Method of Moments

The Method of Moments

Info

Suppose that the parameter has components. For , define the th moment

and the th sample moment

Definition (Method of Moments Estimator)

The method of moments estimator is defined to be the value of such that

This defines a system of equations with unknowns.

Theorem

Let denote the method of moments estimator. Under appropriate conditions on the model, the following statements hold:

1. The estimate exists with probability tending to 1.
2. The estimate is consistent: .
3. The estimate is asymptotically Normal:

where , , and .

The last statement above can be used to find standard errors and confidence intervals. However, bootstrapping provides an easier method.

Example

Let . Then and . By equating these we get the estimator

Example

Let . Then and . We need to solve the equations

This is a system of 2 equations with 2 unknowns. The solution is

Sources

• Wasserman, L. (2010). All of Statistics: A concise Course in Statistical Inference. Chapter 9.2.