Any method of moments problem with one parameter is amenable to the type of question asked. For the Pareto, with lambda known (call it t), the moment equation is t/(a-1) = x with a for alpha and x for x-bar. The solution is a-hat = t/x + 1. Note that for any moment estimation, parameter-hat = g(x), some function of x-bar.
The general approximation for the variance of a function of a random variable is [g'(mu)]^2*Var(x) where mu = E(x). For the Pareto problem, g'(x) = -t/x^2. Also, mu = t/(a-1) and so g'(mu) = -(a-1)^2/t. Because a is unknown, it must be estimated, and the formula given in Jennifer Charlonne's question results.
This is slightly different from the formula/process for functions of parameters. The reason is that we assume that the parameter estimate is unbiased and so mu=parameter and we wind up with g'(parameter). In the above example, we are looking at a function of x-bar, which is not the parameter estimate.
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Stuart Klugman, FSA
Principal Financial Group Professor of Actuarial Science
Drake University
2507 University Avenue
Des Moines, IA 50311 USA
ph: 515-271-4097
e-mail: Stuart.Klugman@drake.edu