Loss Models: From Data to Decisions, 4th Edition

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.

EXAMPLE 19.1

Suppose that n_i = n > 1 for all i and m_ij = 1 for all i and j. That is, for policyholder i, we have the loss vector

Furthermore, conditional on _i = θ_i, X_ij has mean

and variance

and X_i1, …, X_in are independent (conditionally). Also, different policyholders’ past data are independent, so that if i ≠ s, then X_ij and X_st are independent. In this case,

Determine unbiased estimators of the Bühlmann quantities.

An unbiased estimator of μ is

because

For estimation of v and a, we use the following result. ...