Method of moments (statistics) facts for kids

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{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}

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{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}

Reasons to use it

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

Method of moments (statistics) facts for kids

Method

Reasons to use it

See also