在统计学 中，矩估计 （英語：method of moments ）是估计总体 母數 的方法。首先推导涉及感兴趣的参数的总体矩 （即所考虑的随机变量 的幂的期望值 ）的方程。然后取出一个样本并从这个样本估计总体矩。接着使用样本矩取代（未知的）总体矩，解出感兴趣的参数。从而得到那些参数的估计。矩估计是英国统计学家卡尔·皮尔逊 于1894年提出的。
假设问题是要估计表征随机变量
W
{\displaystyle W}
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{\displaystyle f_{W}(w;\theta )}
k
{\displaystyle k}
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{\displaystyle \theta _{1},\theta _{2},\dots ,\theta _{k}}
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{\displaystyle k}
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{\displaystyle \theta }
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{\displaystyle \mu _{1}\equiv E[W]=g_{1}(\theta _{1},\theta _{2},\dots ,\theta _{k}),}
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{\displaystyle \mu _{2}\equiv E[W^{2}]=g_{2}(\theta _{1},\theta _{2},\dots ,\theta _{k}),}
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{\displaystyle \vdots }
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{\displaystyle \mu _{k}\equiv E[W^{k}]=g_{k}(\theta _{1},\theta _{2},\dots ,\theta _{k}).}
设取出一大小为
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{\displaystyle n}
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{\displaystyle w_{1},\dots ,w_{n}}
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{\displaystyle j=1,\dots ,k}
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{\displaystyle {\hat {\mu }}_{j}={\frac {1}{n}}\sum _{i=1}^{n}w_{i}^{j}}
为j阶样本矩，是
μ
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{\displaystyle \mu _{j}}
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{\displaystyle \theta _{1},\theta _{2},\dots ,\theta _{k}}
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{\displaystyle {\hat {\theta }}_{1},{\hat {\theta }}_{2},\dots ,{\hat {\theta }}_{k}}
[來源請求]
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{\displaystyle {\hat {\mu }}_{1}=g_{1}({\hat {\theta }}_{1},{\hat {\theta }}_{2},\dots ,{\hat {\theta }}_{k}),}
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{\displaystyle {\hat {\mu }}_{2}=g_{2}({\hat {\theta }}_{1},{\hat {\theta }}_{2},\dots ,{\hat {\theta }}_{k}),}
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{\displaystyle \vdots }
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{\displaystyle {\hat {\mu }}_{k}=g_{k}({\hat {\theta }}_{1},{\hat {\theta }}_{2},\dots ,{\hat {\theta }}_{k}).}
![{\displaystyle \mu _{1}\equiv E[W]=g_{1}(\theta _{1},\theta _{2},\dots ,\theta _{k}),}](https://wikimedia.org/api/rest_v1/media/math/render/svg/394097a975836199522660360d7bb145236c7876)
![{\displaystyle \mu _{2}\equiv E[W^{2}]=g_{2}(\theta _{1},\theta _{2},\dots ,\theta _{k}),}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8f798100c936f2474c6160c526cc34b085ed35e3)
![{\displaystyle \mu _{k}\equiv E[W^{k}]=g_{k}(\theta _{1},\theta _{2},\dots ,\theta _{k}).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/284aca1e41ed202ba68e2d37005281bf802ab11b)
