Method of Moment Estimator
Background Information
Background Information
Methods of Moments
Examples
Estimators
An estimator is an operation computing the value of an estimate, that targets the parameter, using measurements from a sample.
Data
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.
Methods of Moments
Background Information
Methods of Moments
Examples
Method of Moments
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 \(k=1,\ldots,t\).
Examples
Background Information
Methods of Moments
Examples
Bernoulli Distribution
Let \(X_1, \ldots,X_n\overset{iid}{\sim}\mathrm{Bin}(n,p)\), find the method of moments estimator for \(p\).
Poisson Distribution
Let \(X_1, \ldots,X_n\overset{iid}{\sim}\mathrm{Pois}(\lambda)\), find the method of moments estimator for \(\lambda\).
Uniform Distribution
Let \(X_1, \ldots,X_n\overset{iid}{\sim}U(1,\theta)\), find the method of moments estimator for \(\theta\).
Gamma Distribution
Let \(X_1, \ldots,X_n\overset{iid}{\sim}\mathrm{Gamma}(\alpha,\beta)\), find the method of moments estimator for \(\alpha\) and \(\beta\).
Nomal Distribution
Let \(X_1, \ldots,X_n\overset{iid}{\sim}N(\mu,\sigma^2)\), find the method of moments estimator for \(\mu\) and \(\sigma^2\).
