Problem 1. Let f be a density. Now consider the family of densities {ft : t ∈ R} defined by

ft(x)=f(x−t), x∈R.

Show that the family {ft : t ∈ R} is Hellinger differentiable at each θ if it is Hellinger

differentiable at 0.

Problem 2. Let ft denote the Uniform(0, t) density for t > 0, i.e.,

ft(x) = (1/t)1(0,t)(x).

Verify the identity

∞2 (1+∆−1)2 ∆

√

f1+∆(x) − f1(x) dx = 1 + ∆ + 1 + ∆

for ∆ > 0. Use this to conclude that the family {ft : t > 0} is not Hellinger differentiable at 1.

Problem 3. Let X1,…,Xn be independent Binomial(m,θ) with known m and unknownθ in the interval (0, 1). Find the BUE of θ. Does the BUE achieve the information bound given by the Cram ́er-Rao inequality?

Problem 4. Let X1, . . . , Xn be independent Poisson random variables with unknown inten- sity λ > 0. Does the BUE of λ achieve the information bound?

Problem 5. Let X1, . . . , Xn be independent Gamma(1, θ) random variables. Does the BUE of θ achieve the information bound?

Problem 6. Let X1, . . . , Xn be independent N(θ, θ) random variables. Find the MLE of θ. Does the MLE achieve the information bound?