9.2 Nonparametric estimationTo impl

9.2 Nonparametric estimation
To implement the limited-fluctuation credibility prediction for claim severity and aggregate loss/pure premium, an estimate of the coefficient of variation CX is required. 0 as defined in equation (9.1) is an example of a nonparametric estimator. Note that under the assumption of a random sample, sx and 0 are consistent estimators for the population standard deviation and the population mean, respectively, irrespective of the actual distribution of the random loss variable X. Thus, 0 is a consistent estimator for CX, although it is generally not unbiased.3
For the implementation of the Buhlmann and Buhlmann-Straub credibility models, the key quantities required are the expected value of the process variance, 0, and the variance of the hypothetical means, 0, which together determine the Buhlmann credibility parameter k. We present below unbiased estimates of these quantities. To the extent that the unbiasedness holds under the mild assumption that the loss observations are statistically independent, and that no specific assumption is made about the likelihood of the loss random variables and the prior distribution of the risk parameters, the estimates are nonparametric.
In Section 7.4 we set up the Buhlmann-Straub credibility model with a sample of loss observations from a risk group. We shall extend this set-up to consider multiple risk groups, each with multiple samples of loss observations over possibly different periods. The result in this set-up will then be specialized to derive results for the situations discussed in Chapter 7. We now formally state the assumptions of the extended set-up as follows:
1. Let Xij denote the loss per unit of exposure and mij denote the amount of exposure. The index 0 denotes the 0 risk group, for 0, with 0. Given 0, the index 0 denotes the 0 loss observation in the 0 group, for 0, . . . ,0, where 0, . . . ,r. The number of loss observations 0 in each risk group may differ. We may think of 0 as indexing an individual within the risk group or a period of the risk group. Thus, for the 0 risk group we have loss observations of 0 individuals or periods.
2. Xij are assumed to be independently distributed. The risk parameter of the 0 group is denoted by 0, which is a realization of the random variable 0. We assume 0 to be independently and identically distributed as 0.
3. The following assumptions are made for the hypothetical means and the process variance