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 ni = n > 1 for all i and mij = 1 for all i and j. That is, for policyholder i, we have the loss vector

equation

Furthermore, conditional on i = θi, Xij has mean

equation

and variance

equation

and Xi1, …, Xin are independent (conditionally). Also, different policyholders’ past data are independent, so that if is, then Xij and Xst are independent. In this case,

equation

Determine unbiased estimators of the Bühlmann quantities.

An unbiased estimator of μ is

equation

because

equation

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

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