Let X1, X2, , Xn be i.d uniform random variables on the interval [, + ] where gt; 0 and gt; 0 are both , are unknown.

Let X1, X2, · · · , Xn be i.i.d uniform random variables on the interval [α, α + θ] where α > 0 and θ > 0 are both α, θ are unknown. Recall that in Homework 4, you calculate the method-of-moments estimators for α and θ. In this problem, we consider the maximum likelihood estimates of these parameters.

(a) Determine the likelihood function f(x1, x2, · · · , xn|α, θ)

(b) Determine the maximum likelihood estimates (MLE) ˆαMLE and ˆθMLE of α and θ, respectively, for a sample of size n. Note: for full credit, you should completely justify your answer

(c) For any x ∈ R, compute the probability P(ˆαMLE ≤ x) where the sample size is n. Your answer can depend on the unknown parameters α and θ

(d) Suppose the true value of α is α = α0. For any > 0, what is the limiting probability P( √ n( ˆαMLE − α0) < − epsilon)? as n → ∞? In particular, does √ n(ˆαMLE − α0) have a limiting distribution that is normal, with mean zero and a finite and positive variance? Does this contradict any results that we have discussed on the asymptotic behavior of MLEs?