Method of moments is not unique

Question

For a Poisson distribution both the mean and the variance equal $\lambda$ . So you could estimate $\lambda$ by the sample mean or by the sample variance.

Explain why the method of moments is not unique here, and argue which of the two estimates you should prefer.

Accepted Answer

The non-uniqueness. The method of moments says "match a theoretical moment to its sample counterpart," but it does not say which moment. When a parameter appears in more than one moment, each choice gives a valid, consistent estimator. For the Poisson, \mathbb{E}[X] = \lambda suggests \hat\lambda = \bar X, while \operatorname{Var}(X) = \lambda suggests \hat\lambda = s^2 (the sample variance). Both converge to \lambda, yet they are different estimators and will generally give different numbers on the same data. Which to prefer. Prefer the sample mean. Two reasons, both pointing the same way: -…

Method of moments is not unique

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