THE UNIVERSITY OF NEW SOUTH WALES
DEPARTMENT OF STATISTICS
MID SESSION TEST - Monday, 29th March 2021, 2 pm
MATH3811/3911
Time allowed: 1 hour and 45 minutes (including photocopying and submitting your
(*) subquestion is for 3911 students only
Every subquestion is evaluated separately. Statements of previous subquestions can be
used even if these subquestions were not successfully proven
1. Let X = (X1, X2, . . . , Xn) be a random sample of size n from the density
f(x; θ) =
θ
1− θx
(2θ−1)/(1−θ), x ∈ (0, 1).
Here θ ∈ (0.5, 1) is an unknown parameter.
a) Use any argument to prove that T =
∑n
i=1 logXi is a complete and minimal sufficient statistic
for θ.
b) Show that the expected Fisher information about θ contained in the statistic found in a) is
IT (θ) = IX(θ) =
n
θ2(1− θ)2 .
c) Find the MLE τˆ1 of τ1(θ) = θ and also the MLE τˆ2 of τ2(θ) =
1−θ
θ . Argue that τˆ2 is unbiased
for τ2(θ) and is the UMVUE of τ2(θ). Why do you expect τˆ1 to be biased for τ1(θ) = θ?
d) Justify the asymptotic distribution result

n(τˆ2 − τ2(θ)) −→d N
(
0,
(1− θ)2
θ2
)
.
e) (*) (for 3911 students ONLY). Suppose that besides the sample X = (X1, X2, . . . , Xn) from
the above distribution with a parameter θ1 also a sample Y = (Y1, Y2, . . . , Yn) with parameter
θ2 is available. Show that for the MLE hˆ of h(θ1, θ2) =
θ2
θ1
it holds

n(hˆ− h(θ1, θ2)) −→d N
(
0,
θ22
θ21
[(1− θ1)2 + (1− θ2)2]
)
.
2. Let X1, X2, . . . , Xn be a sample of size n from a distribution with a density
f(x; θ) =
{
3x2/θ3 if 0 < x < θ
0 elsewhere
a) Prove that the density of (X1, X2, . . . , Xn) has a monotone likelihood ratio in the statistic
Z = X(n) = max(X1, X2, . . . , Xn).
b) Find the cumulative distribution function and the density of the statistic Z = X(n).
c) Find the uniformly most powerful α-size test ϕ∗ of H0 : θ ≤ 2 versus H1 : θ > 2 and sketch a
graph of Eθϕ
∗ as accurately as possible.
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