Method of Moments Estimate¶

For this method, we calculate expected value of powers of the random variable to get \(d\) equations for estimating \(d\) parameters (if the solutions exist). For instance, consider \(f_{X}(x) = f(x \lvert \theta, \sigma)\). We can estimate the values of the parameters by solving the two equations

\[ \begin{aligned} E[X] &= \frac{\sum_{i=1}^{n} X_{i}}{n} = \int xf(x \lvert \theta, \sigma) dx\newline E[X^{2}] &= \frac{\sum_{i=1}^{n} X_{i}^{2}}{n} = \int x^{2}f(x \lvert \theta, \sigma) dx \end{aligned} \]

with the solutions to these equations denoted as \(\theta_{MME}\) and \(\sigma_{MME}\). Depending on the distribution, calculation of expected values can be done using moment generating functions. The above estimation is valid when we have a large number of samples, since by the law of large numbers, the sample mean will converge to the true mean.