Method of moments (statistics)

In statistics, the method of moments is a method of estimation of population parameters. The same principle is used to derive higher moments like skewness and kurtosis. It starts by expressing the population moments (i.e., the expected values of powers of the random variable under consideration) as functions of the parameters of interest. Those expressions are then set equal to the sample moments. The number of such equations is the same as the number of parameters to be estimated. Those equations are then solved for the parameters of interest. The solutions are estimates of those parameters. The method of moments was introduced by Pafnuty Chebyshev in 1887 in the proof of the central limit theorem. The idea of matching empirical moments of a distribution to the population moments dates back at least to Pearson. Suppose that the problem is to estimate unknown parameters characterizing the distribution of the random variable . Suppose the first moments of the true distribution (the "population moments") can be expressed as functions of the s: Suppose a sample of size is drawn, resulting in the values . For , let be the j-th sample moment, an estimate of . The method of moments estimator for denoted by is defined to be the solution (if one exists) to the equations: The method described here for single random variables generalizes in an obvious manner to multiple random variables leading to multiple choices for moments to be used. Different choices generally lead to different solutions [5], [6]. The method of moments is fairly simple and yields consistent estimators (under very weak assumptions), though these estimators are often biased. It is an alternative to the method of maximum likelihood. However, in some cases the likelihood equations may be intractable without computers, whereas the method-of-moments estimators can be computed much more quickly and easily. Due to easy computability, method-of-moments estimates may be used as the first approximation to the solutions of the likelihood equations, and successive improved approximations may then be found by the Newton–Raphson method.