Method of moments (statistics)

In statistics, the method of moments is a method of estimation of population parameters.

Suppose that the problem is to estimate ${\displaystyle k}$ unknown parameters ${\displaystyle \theta _{1},\theta _{2},\dots ,\theta _{k}$ describing the distribution ${\displaystyle f_{W}(w;\theta )}$ of the random variable ${\displaystyle W}$.^[1] Suppose the first ${\displaystyle k}$ moments of the true distribution (the "population moments") can be expressed as functions of the ${\displaystyle \theta }$s:

${\displaystyle {\begin{aligned}\mu _{1}&\equiv \operatorname {E} [W]=g_{1}(\theta _{1},\theta _{2},\ldots ,\theta _{k}),\\[4pt]\mu _{2}&\equiv \operatorname {E} [W^{2}]=g_{2}(\theta _{1},\theta _{2},\ldots ,\theta _{k}),\\&\,\,\,\vdots \\\mu _{k}&\equiv \operatorname {E} [W^{k}]=g_{k}(\theta _{1},\theta _{2},\ldots ,\theta _{k}).\end{aligned}$

Suppose a sample of size ${\displaystyle n}$ is drawn, and it leads to the values ${\displaystyle w_{1},\dots ,w_{n}$. For ${\displaystyle j=1,\dots ,k}$, let

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

be the j-th sample moment, an estimate of ${\displaystyle \mu _{j}$. The method of moments estimator for ${\displaystyle \theta _{1},\theta _{2},\ldots ,\theta _{k}$ denoted by ${\displaystyle {\widehat {\theta }_{1},{\widehat {\theta }_{2},\dots ,{\widehat {\theta }_{k}$ is defined as the solution (if there is one) to the equations:^[source?]

${\displaystyle {\begin{aligned}{\widehat {\mu }_{1}&=g_{1}({\widehat {\theta }_{1},{\widehat {\theta }_{2},\ldots ,{\widehat {\theta }_{k}),\\[4pt]{\widehat {\mu }_{2}&=g_{2}({\widehat {\theta }_{1},{\widehat {\theta }_{2},\ldots ,{\widehat {\theta }_{k}),\\&\,\,\,\vdots \\{\widehat {\mu }_{k}&=g_{k}({\widehat {\theta }_{1},{\widehat {\theta }_{2},\ldots ,{\widehat {\theta }_{k}).\end{aligned}$

The method of moments is simple and gets consistent estimators (under very weak assumptions). However, these estimators are often biased.

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