Catastrophe is a loss that has a low probability of occurring but can lead to highcost claims. This paper uses the data of fire accidents from a reinsurance company in Thailand for an experiment. Our study is in two parts. First, we approximate the parameters of a Weibull distribution. We compare the parameter estimation using a direct search method with other frequently used methods, such as the least squares method, the maximum likelihood estimation, and the method of moments. The results show that the direct search method approximates the parameters more precisely than other frequently used methods (to fourdigit accuracy). Second, we approximate the minimum initial capital (MIC) a reinsurance company has to hold under a given ruin probability (insolvency probability) by using parameters from the first part. Finally, we show MIC with varying the premium rate.
The risks of an insurance company can be assessed based on disasters of varying severity. The insurance company evaluates its risks in order to maintain consistency. Insolvency cannot occur if the company knows how to manage the risk process. For instance, if an insurance company does not have sufficient initial capital to pay some claims, then the company can share some of the risks by transferring them to reinsurance. Parameter estimation is an important method to construct a risk model in an insurance business. The wellknown methods are the least squares method, the maximum likelihood estimation, and the method of moments. This research is interested in the estimation of the parameters of a Weibull distribution which are represented by fire accident data. Many authors have studied the different aspects of Weibull parameters. Bergman [
In this paper, we start by introducing the surplus of nonlife insurance. The surplus can be described as
Lundberg [
(1) Claims happening at times
(2) The
(3) The claim severity process
Next, we define the
Next, we denote
Sattayatham et al. [
This can be rewritten as
Reinsurance and investment are a normal activity of insurance companies because reinsurance can reduce the risk (ruin probability) arising from claims, and the investment can make more profit for the company. The process can be controlled by reinsurance, i.e., by choosing the
In the case of an
In case of a
Therefore, the retention level
Recently, Luesamai and Chongcharoen [
In this research, we study a risk model of reinsurance by adding investment (buying bonds or fixed accounts). We present two parts consisting of an approximation of the parameters of Weibull distribution and the calculation of the minimum initial capital of investment discrete time surplus process with Weibull distribution. The first part is the estimation of the Weibull parameters using the direct search technique, the least squares method, the maximum likelihood estimation, and the method of moments. The selected method gives the minimum KS statistic value (to fourdigit accuracy). The second part is a simulation to calculate the ruin probability of the surplus process under the condition that a reinsurance company can invest in riskfree assets (bonds or fixed accounts). The surplus process is of the form
The classical Weibull distribution is useful for reliability engineering. Moreover, it can be extended to the various families of probability distributions which deal with the estimation of model parameters by maximum likelihood and it can also be used to illustrate the potentiality of the extended family with two applications to real data [
Normally, claims that occur infrequently but have high costs will be called catastrophe losses. For example, a fire accident is a type catastrophe loss. Furthermore, a Weibull distribution that shape parameter being less than one and scale parameter being greater than zero is also an example of catastrophe loss. The probability density function of three parameters of a Weibull distribution is of the form
In our work, the costs of all claims
Bergman, Sullivan, and Lauzon proposed the probability estimator
We take the natural logarithm to (
Let
By partial derivative the loglikelihood function
Thus
The
Setting
Since
We calculate a coefficient of variation (CV) of the Weibull distribution from the formula
If we apply the bisection method to (
The KolmogorovSimirnov (KS) test is a distance test. The KS test works well with small samples. Let
Let
Throughout this research, we use the data of the cost of all claim sizes that are greater than twenty million Baht from the fire insurance of Thai Reinsurance Public. The amount
Claims size
15.50  6.70  49.90  6.40  12.40  102.70  44.90 
56.50  138.90  107.30  13.20  13.10  37.70  5.70 
40.00  1.80  40.20  84.30  47.30  112.20  9.20 
28.50  0.90  43.10  3.60  45.80  70.00  2.30 
35.30  7.20  64.60  13.00  2.40  1.40  2.10 
7.50  31.50  4.20  37.20  0.70  24.40  14.20 
20.10  0.40  33.20  9.30  10.80 
Shape and scale parameters for a variety of estimation methods.
Method  Type 




1  Least squares method 1  5.1  0.8405  28.7721 
2  Least squares method 2  5.2  0.8310  28.8602 
3  Least squares method 3  5.3  0.7984  29.1888 
4  Least squares method 4  5.4  0.8580  28.6168 
5  Maximum likelihood estimation    0.8633  28.8668 
6  Method of moments    0.9286  30.0055 
We perform the KS test according to the six methods as shown in Table
The empirical CDF
The direct search technique is a numerical optimization method. The principle of the direct search technique is iterative and random shifting from the beginning solution to a better solution. The direct search technique process can be described as follows.
Begin from the simple initial parameters
Calculate value
where
Calculate the KS statistic value by using
Compare the KS value of Step
If the KS statistic value of Step
If (4.1) is false we set
Iterate until the process is complete (fourdecimal place accuracy).
If value for the KS test is approximated to fourdecimal place accuracy, then the direct search technique is completed. We perform the simulation 1000 times. The results are shown in Table
Parameters
Times 


Value for the KS test 

1  0.5049  31.0142  0.1262 
10  0.5489  30.6153  0.1088 
100  0.7652  29.5447  0.0544 
1000  0.7652  29.5451  0.0544 
Table
The value for the KS test from a variety of estimation methods.
Method  Type 


Value for the KS test 

1  Least squares method 1  0.8405  28.7721  0.0649 
2  Least squares method 2  0.8310  28.8602  0.0636 
3  Least squares method 3  0.7984  29.1888  0.0591 
4  Least squares method 4  0.8580  28.6168  0.0674 
5  Maximum likelihood estimation  0.8633  28.8668  0.0709 
6  Method of moments  0.9286  30.0055  0.0964 
7  Direct search technique  0.7652  29.5450  0.0544 
We consider the surplus process (
The flow chart of a simulation of the ruin probability.
In Figure
Thereafter we perform the simulation of the ruin probability by setting the initial capital
The relation between the initial capital and the ruin probability in the case of
We consider the relationship between the ruin probability
Next, we use the quadratic regression method in (
The parameters





0.1  


0.2 



0.3 



0.4 



0.5 



0.6 



0.7 



0.8 



0.9 



1.0 



Finally, we consider that the ruin probability is not greater than
For example, we can apply (
The MIC of a reinsurance company has to hold for ensuring that the ruin probability is not greater than

Premium rate  MIC million Baht  MIC million Baht 





0.1  4.9027  1642.3413  1622.0112 
0.2  5.3484  1627.1924  1606.6316 
0.3  5.7941  1615.2543  1594.3263 
0.4  6.2398  1602.4812  1581.1251 
0.5  6.6855  1589.8425  1567.9948 
0.6  7.1312  1582.7270  1560.1270 
0.7  7.5769  1574.8999  1551.4706 
0.8  8.0226  1563.0873  1538.9719 
0.9  8.4683  1555.8489  1530.7069 
1.0  8.9140  1546.0777  1520.0044 
Table
This work presents the use of the direct search technique to estimate the parameters of a Weibull distribution. The direct search technique is compared with the least squares method, the maximum likelihood estimation, and the method of moments. The results show that the direct search technique is more accurate than the other methods (accuracy of four decimal points). Fire accident data is used for special cases exceeding 20 million Baht. The results show that if the safety loading is increased, the MIC decreases under the given
The data used to support the findings of this study are included within the article.
The authors declare that there are no conflicts of interest regarding the publication of this paper.
This paper is supported by the Faculty of Science and Engineering, Kasetsart University, Chalermphrakiat Sakon Nakhon Province Campus, and Mathematics and Statistics Program, Faculty of Science and Technology, Sakon Nakhon Rajabhat University, Sakon Nakhon, 47000, Thailand.