School of Mathematics and Statistics
MATH5905 Statistical Inference
Term One 2024
Assignment Two
Given: Friday 29 March 2024
Due date: Sunday, 14 April 2024
Problem 1
Let X = (X1, X2, . . . , Xn) be i.i.d. random variables, each with a density
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 find E(log(Xi)2).
Note: Density transformation formula: For Y = W(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 and show that it is unbiased for h(θ). Is it also the UMVUE of θ? Justify your answer.
d) What is the MLE of ˜h(θ) = √
θ ? Determine the asymptotic distribution of the MLE of ˜h(θ) = √
θ.
e) Prove that the family L(X, θ) has a monotone likelihood ratio in T = (log(Xi))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
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) Using a) (or otherwise), completely determine find the uniformly most powerful α-size test of H0 : θ ≤ 3 versus H1 : θ > 3. Justify all steps in your argument.
c) Find the power function of the test in b) and sketch the graph of Eθφ
∗ as accurately as possible.
d) Let X(1) < X(2) < · · · < X(n) be the order statistics from this distribution. Show that X(1)/X(n) and X(n) are independent random variables.
Hint: Find the joint density of X(1) and X(n) first, then use the density transformation formula for vector of two variables.
Problem 3
Assume X = (X1, X2, . . . , Xn) is a random sample of size n from the density
f(x, θ) = (1 + θ)x
θ
, x ∈ (0, 1),
where θ > −1 is an unknown parameter.
1. Why is T = log(Xi) complete and minimal sufficient for θ?
2. If h(θ) = 1+θ/1 argue that the MLE ˆh of h(θ) is unbiased for h(θ).
3. State the asymptotic distribution of √
n(ˆh − h(θ)).
4. Suppose that besides the sample X = (X1, X2, . . . , Xn) from the above distribution with parameter θ = θ1, another independent sample Y = (Y1, Y2, . . . , Yn) is available with parameter θ2. Show that for the MLE ˆg of g(θ1, θ2) = θ2+1/θ1+1, it holds
Problem 4
Suppose X(1) < X(2) < X(3) < X(4) are the order statistics based on a random sample of size n = 4 from the exponential density f(x) = 2e
−2x
, x > 0.
1. Find the numerical value of E(X(2)).
2. Find the density of the midrange M = 2/1 (X(1) + X(4)).
3. Find P(M > 1/2).
Hint: To completely solve this problem, you might need to use a computer package to approximate some integrals. R.e., the integrate function in R might be handy.