Answer to Question #122037 in Statistics and Probability for Naeema

Question

Suppose that Y1, Y2,...,Yn form a random sample from a PARETO (α, β) distribution, where α and β are unknown. Show that the maximum Likelihood Estimates of α and β (i.e. α-hat and β-hat) can be found by solving the following set of equations:

(n/α) + n ln(β-hat) = Σni=1 ln[β-hat + Yi]
(nα/β-hat) = (α-hat+1) Σni=1 [β-hat + Yi]^-1

NOTE: The pdf of a PARETO (α, β) distribution is given by

f(y) = {(αβα)/(β+y)^(α+1) if y>0}
{0 if y≤0}

Answer 1

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