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On Asymptotic Optimality in Linear Empirical Bayes Premium

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
Submitted On: 04-01-2010 | Views:	Article Tools:

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, entropy~1 loss function and entropy~2 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 (θ_i,(?),…,(?),) such thatθ_i is unobservable. Conditional onθ_i the random losses (?) are independent and satisfy the following assumptions:where m_(ij) 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 premiumμ_1(θ_1) for each i∈{1,…m}.In this paper, we modified the Biihlmann-Straub model: Let m_1,m_2,…denotethe year risk numbers. The random losses X_j(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, entropy~1 loss function and entropy~2 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(X_1,X_2,…,X_n) is a decision estimator ofθ,△denotes a decision-space.Theorem 2.1 (Linear Bayes premium under quadratic loss function) Let X_i, i=l, 2,…denote total claims of a policyholder in the ith policy period. The distribution of X_i depends on the parameterθ∈(?). Denote X_n= ( X_1, X_2,…, X_n)∈(?) as the policyholder's claim experience in the first n periods, and the claim numbersare m_1,m_2,…,m_n. For the loss function U_1(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(X_1, X_2,…, X_n) 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 X_i, i=l, 2,…denote total claims of a policyholder in the ith policy period. The distributionof X_i depends on the parameterθ∈(?).Denote X_n=( X_1, X_2,…, X_n)∈(?) as the policyholder's claim experience in the first n periods,and the claim numbers are m_1,m_2,…,m_n.For the balance loss function U_2(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 Entropy~1 loss function. Let the Entropy~1 loss function is: whereδis a decision estimator ofθand△denotes a decision-space.We call U_3 as Entropy~2 loss function.Theorem 2.3 (Linear Bayes premium under Entropy~1 loss function) Let X_i, i=l, 2,…denote total claims of a policyholder in the ith policy period. The distribution of X_i depends on the parameterθ∈(?). Denote X_n= ( X_1, X_2,…, X_n)∈(?) as the policyholder's claim experience in the first n periods.The claim numbers are m_1, m_2,…, m_n.For the loss function U_3(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 Entropy~2 loss function. Let the Entropy~2 loss function is: whereδis a decision estimator ofθand△denotes a decision-space.Theorem 2.4 (Linear Bayes premium under Entropy~2 loss function) Let X_i, i=l, 2,…denote total claims of a policyholder in the ith policy period. The distribution of X_i depends on the parameterθ∈(?).Denote X_n= ( X_1, X_2,…, X_n)∈(?) as the policyholder's claim experience in the first n periods.The claim numbers are m_1,m_2,…, m_n.For the loss function U_4(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 e_i is the linear Bayes estimator ofπ_i,and (?), is the linear empirical Bayes estimator ofπ_i.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|X_ij|_(2+δ)≤B for all i, j. Let (?), be an estimator ofμ, and Let (?) be an estimator ofη= (?) and let (?) be an estimator ofη_2 = (?) 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 Entropy~1 loss function is (?) and the one under Entropy~2 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 letη_2 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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