# 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 *k*th raw moment, and let π_{g} (θ) be the 100*g*th 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 *k*th moment and let be the empirical estimate of the 100*g*th 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 ...

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