矩估计

假设问题是要估计表征随机变量${\displaystyle W}$的分布${\displaystyle f_{W}(w;\theta )}$的${\displaystyle k}$个未知参数${\displaystyle \theta _{1},\theta _{2},\dots ,\theta _{k}}$。如果真实分布（"总体矩"）的前${\displaystyle k}$阶矩可以表示成这些${\displaystyle \theta }$的函数:

${\displaystyle \mu _{1}\equiv E[W]=g_{1}(\theta _{1},\theta _{2},\dots ,\theta _{k}),}$

${\displaystyle \mu _{2}\equiv E[W^{2}]=g_{2}(\theta _{1},\theta _{2},\dots ,\theta _{k}),}$

${\displaystyle \vdots }$

${\displaystyle \mu _{k}\equiv E[W^{k}]=g_{k}(\theta _{1},\theta _{2},\dots ,\theta _{k}).}$

设取出一大小为${\displaystyle n}$的样本，得到${\displaystyle w_{1},\dots ,w_{n}}$。对于${\displaystyle j=1,\dots ,k}$，令

${\displaystyle {\hat {\mu }}_{j}={\frac {1}{n}}\sum _{i=1}^{n}w_{i}^{j}}$

为j阶样本矩，是${\displaystyle \mu _{j}}$的估计。${\displaystyle \theta _{1},\theta _{2},\dots ,\theta _{k}}$的矩估计量记为${\displaystyle {\hat {\theta }}_{1},{\hat {\theta }}_{2},\dots ,{\hat {\theta }}_{k}}$，由这些方程的解（如果存在）定义：

${\displaystyle {\hat {\mu }}_{1}=g_{1}({\hat {\theta }}_{1},{\hat {\theta }}_{2},\dots ,{\hat {\theta }}_{k}),}$

${\displaystyle {\hat {\mu }}_{2}=g_{2}({\hat {\theta }}_{1},{\hat {\theta }}_{2},\dots ,{\hat {\theta }}_{k}),}$

${\displaystyle \vdots }$

${\displaystyle {\hat {\mu }}_{k}=g_{k}({\hat {\theta }}_{1},{\hat {\theta }}_{2},\dots ,{\hat {\theta }}_{k}).}$

• 广义矩估计
• 點估計
• 估計量的偏誤