Method of Moments Foundations

Overview

The Method of Moments (MoM) equates population moments to sample moments to estimate parameters.

Definition

For a distribution with \(p\) parameters \(\theta_1, \ldots, \theta_p\), the MoM sets:

\[ \mu_k'(\theta_1, \ldots, \theta_p) = m_k' = \frac{1}{n}\sum_{i=1}^n X_i^k, \quad k = 1, \ldots, p \]

Properties

• Consistency: MoM estimators are consistent under mild conditions
• Simplicity: Often yields closed-form solutions
• Not necessarily efficient: May have larger variance than MLE
• May produce inadmissible estimates: e.g., negative variance estimates