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MATH3811
STATISTICAL INFERENCE
(1) TIME ALLOWED – 2 Hours
(2) TOTAL NUMBER OF QUESTIONS – 4
(3) ANSWER ALL QUESTIONS
(4) THE QUESTIONS ARE NOT OF EQUAL VALUE
(5) TOTAL NUMBER OF MARKS – 100
(6) THIS PAPER MAY BE RETAINED BY THE CANDIDATE
All answers must be written in ink. Except where they are expressly required pencils
may only be used for drawing, sketching or graphical work.
JUNE 2006 MATH3811 Page 2
1. [28 marks] Suppose a sample X = (X1, X2, . . . , Xn) from the density
f(x, θ) =
{
(1 + θ)xθ, x ∈ (0, 1),
0 elsewhere
is given where θ > −1 is the unknown parameter.
a) Calculate E(X1).
b) Find a minimal sufficient statistic T for θ. Give reasons for your answer.
c) Calculate the Fisher information about θ contained in the statistic T that
you found in b). Show that it is equal to n
(1+θ)2
.
d) Find the MLE of θ and also the MLE of h(θ) = 1
1+θ
.
e) Find the Cramer-Rao lower bound for the variance of an unbiased esti-
mator of h(θ) = 1
1+θ
.
f) Write down the score function. Find out if an UMVUE of h(θ) = 1
1+θ
exists and if it exists, write it down. Is there an unbiased estimator of
h(θ) = θ whose variance attains the Cramer-Rao bound? Explain your
answers.
g) State the asymptotic distribution of
√
n(hˆmle − h(θ)) for h(θ) = 11+θ .
Hint: The “delta method” implies that for any smooth function h(θ) :
√
n(h(θˆmle)− h(θ0))→d N(0, (∂h
∂θ
(θ0))
2I−1(θ0)),
I(θ) = E{ ∂
∂θ
[lnf(x, θ)]}2 = E{− ∂
2
∂θ2
[lnf(x, θ)]}.
2. [28 marks]
a) Let X = (X1, X2, . . . , Xn) be i.i.d. observations, each from a Poisson
distribution:
f(y|λ) = e(−λ) · λy/(y!), y = 0, 1, 2, . . . , λ > 0.
The prior on λ is believed to be Gamma(2,3).
(Note that, for any values of α > 0, β > 0, the density of the Gamma(α, β)
distribution is given by
τ(λ) =
{
1
βα·Γ(α) · λα−1 exp(−λ/β) , λ > 0;
0 else
and Γ(α) =
∫∞
0 e
−xxα−1dx is the Gamma function with the property
Γ(α+ 1) = αΓ(α).)
Please see over . . .
JUNE 2006 MATH3811 Page 3
i) Find the posterior density h(λ|X) of λ given X = (X1, X2, . . . , Xn).
Show that, like the prior, the posterior density is also a member of
the Gamma family.
ii) Find the Bayesian estimator of λ for quadratic error-loss with respect
to the prior τ(λ).
b) In a preliminary testing of a random number generator, the following ten
values were generated:
x1 = 0.621, x2 = 0.503, x3 = 0.203, x4 = 0.477, x5 = 0.710,
x6 = 0.581, x7 = 0.329, x8 = 0.480, x9 = 0.554, x10 = 0.382.
Carry out a Kolmogorov-Smirnov test to determine if the hypothesis of
a Uniform [0,1] distribution can be accepted. Use α = 0.05.
You can use the following extract of Table F for one-sample Kolmogorov
tests:
n 0.2 0.1 0.05 0.02 0.01
8 .358 .410 .454 .507 .542
9 .339 .387 .430 .480 .513
10 .323 .369 .409 .457 .489
(Each table entry is the value of the Kolmogorov statistic Dn for sample
of size n = 8, 9, 10 with the corresponding p-value given on the top row.)