# Is there any method to quantify parameter estimation uncertainty of method of moments fitting technique?

If I want to fit a distribution (let's say we can be certain about the type) to observations using maximum-likelihood method, I have many options to express the parameter estimation uncertainty due to scarcity of data, e.g. constructing confidence intervals using (following [Coles, 2001]):

• asymptotic normality [Coles, 2001, Ch. 2.6.4]
• deviance function [Coles, 2001, Ch. 2.6.5]
• profile likelihood [Coles, 2001, Ch. 2.6.6]

According to Coles all of these approaches are applicable only to maximum-likelihood estimation.

So my question: Is there any method to do similar quantification of parameter estimation uncertainty using method of moments?

Is it possible to derive asymptotic distributions as in case of maximum-likelihood approach?

I think bootstrapping might be used for this purpose, but it is a quite expensive approach. Any advice, reference is greatly appreciated.

S. Coles. 2001. An Introduction to Statistical Modeling of Extreme Values. Springer. ISBN 978-1-4471-3675-0.

## 1 Answer

According to Coles all of these approaches are applicable only to maximum-likelihood estimation.

This is simply not true. GMM estimators are asymptotically normal. Deviance has a direct comparison in tests based on the $J$ statistic. I am not aware of an equivalent to profile likelihoods, but Imbens and Spady appear to construct something similar with GMM by using empirical likelihoods.

A good reference for the asymptotic behavior of GMM estimators and their comparison to ML would be "Large Sample Estimation and Hypothesis Testing" by Newey and McFadden, which is available here.

• My bad, I think I added the word 'only', it is not in the book explicitly and I was referring to those particular chapters in the book with the formulas presented there and not to the approaches/methods in general. Thank you for the answer, references. Nov 11, 2014 at 19:08