Loss Models: From Data to Decisions, 4th Edition

13.1 Method of moments and percentile matching

For these methods we assume that all n observations are from the same parametric distribution. In particular, let the distribution function be given by

where θ^T is the transpose of θ. That is, θ is a column vector containing the p parameters to be estimated. Furthermore, let be the kth raw moment, and let π_g (θ) be the 100gth percentile of the random variable. That is, F. If the distribution function is continuous, there will be at least one solution to that equation.

For a sample of n independent observations from this random variable, let be the empirical estimate of the kth moment and let be the empirical estimate of the 100gth percentile

Definition 13.1 A method-of-moments estimate of θ is any solution of the p equations

The motivation for this estimator is that it produces a model that has the same first p raw moments as the data (as ...