(MLE for Uniform) If the random variable U has the uniform distribution with parameter , then its p.d.f. is 1( U ). Given

(MLE for Uniform) If the random variable U has the uniform distribution with parameter θ₀, then its p.d.f. is 1(θ₀ ≤ U ≤ θ₀). Given a sample of N realizations {U₁, ..., U_N}, the MLE for θ₀ is the largest observed value $\hat{θ}_N$ = U_(N).

(a) Find the p.d.f. of $\hat{θ}_N$. [HINT: Use (13.9).]

(b) Show that the mean and variance of $\hat{θ}_N$ are N/(N+1)θ₀ and [N/[(2+N)(N+1)²]]θ₀².

(c) Is $\hat{θ}_N$ consistent? How could you correct the bias in $\hat{θ}_N$?

(d) How would you standardize $\hat{θ}_N$ to find an asymptotic approximation to its distribution?

(e) Show that the limiting distribution of your standardized statistic is an exponential distribution.