Let [mathjaxinline]X[/mathjaxinline] be a random variable that takes on values [mathjaxinline]-1[/mathjaxinline] and [mathjaxinline]+1[/mathjaxinline] with probabilities [mathjaxinline]p[/mathjaxinline] and [mathjaxinline]1-p[/mathjaxinline], respectively. Let [mathjaxinline]\widehat{m}_1[/mathjaxinline] be the sample average of [mathjaxinline]n[/mathjaxinline] i.i.d. observations of [mathjaxinline]X[/mathjaxinline].
What is the method of moments estimator [mathjaxinline]\widehat{p}_ n^{\text {MM}}[/mathjaxinline]?
The first moment of [mathjaxinline]X[/mathjaxinline] is given by:
[mathjaxinline]\mu_1 = (-1) \cdot p + (+1) \cdot (1-p) = 1-2p[/mathjaxinline]
Setting this equal to the sample average and solving for [mathjaxinline]p[/mathjaxinline], we get:
[mathjaxinline]\widehat{p}_ n^{\text {MM}} = \frac{1 - \widehat{m}_1}{2}[/mathjaxinline]
The method of moments estimator [mathjaxinline]\widehat{p}_n^{\text{MM}}[/mathjaxinline] can be obtained by equating the sample moments to the corresponding population moments and solving for [mathjaxinline]\widehat{p}_n^{\text{MM}}[/mathjaxinline].
In this case, the sample moment is the sample average [mathjaxinline]\widehat{m}_1[/mathjaxinline], and the population moment is the expected value of [mathjaxinline]X[/mathjaxinline].
The expected value of [mathjaxinline]X[/mathjaxinline] can be calculated using the formula:
[mathjax]\begin{align*}
\text{E}(X) &= -1 \cdot \text{Pr}(X = -1) + 1 \cdot \text{Pr}(X = 1) \\
&= -1 \cdot p + 1 \cdot (1-p) \\
&= 1 - 2p
\end{align*}[/mathjax]
Equating the sample moment [mathjaxinline]\widehat{m}_1[/mathjaxinline] to the population moment [mathjaxinline]1 - 2p[/mathjaxinline], we have:
[mathjax]\begin{align*}
\widehat{m}_1 &= 1 - 2\widehat{p}_n^{\text{MM}}
\end{align*}[/mathjax]
Solving for [mathjaxinline]\widehat{p}_n^{\text{MM}}[/mathjaxinline], we get:
[mathjax]\begin{align*}
\widehat{p}_n^{\text{MM}} &= \frac{1 - \widehat{m}_1}{2}
\end{align*}[/mathjax]