Simplest method of moments estimator
Problem
Let’s consider a very simple example. I have a random sample \(y_1, \ldots ,y_n\) and I want to estimate \(μ=E\left( y_i \right)\) using method of moments.
a. Find an elementary zero function \(f\left( y_i,μ \right)\)
b. Find the corresponding sample moments
c. Find the method of moments estimator of \(μ\) .
Solution
a. \(f\left( y_i,μ \right)=y_i-μ\) . \(E\left( f\left( y_i,μ \right) \right)=E\left( y_i-μ \right)=E\left( y_i \right)-μ=0\)
b.
\[ \frac{1}{n}\sum_{i=1}^{n}{ f\left( y_i,μ \right) }= \frac{1}{n}\sum_{i=1}^{n}{ \left( y_i-μ \right) }= \frac{1}{n}\sum_{i=1}^{n}{ y_i }-μ\]
c. Solve
\[ \frac{1}{n}\sum_{i=1}^{n}{ y_i }-{\hat{μ}}_{MM}=0\]
Solution:
\[{\hat{μ}}_{MM}= \frac{1}{n}\sum_{i=1}^{n}{ y_i }=\bar{y}\]
The method of moments estimator is the very natural sample mean.