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 θT 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 πg (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 ...