Method of moments (statistics)

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:

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

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