Method of Moments

Idea

Match sample moments with population moments to estimate parameters.

Moments

• First moment: $E[X]$ (mean)
• Second moment: $E[X^2]$
• $k$th moment: $E[X^k]$

Sample Moments

• $\bar{X} = \frac{1}{n}\sum_{i=1}^n X_i$ (first sample moment)
• $\frac{1}{n}\sum_{i=1}^n X_i^2$ (second sample moment)

Method

1. Express population moments in terms of parameters
2. Solve by matching with sample moments

Example: Normal Distribution

$X \sim N(\mu, \sigma^2)$. Parameters: $\mu, \sigma$.

First moment: $E[X] = \mu = \bar{X}$

Second moment: $E[X^2] = \mu^2 + \sigma^2 = \frac{1}{n}\sum X_i^2$

Solution:

• $\hat{\mu} = \bar{X}$
• $\hat{\sigma}^2 = \frac{1}{n}\sum X_i^2 - \bar{X}^2 = S_n^2$

Example: Exponential

$X \sim \text{Exp}(\lambda)$. Parameter: $\lambda$.

$E[X] = \frac{1}{\lambda} = \bar{X}$

$\hat{\lambda} = \frac{1}{\bar{X}} = \frac{n}{\sum X_i}$

Example: Uniform

$X \sim \text{Unif}[0, \theta]$. Parameter: $\theta$.

$E[X] = \frac{\theta}{2} = \bar{X}$

$\hat{\theta} = 2\bar{X}$

Properties

• Simple to compute
• Consistent (moment estimators converge)
• Not always efficient
• May not work if moments don’t exist

Summary

Step	Action
1	Write population moments in parameters
2	Equate to sample moments
3	Solve for parameters