An estimator is an operation computing the value of an estimate, that targets the parameter, using measurements from a sample.
Let \(X_1,\ldots,X_n\overset{iid}{\sim}F(\boldsymbol \theta)\) where \(F(\cdot)\) is a known distribution function and \(\boldsymbol\theta\) is a vector of parameters. Let \(\boldsymbol X = (X_1,\ldots, X_n)^\mathrm{T}\), be the sample collected.
Let the \(k\)th moment be defined as \(\mu_k\) and the corresponding \(k\)th moment average \(\frac{1}{n}\sum^n_{i=1}X_i^{k}\):
\[ \mu_k = \frac{1}{n}\sum^n_{i=1}X_i^k. \]
The parameter estimates are for \(t\) parameters are the solutions for \(\mu_k\) for \(k=1,\ldots,t\).
Let \(X_1, \ldots,X_n\overset{iid}{\sim}\mathrm{Bin}(n,p)\), find the method of moments estimator for \(p\).
Let \(X_1, \ldots,X_n\overset{iid}{\sim}\mathrm{Pois}(\lambda)\), find the method of moments estimator for \(\lambda\).
Let \(X_1, \ldots,X_n\overset{iid}{\sim}U(1,\theta)\), find the method of moments estimator for \(\theta\).
Let \(X_1, \ldots,X_n\overset{iid}{\sim}\mathrm{Gamma}(\alpha,\beta)\), find the method of moments estimator for \(\alpha\) and \(\beta\).
Let \(X_1, \ldots,X_n\overset{iid}{\sim}N(\mu,\sigma^2)\), find the method of moments estimator for \(\mu\) and \(\sigma^2\).