Method of Moments

Let X1, … , X n be a sample from a distribution Po that depends on a parameter ranging over some set. The method of moments consists of estimating by the solution of a system of equations

for given functions Thus the parameter is chosen such that the sample moments (on the left side) match the theoretical moments. If the parameter is k-dimensional one usually tries to match k moments in this manner. The choices lead to the method of moments in its simplest form.

Moment estimators are not necessarily the best estimators, but under reasonable conditions they have convergence rate and are asymptotically normal. This is a consequence of the delta method. Write the given functions in the vector notation and let be the vector-valued expectation Then the moment estimator solves the system of equations

For existence of the moment estimator, it is necessary that the vectorbe in the range of the function If is one-to-one, then the moment estimator is uniquely determined as

If is asymptotically normal and is differentiable, then the right side is asymptotically normal by the delta method.

The derivative of at is the inverse of the derivative of eat Because the function is often not explicit, it is convenient to ascertain its differentiability from the differentiability of This is possible by the inverse function theorem. According to this theorem a map that is (continuously) differentiable throughout an open set with nonsingular derivatives is locally one-to-one, is of full rank, and has a differentiable inverse. Thus we obtain the following theorem.