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 [math]\displaystyle{ k }[/math] unknown parameters [math]\displaystyle{ \theta_{1}, \theta_2, \dots, \theta_k }[/math] describing the distribution [math]\displaystyle{ f_W(w; \theta) }[/math] of the random variable [math]\displaystyle{ W }[/math]. Suppose the first [math]\displaystyle{ k }[/math] moments of the true distribution (the "population moments") can be expressed as functions of the [math]\displaystyle{ \theta }[/math]s:

[math]\displaystyle{ \begin{align} \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{align} }[/math]

Suppose a sample of size [math]\displaystyle{ n }[/math] is drawn, and it leads to the values [math]\displaystyle{ w_1, \dots, w_n }[/math]. For [math]\displaystyle{ j=1,\dots,k }[/math], let

[math]\displaystyle{ \widehat\mu_j = \frac{1}{n} \sum_{i=1}^n w_i^j }[/math]

be the j-th sample moment, an estimate of [math]\displaystyle{ \mu_j }[/math]. The method of moments estimator for [math]\displaystyle{ \theta_1, \theta_2, \ldots, \theta_k }[/math] denoted by [math]\displaystyle{ \widehat\theta_1, \widehat\theta_2, \dots, \widehat\theta_k }[/math] is defined as the solution (if there is one) to the equations:

[math]\displaystyle{ \begin{align} \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{align} }[/math]

Reasons to use it

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