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]
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]?
1 answer