Linear Bayes Estimation

As mentioned at the end of Chapter 5, it is generally difficult, if not impossible, to calculate the Bayes estimator $\hat\mu_{\rm B}= E(\mu(\theta_i)\mid \bf X_i)$ of the net premium $\mu(\theta_i)=E(X_{i,t}\mid \theta)$ in the ith policy based on the data $\bf X_i=(X_{i,1},\ldots,X_{i,n_i})'$. As before, we write $X_{i,t}$ for the claim size/claim number in the ith policy in the tth period. One way out of this situation is to minimize the risk,

$$\rho(\hat \mu)= E\left[(\mu(\theta_i)- \hat\mu)^2\right]\,$$

not over the whole class of finite variance measurable functions $\hat\mu$ of the data $\bf X_1,\ldots,\bf X_r$, but over a smaller class. In this section we focus on the class of linear functions

$${\mathcal L}= \left\{\hat \mu:\, \hat\mu=a_{0}+\sum_{i=1}^r \sum_{t=1}^{n_i} a_{i,t}\, X_{i,t}\,,\quad a_0\,,a_{i,t}\in {\mathcal R} \right\}\,$$

((6.0.1))

.

If a minimizer of the risk $\rho(\hat\mu)$ in the class ${\mathcal L}$ exists, we call it a linear Bayes estimator for $\mu(\theta_i)$, and we denote it by $\hat\mu_{\rm LB}$.