Method of moments for a heavy-tailed Pareto

Question

Loss sizes follow a Pareto distribution with known minimum $1$ and unknown tail index $\alpha > 1$ , whose mean is $\alpha/(\alpha - 1)$ . From your data the sample mean is $3$ .

Use the method of moments to estimate $\alpha$ , and explain why you should not try to use the second moment here.

Accepted Answer

Match the mean. Set the model mean equal to the sample mean: \frac{\alpha}{\alpha - 1} = 3. Solve: \alpha = 3(\alpha - 1) = 3\alpha - 3, so -2\alpha = -3 and \hat\alpha = \frac{3}{2} = 1.5. The trap in the second moment. A tempting alternative is to match the variance, but for a Pareto the variance is only finite when \alpha 2. Our estimate \hat\alpha = 1.5 is below 2, which means the true second moment is infinite: the theoretical variance does not exist, so there is nothing to match it to. Worse, the sample variance would still compute to some finite number, but it would be wildly unstable …

Method of moments for a heavy-tailed Pareto

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