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Automobile insurance is one of the most popular research areas, and there are a lot of different methods for it .We uses linear empirical Bayesian estimation for the study of automobile insurance, giving the estimator of the policy’s future claim size. Thus, a new point of view is given on the pricing of automobile insurance.

Automobile insurance is one of the most important insurance in property insurance. Recently, automobile insurance premium income grows steadily in our country. Since 2000, automobile insurance premium income accounted for the proportion of insurance premium income and has been maintained at above 60%. Cao Jing, Li Ping, Gao Yuan (2006) [

In the current, actuary pricing risk premium in auto insurance usually use the claim number mean and optimal estimation to calculate the claim amount. We commonly use the following methods to study the optimal estimation model of the claim size. For example: Poisson-inverse Gauss distribution, Mixed Pareto distribution Model, Mixed two-parameters Exponential distribution Model, Three parameters of Mixed Gamma distribution Model, Two dimensional Risk Model, Two negative binomial model, compound PNB distribution model, and so on. For the claim amount, there are only a few optimal estimation models, and now the relatively mature model is Pareto model [

Based on linear empirical Bayesian method (L.E.B method) [

First we introduce two Lemmas [

Lemma 1: [

is right for all functions, where is a vector, Representing the functions of with the number, and the second-order moments of are existed with.

The Explaining of equation (1) is that: conditional expectation is the closest function to in all functions of, and the Close degree can be measured with the mean square error between and. However, it’s difficult to calculate the conditional expectation, so if the linear functions are taken into account, the problem will be solved. This is the following conclusion:

Lemma 2: [

in which

The results show that is the closest function to y in all linear functions of x. with Lemma 2, if we replace y with the parameter vector of statistical problems, and regard x as a sample, then the estimation of parameter will be

And it will be the minimum mean square error estimation. From Equation (4), if we can find the values of, , and in the right equation, the estimation of parameter vector can be calculated. If we suppose variable X-normal distribution for which the variance is known, with the above results, we can give the estimation expression of parameter. With is known in statistical problems, and is a conditional probability density with parameter, so and can be evaluated.

Theorem 1: If the conditional probability density of normal distribution is, then have the following two properties:

And the estimation of parameter vector will be

Proof: With the conditions in the theorem, we can get

So the variance

Now we simultaneously substitute Equation (10) and Equation (8), Equation (9) on Equation (4), and we gain parameter estimation:

From Equation (7), as long as we have the history data of the sample, we can figure out that the sample mean is, and the sample variance is.

With classical statistics methods, we can estimate with, while we can estimate with, then

.

Assuming that the amount claim is subject to lognormal distribution with the parameters, with the variance is known, and its probability density function is

for which. With Equation (11), the expectations of the future claimed amount for a single insurance slip in insurance companies will be

Thus if we get the optimal estimation of the parameter, we can calculate the expectation of Y.

Suppose the Claim history of a single insurance slip in the insurance company is, and assume the current new observation is a vector y. Now we Calculate the estimation of parameter by using the L.E.B method in the classic statistics. Because when a single insured claim occurs, the claim amount of money Y is subject to log-normal distribution with the parameters, that is. So we can get that

,.

When variance is known, we get the estimation of the parameter with

For which

so when we use L.E.B. method to calculate the estimation of the parameter, is equal to the weighted average value between the true value and the experience value, and the weight is respectively

and. So when the Claim history of a single insurance slip is, and assume the current new observation is a vector, from Equation (12) we can get that the expected estimation of the future claimed amount for a single insurance slip is

Now suppose that automobile insurance claim number X obey mixed three parameter gamma distribution, and the historic claim frequency information for insurance slip is, we can calculate the optimal claim frequency estimation for (seeing Reference [

While it is assumed that the claim amount is subject to lognormal distribution with parameter, and when the history Claim of a single insurance slip is, and assume the current new observation is a vector y, we can get that the expected estimation of the future claimed amount is based on L.E.B. method, so we can calculate the Optimal estimation of future premiums with

.

In recent years, with the rapid increase of motor vehicles, automobile insurance premium income accounted for the proportion of insurance income has been gradually improved, thus insurance premium price becomes particularly important. This paper mainly studies the policy’s future claim amount estimation value, and on this basis, predicting the premium price through the establishment of model. Through the use of linear empirical Bayesian method (L.E.B method), given the parameter estimation of in the lognormal distribution with its parameter. the result is that the minimum mean square error estimation is the weighted average value of the observed value and the experience valuewith its weights and. At the same timeaccording to (12), we calculate the policy’s future claim amount expected value estimation is

Finally, in the hypothesis that claim number X obey three parameters with mixed gamma distribution, gets the optimal premium predictive value for insurance premium, provides certain theory basis in the pricing problem.

This work was supported by the Sichuan Provincial Office of education projects for Humanities and Social Sciences LY09-10.