矩法估计的一般步骤

设总体 $X\sim F_X\left(\cdot ,{\theta }_1,{\theta }_2,\cdots ,{\theta }_l\right)$, 先计算 $X$ 的从 1 到 $l$ 的各阶矩, 得到各阶矩与参数的关系

$$\{\begin{array}{ll}E\left[X\right]=& g_1\left({\theta }_1,{\theta }_2,\cdots ,{\theta }_l\right),\\ E\left[X^2\right]=& g_2\left({\theta }_1,{\theta }_2,\cdots ,{\theta }_l\right),\\ & \cdots \cdots \\ E\left[X^l\right]=& g_l\left({\theta }_1,{\theta }_2,\cdots ,{\theta }_l\right).\end{array}\text{\hspace{1em}(7.1.2)}$$

由这些关系式中解出

$$\{\begin{array}{c}{\theta }_1=h_1\left(E\left[X\right],E\left[X^2\right],\cdots ,E\left[X^l\right]\right),\\ {\theta }_2=h_2\left(E\left[X\right],E\left[X^2\right],\cdots ,E\left[X^l\right]\right),\\ \cdots \cdots \\ {\theta }_l=h_l\left(E\left[X\right],E\left[X^2\right],\cdots ,E\left[X^l\right]\right).\end{array}\text{\hspace{1em}(7.1.3)}$$

最后, 将 (7.1.3) 右端函数中的总体各阶原点矩用样本各阶原点矩代替, 再将 ${\theta }_1,{\theta }_2,\cdots ,{\theta }_l$ 换成 ${\widehat{{\theta }_1}}_M,{\widehat{{\theta }_2}}_M,\cdots ,{\widehat{{\theta }_l}}_M$ 即得.