19.2 Nonparametric estimation
In this section we consider unbiased estimation of μ v, and a. To illustrate the ideas, let us begin with the following simple Bühlmann-type example.
Suppose that ni = n > 1 for all i and mij = 1 for all i and j. That is, for policyholder i, we have the loss vector
Furthermore, conditional on i = θi, Xij has mean
and Xi1, …, Xin are independent (conditionally). Also, different policyholders’ past data are independent, so that if i ≠ s, then Xij and Xst are independent. In this case,
Determine unbiased estimators of the Bühlmann quantities.
An unbiased estimator of μ is
For estimation of v and a, we use the following result. ...