameters using this distribution assumption. Hypothetical Mean and the Expected Value for 2-Poisson Model As before, the hypothetical mean for 2-Poisson model can be determined using Equation (9). The 2-Poisson distribution gives that 𝑋|Θ ∽ Poisson(𝜃) and Θ ∽ 𝑢(𝜃) = { 𝑝 ∶ 𝜃 = 𝜆1 1 − 𝑝 ∶ 𝜃 = 𝜆2. (23) Since 𝑋|Θ ∽ Poisson(𝜃), then according to Equation (2), 𝜇(𝜃) = 𝐸(𝑋|Θ = θ) = 𝜃. (24) The expected value of the hypothetical mean (𝜇) is 𝜇 = 𝐸(𝜇(Θ)) = 𝐸(Θ) = 𝑝𝜆1 + (1 − 𝑝)𝜆2 = 𝑝(𝜆1 − 𝜆2) + 𝜆2. (25) Furthermore, the variance of the hypothetical mean can be determined using Equation (11) and (23) as follows: 𝑎 = 𝑉𝑎𝑟(𝜇(𝜃)) = 𝑉𝑎𝑟(Θ) = 𝐸(Θ2) − (𝐸(Θ)) 2 = 𝜆1 2 𝑝 + 𝜆2 2(1 − 𝑝) − (𝑝(𝜆1 − 𝜆2) + 𝜆2) 2 = 𝜆1 2 𝑝 + 𝜆2 2(1 − 𝑝) − (𝑝2(𝜆1 − 𝜆2) 2 + 2𝑝(𝜆1 − 𝜆2)𝜆2 + 𝜆2 2 ) = (𝜆1 2 − 𝜆2 2 )𝑝 + 𝜆2 2 − [(𝑝(𝜆1 − 𝜆2) + 2)𝑝(𝜆1 − 𝜆2) + 𝜆2 2 ] = (𝜆1 2 − 𝜆2 2 )𝑝 (𝜆1 − 𝜆2)(𝜆1 + 𝜆2) − [(𝑝(𝜆1 − 𝜆2) + 2)𝑝(𝜆1 − 𝜆2)] = [(𝜆1 + 𝜆2) − (𝑝(𝜆1 − 𝜆2) + 2)𝑝(𝜆1 − 𝜆2)] 𝑎 = [(1 − 𝑝)(𝜆1 + 𝜆2) − 2]𝑝(𝜆1 − 𝜆2). (26) Variance Process and the Expected Value for 2-Poisson Model The variance process and the expected value can be determined using Equation (12) Bühlmann's Credibility Model with Claims of Negative Binomial and 2-Poisson Distribution Ikhsan Maulidi 498 and (13). Since 𝑋|Θ ∽ Poisson(𝜃) then according to Equation (3), we get 𝑣(𝜃) = 𝑉𝑎𝑟(𝑋|Θ = θ) = 𝜃. The expected value of the variance process (𝑣) can be obtained using Equation (23), which is as follows: 𝑣 = 𝐸(𝑣(Θ)) = 𝐸(Θ) 𝑣 = 𝑝𝜆1 + (1 − 𝑝)𝜆2 𝑣 = 𝑝(𝜆1 − 𝜆2) + 𝜆2. (27) Bühlmann’s Credibility Coefficient for 2-Poisson Model The last part of determining Bühlmann's credibility premium is the need to assign a credibility coefficient value. By using Equation (14), (26), and (27), it is obtained 𝐾 = 𝑣 𝑎 𝐾 = 𝑝𝜆1 + (1 − 𝑝)𝜆2 [(1 − 𝑝)(𝜆1 + 𝜆2) − 2]𝑝(𝜆1 − 𝜆2) . (28) Bühlmann's Credibility Premium for 2-Poisson Model Bühlmann's credibility premium is obtained by using Equation (15) and (28) so that the Bühlmann credibility value for the 2-Poisson model is 𝑃𝐶 = 𝑍�̅� + (1 − 𝑍) μ, (29) with 𝑍 = 𝑁 𝑁 + 𝐾 = 𝑁 𝑁 + 𝑝𝜆1+(1−𝑝)𝜆2 [(1−𝑝)(𝜆1+𝜆2)−2]𝑝(𝜆1−𝜆2) , �̅� = ∑ 𝑥𝑖 (𝑛𝑥𝑖 ) 𝑁 𝑖=1 (𝑛𝑥𝑖 ) . The 𝑍 variable is also known as the Bühlmann’s credibility factor for the frequency of claims with 2-Poisson distribution. The value of μ can be determined by Equation (25) and �̅� is the average of the observed data. Application on Data The data used for the application of the model is data on the distribution of claims (𝑛𝑥 ) on the motor vehicle insurance portfolio in Singapore [22]. The number of claims occurred from 1993 to 2001 which can be seen in Table 1. Table 1. Portfolio of the number of claims from observation results (millions of dollars) 𝑥 𝑛𝑥 0 178.080 1 19.224 2 1.859 3 177 4 11 5 1 >5 0 Total 𝑛𝑥 = 199.352 Before applying to the model, it is first tested whether the claim frequency data in Table 1 has a Negative Binomial and 2-Poisson distribution or not. Testing the distribution of data was carried out using the chi-square test. Bühlmann's Credibility Model with Claims of Negative Binomial and 2-Poisson Distribution Ikhsan Maulidi 499 Negative Binomial Distribution Test The Negative Binomial distribution has two estimators for the parameters (�̂� and �̂�) based on Equation (7). Parameter estimates can be obtained using the moment method. Based on Table 1, the average value of the number of claims �̅� = 0,1179923, �̅�2 = 0,01392218, and variance 𝑆2 = 0,12881027. The values of �̂� and �̂� are �̂� = �̅�2 𝑆2 − �̅� = 0,01392218 0,01081797 = 1,28694966, �̂� = �̅� 𝑆2 − �̅� = 0,1179923 0,01081797 = 10,9070648. The distribution test steps carried out are as follows: a. Hypothesis formulation. 𝐻0 ∶ data has Negative Binomial distribution. 𝐻1 ∶ data is not distributed Negative Binomial. b. Calculates the probability for each claim frequency and the expected value. The probability is calculated for each claim frequency (𝑝𝑥 ) for each 𝑥 in the table data and based on the parameter estimator values and the Negative Binomial distribution formula, then for 𝑥 = 0: 𝑝𝑥 = ( 𝛼 + 𝑥 − 1 𝑥 ) ( 𝜏 1 + 𝜏 ) 𝛼 ( 1 1 + 𝜏 ) 𝑥 𝑝0 = ( 1,28694966 + 0 − 1 0 ) ( 10,9070648 1 + 10,9070648 ) 1,28694966 ( 1 1 + 10,9070648 ) 0 𝑝0 = 0,89324646. For the expected value (𝑛𝑝𝑥 ): 𝑛𝑝0 = 199.352(0,89324646) = 178.070,47. In the same way, it will produce a portfolio in Table 2. Table 2. Portfolio of the number of claims with Negative Binomial distribution 𝑥 𝑛𝑥 𝑛𝑝𝑋 (𝑥) 0 178.080 178.070,47 1 19.224 19.246,38 2 1.859 1.848,29 3 177 170,07 4 11 15,31 5 1 1,36 >5 0 0,12 Total 199.352 199.352 c. Determine the value of the chi-square test statistic. The chi-square test statistic determined by Equation (17) is obtained 𝜒𝑐𝑎𝑙𝑐𝑢𝑙𝑎𝑡𝑒𝑑 2 = 1,6912. With a 95% confidence interval, then 𝛼 = 0,05 and 𝜒𝑡𝑎𝑏𝑙𝑒 2 with degrees of freedom 𝑚 − 𝑟 = 5 − 2 = 3 is 7,8147. Based on the values of 𝜒𝑐𝑎𝑙𝑐𝑢𝑙𝑎𝑡𝑒𝑑 2 and 𝜒𝑡𝑎𝑏𝑙𝑒 2 , it can be concluded that 𝜒𝑐𝑎𝑙𝑐𝑢𝑙𝑎𝑡𝑒𝑑 2 < 𝜒𝑡𝑎𝑏𝑙𝑒 2 and the null hypothesis is accepted. Thus, the data used in this study has met the requirements for a Negative Binomial distribution. 2-Poisson Distribution Test There are three parameters that are assumed to have 2-Poisson distribution based on Bühlmann's Credibility Model with Claims of Negative Binomial and 2-Poisson Distribution Ikhsan Maulidi 500 Equation (8). By using the moment method, it is obtained that 𝑝 ̂ = 0,77481, 𝜆1̂ = 0,06191, and 𝜆2̂ = 0,31092. The distribution test steps carried out are as follows: a. Hypothesis formulation. 𝐻0 ∶ data has 2-Poisson distribution. 𝐻1 ∶ data is not distributed 2-Poisson. b. Calculates the probability for each claim frequency and the expected value. The probability is calculated for each claim frequency (𝑝𝑥 ) for each 𝑥 in the table data and based on the parameter estimator values and the 2-Poisson distribution formula, the portfolio is obtained in Table 3. Table 3. Portfolio of the number of claims with 2-Poisson distribution 𝑥 𝑛𝑥 𝑛𝑝𝑋 (𝑥) 0 178.080 178.081,52 1 19.224 19.217,82 2 1.859 1.868,36 3 177 170,53 4 11 12,89 5 1 0,79 >5 0 0,04 Total 199.352 199.352 c. Determine the value of the chi-square test statistic. The chi-square test statistic determined by Equation (17) is obtained 𝜒𝑐𝑎𝑙𝑐𝑢𝑙𝑎𝑡𝑒𝑑 2 = 0.6249634. With a 95% confidence interval, then 𝛼 = 0,05 and 𝜒𝑡𝑎𝑏𝑙𝑒 2 with degrees of freedom 𝑚 − 𝑟 = 4 − 3 = 1 is 3,8414. Based on the values of 𝜒𝑐𝑎𝑙𝑐𝑢𝑙𝑎𝑡𝑒𝑑 2 and 𝜒𝑡𝑎𝑏𝑙𝑒 2 , it can be concluded that 𝜒𝑐𝑎𝑙𝑐𝑢𝑙𝑎𝑡𝑒𝑑 2 < 𝜒𝑡𝑎𝑏𝑙𝑒 2 and the null hypothesis is accepted. Thus, the data used has met the requirements for a 2-Poisson distribution. Parameter Estimation of Bühlmann's Credibility Premium Based on the estimation of distribution parameter values that have been obtained, it can be determined the parameter estimation of Bühlmann's credibility premium. Determination of the estimated parameter value using Equation (18)-(29). The alleged results are presented in the Table 4. Table 4. Parameter estimation of Bühlmann's credibility premium from the data used Parameter Estimation Negative Binomial 2-Poisson �̅� 0,1179 0,1179 �̂� 0,1180 0,1179 �̂� 0,0108 0,3696 𝑣 0,1180 0,1179 �̂� 10,9071 0,3191 �̂� 0,9999 0,9494 𝑃𝐶 0,11790 0,11799 Based on the results in Table 4, it can be seen that the estimated frequency of claims in the next period assuming the data is Negative Binomial and 2-Poisson distribution is 0.11790 and 0.11799, respectively. This means that in the Negative Binomial model it is estimated that there will be 11.79% of policyholders who will make insurance claims in the Bühlmann's Credibility Model with Claims of Negative Binomial and 2-Poisson Distribution Ikhsan Maulidi 501 next period, while in the 2-Poisson model it is estimated that there will be 11.799% of policyholders who will make insurance claims. The two models give fairly close results, this is because the Bühlmann's credibility factor for the two distribution models is quite large, namely each 0.9999 and 0.9494. CONCLUSIONS Bühlmann's credibility formula has been given for data with Negative Binomial and 2- Poisson distribution. Both distributions are mixed distributions. Mixed distributions are quite well used in determining premiums with credibility. This is because in a mixed distribution, the distribution of claim frequency often depends on the distribution of risk. 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