Here's a problem from a well-known text (Statistical Inference, 2nd
edition, by Casella & Berger): problem 7.6 on pg 355 (of the 2nd
edition): take a random sample (iid, size n) from a distribution with
pdf:

f(x) = theta*x^(-2), for 0 < theta <= x < infinity.

Parts (a) and (b) of this problem are straightforward:
(a) What is a sufficient statistic for theta?
(b) Find the MLE of theta.

(In both cases, the answer is the first order-statistic.)

Part (c): Find the method of moments estimator of theta.

Huh? The density has no moments. EX = theta*Integral(x^(-1), from
theta to infinity), diverges.

Is this just a trick question (i.e., the answer is that there is no
method-of-moments estimator), or am I overlooking something?


What you did is exactly what Casella and Berger say in their solns manual.