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:

$\begin{array}{cc}{\mu }_1& \equiv \mathrm{E}[W]=g_1({\theta }_1,{\theta }_2,\dots ,{\theta }_k),\\ {\mu }_2& \equiv \mathrm{E}[W^2]=g_2({\theta }_1,{\theta }_2,\dots ,{\theta }_k),\\ & ⋮\\ {\mu }_k& \equiv \mathrm{E}[W^k]=g_k({\theta }_1,{\theta }_2,\dots ,{\theta }_k).\end{array}$

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

${\hat{\mu }}_j=\frac{1}{n}\sum _{i=1}^n{w}_i^j$

be the j-th sample moment, an estimate of ${\mu }_j$. The method of moments estimator for ${\theta }_1,{\theta }_2,\dots ,{\theta }_k$ denoted by ${\hat{\theta }}_1,{\hat{\theta }}_2,\dots ,{\hat{\theta }}_k$ is defined as the solution (if there is one) to the equations:Template:Citation needed

$\begin{array}{cc}{\hat{\mu }}_1& =g_1({\hat{\theta }}_1,{\hat{\theta }}_2,\dots ,{\hat{\theta }}_k),\\ {\hat{\mu }}_2& =g_2({\hat{\theta }}_1,{\hat{\theta }}_2,\dots ,{\hat{\theta }}_k),\\ & ⋮\\ {\hat{\mu }}_k& =g_k({\hat{\theta }}_1,{\hat{\theta }}_2,\dots ,{\hat{\theta }}_k).\end{array}$

Reasons to use it

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