Method of moments (statistics)

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

Method

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

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

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

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

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

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

Reasons to use it

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