Statistics - Pareto Distribution

Question

I'm taking an Mathematical Stat course at university and I came up with this question.

Question can be found via(the question involves hard to type notations, so I thought of uploading it and sharing: drive(dot)proton(dot)me/urls/FVHX8Y08RR#jb7eaDEeff3H

As for this question(the one I've marked in red colour), we need to find a parameter estimation for alpha.

I have completed part (i) and my immediate next approach for solving part (ii) was to find E(x) and E(x^2) and then find estimator for alpha.

However, here I'm facing trouble with finding E(x^2) using integration. I'm getting the answer for integration related to E(x^2) as follows: E(x^2) = [ alpha*((beta)^alpha)/(2-alpha) ]*[((infinity)^(2-alpha)) - ((beta)^(2-alpha))

Obviously alpha cannot be equal to 2 in this case given above.

Also, from part 1, we know E(x) will only hold when alpha>1.

Your help on finding a parameter estimator for alpha is highly appreciated!

Answer 1

It turns out, you do not need the second moment to determine alpha. To get a method of moment estimator for alpha, you only need to describe alpha in terms of E[X] and beta (since beta is known). Part (i) already gives an equation for the E[X] in terms of alpha and beta, so all you will need to do is solve for alpha, which can be done algebraically. Then, by the law of large numbers, you can replace E[X] by the average of X.