Suppose X1,..., Xn are i.i.d. with common density function p(x) = c x +1 , 0 < c < x, > 0 . Here,

Suppose X1,..., Xn are i.i.d. with common density function pθ(x) = θcθ

x θ+1 , 0 < c < x, θ > 0 .

Here, c is fixed and known and θ is unknown.

(i) Show that the maximum likelihood estimator ˆ

θn is well-defined and determine the limiting distribution of √n(θˆ

n − θ) under θ.

(ii) What is the score test for testing the null hypothesis θ = θ0 vs. θ = θ0?