Ruin Probabilities in the Mixed Claim Frequency Risk Models

Mixture models are a fundamental tool in applied statistics, for most mixture models, including the widely used mixtures of Gaussians and hidden Markov models (HMMs); the current practice relies on the Expectation-Maximization (EM) algorithm, a local search heuristic for maximum likelihood estimation; an efficient method of moments approach to parameter estimation for a broad class of high-dimensional mixture models with many components was developed [1– 3]. Finite mixture models have a long history in statistics; a detailed review of mixture models and models-based clustering was provided in [4], for a finite mixture of regressions model, [5, 6] develop an efficient EM algorithm for numerical optimization with provable convergence properties. In this paper we consider two mixture models. In Section 2 we set up the binomial-Poisson model, and some important results about ruin probabilities are obtained by martingale approach. In Section 3 we also set up another Poisson-dualistic model; in this sectionwe obtain some important probabilistic properties and estimates for ruin probability. In the classical risk model, the surplus of an insurance company over the interval (0, t] is X(t), which is defined by


Introduction
Mixture models are a fundamental tool in applied statistics, for most mixture models, including the widely used mixtures of Gaussians and hidden Markov models (HMMs); the current practice relies on the Expectation-Maximization (EM) algorithm, a local search heuristic for maximum likelihood estimation; an efficient method of moments approach to parameter estimation for a broad class of high-dimensional mixture models with many components was developed [1][2][3].Finite mixture models have a long history in statistics; a detailed review of mixture models and models-based clustering was provided in [4], for a finite mixture of regressions model, [5,6] develop an efficient EM algorithm for numerical optimization with provable convergence properties.
In this paper we consider two mixture models.In Section 2 we set up the binomial-Poisson model, and some important results about ruin probabilities are obtained by martingale approach.
In Section 3 we also set up another Poisson-dualistic model; in this section we obtain some important probabilistic properties and estimates for ruin probability.
In the classical risk model, the surplus of an insurance company over the interval (0, ] is (), which is defined by where () is to be interpreted as the number of claims on the company during the interval (0, ]; assume that {();  ≥ 0} is an homogeneous Poisson process with intensity .In the complex problems of actual operation in insurance business, insurance company classifies the risk by its some characteristics, but the claim frequency  of the individual policy which has been classified into the same kind of portfolio may be different; that is, this is the so-called nonhomogeneity.For a nonhomogeneous portfolio, we can assume that  is a random variable; thus the mixed Poisson distribution model can be derived.In general, just as reported in [6] if the number for claim  is a discrete distribution with parameter  and the distribution sequence is and the parameter  is random variable or random vector, its probability distribution function is Then the corresponding risk model is a mixed claim frequency risk model.A sequence {  } ∞ 1 is an independent and identically distributed nonnegative random variable, having the common distribution function , with (0) = 0, mean value , and variance  2 ; the above random process and random sequence are mutually independent; ,  ( > 0, 0 <  < 1) are both constants.

Binomial-Poisson Model
Let Then the process defined by ( 4) is a binomial-Poisson mixed claim frequency risk model.The conditional moment generating function of  is Therefore Then {();  ≥ 0} is a homogeneous Poisson process with intensity .Thus the model defined by ( 4) is a classical risk model.

The Meaning of the Model in the Insurance Practice.
Suppose that () is the number of accidents during the interval (0, ]; the number of claims per accident  is 0-1 distribution with ; that is, ( = 1) = , ( = 0) = 1− (for example, in the deductible insurance, the probability of loss exceeding the amount of deductible is ); then the number of claims () during the interval (0, ] is a conditional binomial distribution with parameters , .Obviously the process {();  ≥ 0} is a -sparse process of the process {();  ≥ 0}.{();  ≥ 0} is a homogeneous Poisson process with intensity , if only {();  ≥ 0} is a homogeneous Poisson process with intensity . is the premium rate, {  } ∞ 1 is the size of claim amount per accident, and ∑ () =1   is the total claim process.

Several Conclusions about the Ultimate Ruin Probability.
We can get easily In order to stabilize the operation of company, we should ensure that premiums received in a unit of time meet  > .
The relative safety loading  is defined by We can now define the ruin probability () of a company facing the risk process (4) and having initial capital .Consider  () =  { +  () < 0 for some  > 0} . (9) Let We assume that there exists  ∞ > 0 such that ℎ() ↑ +∞ when  ↑  ∞ , that is, a light-tailed distribution .
where  is given by (8).
Theorem 4. Consider where  is the positive solution of ℎ() = /, which is called adjustment coefficient.And  is given by (8).
Theorem 5. Ruin probability: where  is the positive solution of ℎ() = /, which is called adjustment coefficient.
The above theorem can be derived directly by the corresponding results in classical risk theory [7].

The Meaning of the Model in the Insurance
Practice.We assume that a portfolio is composed of high risk and low risk insurance policy, where high risk policy is accounting for  1 and low risk policy is accounting for  2 = 1 −  1 ; the Poisson parameter of these two kinds of policy, respectively, is  1 ,  2 (corresponding high risk cover for  1 , obviously  1 >  2 ). is the premium rate, {  } ∞ 1 is the size of claim amount per accident, and ∑ () =1   is the total claim process.

Some Probabilistic Properties of Model.
The probabilistic properties of the number of claims () are as follows.
Property 1. Claim frequency distribution is Proof.The above distribution can be derived by The proof is ended.
Property 2. The mean and variance of () are Proof. Consider The proof is ended.
Since the mean and variance of the Poisson distribution are always equal, if the sample variance of the portfolio's claim of a random variable is greater than the number of the sample mean, we can conclude that the existence of this policy combination of a degree of nonhomogeneity, and because Var[()] − [()] = Var() can reflect the variance of the degree of nonhomogeneity of portfolio, if the variance is much more greater than the mean, that is, the bigger ( 1 −  2 ) 2  1  2 is, the more serious the nonhomogeneity is. Proof.Consider The proof is ended.Proof.Consider The proof is ended.
Proof.We only need to prove that when the conditional distribution of {  } about  is exponentially distributed, {  } follow the mixed exponential distribution.Then, when  > 0, the distribution function of {  } is Thus the density function of {  } is The proof is ended.
Theorem 9.It occur () claims over (0, ]; then the occurrence probability of intensity  =  1 for claim is Proof.It can be derived by The proof is ended.
Thus we can get Theorem 10.Assume that () is the time interval from the moment t to the next claim; then the conditional distribution of () is Proof.Obviously () is the occurrence time after the moment , when  is a constant; then () follows the exponential distribution with .Then the result can be derived by The proof is ended.
The probabilistic properties of total amount of claims () = ∑ () =1   are as follows.
Property 4. The mean and variance of total amount of claims The proof is ended.
where   () is the moment generation function of the individual claim amount.
Proof.We can get the following from (19): The proof is ended.

Estimation of Lundberg Exponential Upper Bounds for the Ultimate Ruin Probability. Obviously, 𝐸[𝑌(𝑡)] = [𝑐 − (𝜆
It seems very natural to assume the premium rate per unit of time  > ( 1  1 +  2  2 ).Further, let  >  1 , in order to stabilize the operation of the company.
The relative safety loading: Obviously, 1 can be seen as the corresponding relative safety loading of portfolio consisted by high risk policy.Obviously in terms of premium rate and the average individual claim amount are equal, the relative safety loading of the corresponding portfolio of model ( 14) should be greater.
Let  be the initial capital; then the ruin moment is defined as Obviously   is a   -stopping time.
Proof.We get from (43) that The proof is ended.
Choose  0 < ∞ and consider  0 ∧   which is a bounded -stopping time.We get from Doob's stopping theorem that Using +(  ) ≤ 0 on {  < ∞}, the lower bound was shown to be given by Thus we have (50) We now want to choose  as large as possible under the restriction {sup ≥0 exp[(ℎ() − )]} < ∞.Let  denote that value, named adjustment coefficient of surplus process (14). Since where   () =   ℎ() − ,  = 1, 2, the two terms of the right side in the above equality are both positive, which corresponds to restrict Since thus we just need to restrict Then we have Theorem 12.The ultimate ruin probability meets the inequality where  is the only positive solution of  1 () = 0, named adjustment coefficient.
The proof is ended.

Boundary of the
Proof.From [7] we know  < 2( −  1 )/ 1  2 ; then we can get the conclusion.The proof is ended.
Form the above theorem and the expression of the relative safety loadings ,  1 , we get The result shows that the upper bound of  can be only defined by the one-or two-order moment of individual claim amount and the relative safety loading; while having nothing to do with the Poisson parameter, this case have its convenience in use.
Theorem 14 (see [7]).If the individual claim amount {  } has the upper bound , that is,   ≤ , we have The result indicates that, under the conditions of the theorem, the lower bound of  can be only defined by the relative safety loading  1 and the upper bound of individual claim amount.
When the exact value of  cannot be determined by the equation  1 () = 0, we can solve its approximate solution by the numerical method, and using these two boundaries as the initial values for iteration, we can quickly find the approximate solution satisfying requirements of certain accuracy.
3.6.The Probability of Survival.Except for a few special cases, generally, complicated calculation is needed to get the adjustment coefficient values.Through the following discussion on integrodifferential equations satisfied by survival probability, we can avoid the computation of the adjustment coefficient.
Since the number of claim process {();  ≥ 0} about  is a conditional renewal process and will not be ruined during the period of (0,  1 ), then we get  (66) Proof.From (65) and the hypothesis of Φ 1 () and Φ 2 (), by taking derivation, we can get the above result.
The proof is ended.
Φ 1 () and Φ 2 () are calculated using the above equations; thus the expression of Φ() can be obtained.According to the above method to obtain the survival probability, we can immediately get the expression of ruin probability, and then the calculation of the adjustment coefficient can be avoided.

3. 1 .
The Setting-Up of the Model Definition 6.The random variable  follows two-point distribution; that is,
Adjustment Coefficient .The exact value of adjustment coefficient  generally cannot be determined by  1 () = 0, but because of its great significance to estimate the upper bound of the ruin probability, we estimate the boundary of the adjustment coefficient .