Statistics - Pareto Distribution

Question

Hello everyone! 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: https://drive.proton.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

Hello Ashley!

Today I got email from Wyzant about this problem.

I believe you complicated the problem, the solution is easier.

You found that E[X] = αβ/(α-1) if α>1.

Suppose α>1, otherwise the 1st moment is undefined and we cannot use method of moments.

Next, the parameter β is known ( https://drive.proton.me/urls/FVHX8Y08RR#jb7eaDEeff3H), so

E[X]/β = α/(α-1) = 1 + 1/(α-1)

Use method of moments: replace the theoretical 1st moment E[X] with the empirical 1st moment x¯ = ∑x_i/n

We get the equation x¯ / β = 1 + 1/(α-1)

solve this equation for the the parameter α.

1/(α - 1) = x¯ / β - 1 = (x¯- β) / β

α - 1 = β / (x¯- β)

α = β / (x¯- β) + 1 = x¯/(x¯- β)

This expression x¯/(x¯- β) is the moment estimator for the parameter α.