Advantages and Disadvantages of This Method
In some respects, when estimating parameters of a known family of probability distributions, this method was superseded by Fisher's method of maximum likelihood, because maximum likelihood estimators have higher probability of being close to the quantities to be estimated.
However, in some cases, as in the above example of the gamma distribution, the likelihood equations may be intractable without computers, whereas the method-of-moments estimators can be quickly and easily calculated by hand as shown above.
Estimates by the method of moments 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. In this way the method of moments and the method of maximum likelihood are symbiotic.
In some cases, infrequent with large samples but not so infrequent with small samples, the estimates given by the method of moments are outside of the parameter space; it does not make sense to rely on them then. That problem never arises in the method of maximum likelihood. Also, estimates by the method of moments are not necessarily sufficient statistics, i.e., they sometimes fail to take into account all relevant information in the sample.
When estimating other structural parameters (e.g., parameters of a utility function, instead of parameters of a known probability distribution), appropriate probability distributions may not be known, and moment-based estimates may be preferred to Maximum Likelihood Estimation.
Read more about this topic: Method Of Moments (statistics)
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