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The Method of Moments

理論統計学

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}.