The Method of Moments - yasuhisa's blog

Definition

理論統計学のレジメのP250より。

Let $f_X(x;\theta_1,\cdots,\theta_p)$ be ad density of a random variable X which has p parameters $\theta_1,\cdots,\theta_p$ . The r-th population moments about 0 $E(X^r)$ are in general known function of the p parameters, $\theta_1,\cdots,\theta_p$ . Let the jth sample moment about 0 be
$M_j = \frac{1}{n}\sum^n_{i=1}X_i^j$ .
Forming a set of p simultaneous equations of p parameters $\theta_1,\cdots,\theta_p$ as
$E(X) = \frac{1}{n}\sum^n_{i=1}X_i$
$E(X^2) = \frac{1}{n}\sum^n_{i=1}X_i^2$
$E(X^p) = \frac{1}{n}\sum^n_{i=1}X_i^p$
adn solving with respect to $\theta_1,\cdots,\theta_p$ obtains the method of moments estimators $\hat{\theta_1},\cdots,\hat{\theta_p}$ .