parameter estimation by the method of moments

Question

Let $X_1,X_2,\dots X_n$ be a random sample from the density $$f(x;\theta)=\theta x^{-2}, \quad 0<\theta \le x<\infty$$ Find the method of moments estimator of $\theta$.

My attempt

$ E(X) = \int_{\theta}^{\infty}\theta x^{-2} = 1 $

I think that the integral bounds aren't correct.

Answer 1

The density is nice because

$$\int_{\theta}^{\infty}f(x)dx=1$$

But

$$E(X^n)=\int_{\theta}^{\infty}x^nf(x)dx=\infty$$

That means your density does not have finite moments and thus it is not possibile to estimate the parameter with moments

... but you can estimate it with Max likelihood

Answer 2

The integral bounds are correct since you are finding the expectation of $X$.

Another approach

What you need to maximize is $$ f(x_1,\cdots,x_n|\theta)=\prod_{i=1}^n f(x_i|\theta)=\theta^n(x_1\cdots x_n)^{-2} $$ where $\theta\le \min_i x_i$.