Method of Moments

ProbabilityTheoryStatistics The Moment Generating Function of a random variable, when it exists, uniquely determines it’s distribution. Therefore the Law of Large Numbers gives a criteria for us to construct consistent estimators, if we can find an such that The method of moments estimator is any estimator that solves the system of equations (for $k \in {1, \cdots, m}$$$\frac1n \sum_{i=1}^n X_i^k = \mathbb{E}{\hat{\theta}{\text{MOM}}}[X^k]\rho(x, \theta) := \sum_{k=1}^m (x^k - \mathbb{E}{\theta}[X^k])^2\mathbb{E}{\hat{\theta}{\text{MOM}}}[\rho(X, \theta)] = \sum{k=1}^m \text{Var}{\hat{\theta}{\text{MOM}}}(X^k) + (\mathbb{E}{\hat{\theta}{\text{MOM}}}[X^k] - \mathbb{E}_{\theta}[X^k])^2$$