| People often use credibility model to calculate the premium in the practical work of insurance.However, the structural parameters are often unknown,so we need to estimate the structural parameters based on the past data and this approach is also called linear empirical Bayes method.The estimator obtained by linear empiricalBayes method is called linear empirical Bayes premium.The most important standard to judge the estimator good or not is to exam wether it is asymptoticallyoptimal. There have been a lot of researches about the asymptotic optimality theory,but all of them are based on the Biihlmann-Straub model and quadratic loss function;In this paper, the author will discuss the linear empirical Bayes premiumunder quadratic loss function, balance loss function, entropy1 loss function and entropy2 loss function in a general Biihlmann-Straub model.In the empirical Bayes estimation of premiums in a Biihlmann-Straub model one is faced with m independent risk contracts.In the ith contract there is a random vector (θi,(?),…,(?),) such thatθi is unobservable. Conditional onθi the random losses (?) are independent and satisfy the following assumptions:where mij are unknown positive numbers.The problem is to use the data from all of the m contracts in order to obtain asymptotically optimal estimates of the pure premiumμ1(θ1) for each i∈{1,…m}.In this paper, we modified the Biihlmann-Straub model: Let m1,m2,…denotethe year risk numbers. The random losses Xj(j≥1) satisfy the following assumptions for each j, We call the above model as general Buhlmann-Straub model. We will discuss the linear Bayes premium under quadratic loss function, balance loss function, entropy1 loss function and entropy2 loss function and their asymptotic optimality in the above model.In Chapter 2,we inferred the linear empirical Bayes premiums in the general Buhlmann-Straub model.In Section 2.1,we give the Buhlmann-Straub model and general B(?)hlmann-Straubmodel.In Section 2.2, the author considered the linear Bayes premium under quadratic loss function. Let the quadratic loss function is:whereδ=δ(X1,X2,…,Xn) is a decision estimator ofθ,△denotes a decision-space.Theorem 2.1 (Linear Bayes premium under quadratic loss function) Let Xi, i=l, 2,…denote total claims of a policyholder in the ith policy period. The distribution of Xi depends on the parameterθ∈(?). Denote Xn= ( X1, X2,…, Xn)∈(?) as the policyholder's claim experience in the first n periods, and the claim numbersare m1,m2,…,mn. For the loss function U1(θ,δ),the linear Bayes. premium of the (n+l)th year is:where (?) is the sample mean.In Section 2.3, we considered the linear Bayes premium under balance loss function. Let the balance loss function is:where 0≤α≤1 is known,δ=δ(X1, X2,…, Xn) is a decision estimator ofθ, and△denotes a decision-space.We call this loss function as Balance Loss Function. Theorem 2.2 (Linear Bayes primium under balance loss function) Let Xi, i=l, 2,…denote total claims of a policyholder in the ith policy period. The distributionof Xi depends on the parameterθ∈(?).Denote Xn=( X1, X2,…, Xn)∈(?) as the policyholder's claim experience in the first n periods,and the claim numbers are m1,m2,…,mn.For the balance loss function U2(θ,δ), the linear empiRIcal Bayes premium of the (n+l)th year is where (?) is sample mean,In Section 2.4, the author considered the linear Bayes premium under Entropy1 loss function. Let the Entropy1 loss function is: whereδis a decision estimator ofθand△denotes a decision-space.We call U3 as Entropy2 loss function.Theorem 2.3 (Linear Bayes premium under Entropy1 loss function) Let Xi, i=l, 2,…denote total claims of a policyholder in the ith policy period. The distribution of Xi depends on the parameterθ∈(?). Denote Xn= ( X1, X2,…, Xn)∈(?) as the policyholder's claim experience in the first n periods.The claim numbers are m1, m2,…, mn.For the loss function U3(θ,δ), the linear empirical Bayes premium of the (n+l)th year iswhere (?) is sample mean,In Section 2.5,we considered the linear Bayes premium under Entropy2 loss function. Let the Entropy2 loss function is: whereδis a decision estimator ofθand△denotes a decision-space.Theorem 2.4 (Linear Bayes premium under Entropy2 loss function) Let Xi, i=l, 2,…denote total claims of a policyholder in the ith policy period. The distribution of Xi depends on the parameterθ∈(?).Denote Xn= ( X1, X2,…, Xn)∈(?) as the policyholder's claim experience in the first n periods.The claim numbers are m1,m2,…, mn.For the loss function U4(θ,δ), the linear empirical Bayes premium of the (n+l)th year iswhere (?) is sample mean,In Chapter 3, we give a definition of asymptotic:Definition 3.l The premium estimators (?), are called asymptotically optimal ifwhere ei is the linear Bayes estimator ofπi,and (?), is the linear empirical Bayes estimator ofπi.In Section 3.1, Theorem 3.1 gives provide sufficient conditions of asymptotic optimality in Biihlmann-Straub model.Theorem 3.1 Suppose for some (?) for all i, j.Let (?) be an estimator ofμand let (?)≥0 be an estimator ofη.Let (?) with (?). Then the following two conditions are sufficient for asymptotic optimality of (?), as described in Definition 3.1:In Section 3.2, Theorem 3.2 gives provide sufficient conditions of asymptotic optimality in a general Biihlmann-Straub model.Theorem 3.2: Suppose for some b <∞andδ> 0,E|Xij|2+δ≤B for all i, j. Let (?), be an estimator ofμ, and Let (?) be an estimator ofη= (?) and let (?) be an estimator ofη2 = (?) Let (?) with (?) Then the following three conditions are sufficient for asymptotic optimality of (?) as described in Definition 3.1:Since in B(?)hlmann-Straub model, the linear Bayes credibility under balance loss function is (?), The one under Entropy1 loss function is (?) and the one under Entropy2 loss function is (?),they all have the same or similar form with the one under quadratic loss function. Therefore, their asymptotic optimality's sufficient conditions can also be given by theorem 3.2.The estimators v and (?) are unbiased and provide an estimator ofηgiven by:Since (?) andwe have Sundt(1983) proposes estimatingμbyObserved that with(?), we have (?) is equal to (?)when (?) > 0, and (?) is equal to (?), when (?) = 0.Hence interpreting 0/0 as 0 we can write Sundt's estimator ofμasFor an alternative estimator ofη,letand letη2 be given bywhere g =(?)In Chapter 3, we showed the asymptotic optimality of the estimator proposed by Sundt(1983) above in Theorem 4.1, Theorem 4.2 and Theorem 4.3... |
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