Problem 1

Let X = (X1 , X2 , . . . , Xn) be i.i.d. random variables, each with a density

f(x,θ) = w(e)he({−)12re([)ln x) ] 2 },x > 0

where θ > 0 is a parameter.  (This is called the log-normal density.)

a) Show that Yi = log Xi is normally distributed and determine the mean and variance of

this normal distribution. Hence calculate E(log2 Xi).

Note: Density transformation formula: For Y = W(X) :

fY (y) = fX (W−1 (y))| | = fX (x)| |.

b) Find the Fisher information about θ in one observation and in the sample of n observa- tions.

c) Find the Maximum Likelihood Estimator (MLE) of h(θ) = θ 2 and show that it is unbiased for h(θ).

d) Determine the asymptotic distribution of the MLE of h(θ) = θ 2 .

e) Prove that the family L(X,θ) has a monotone likelihood ratio in T = (lnXi)2 .

f) Argue that there is a uniformly most powerful (UMP) α−size test of the hypothesis H0 : θ ≤ θ0 against H1 : θ > θ0 and exhibit its structure.

g) Using f)  (or otherwise), find the threshold constant in the test and hence determine completely the uniformly most powerful α − size test φ∗ of

H0 : θ ≤ θ0 versus H1 : θ > θ0 .

Problem 2

Let X = (X1,X2 , . . . ,Xn) be a sample of n observations each with a uniform in [0,θ) density

f(x,θ) =

where θ > 0 is an unknown parameter. Denote the joint density by L(X,θ).

a) Show that the family {L(X,θ)},  θ > 0 has a monotone likelihood ratio in X(n) .

b) Show that the uniformly most powerful α-size test of H0 : θ ≤ 2 versus H1 : θ > 2 is

given by

φ∗ (X) =

Justify all steps in your argument.