STAT0008 Statistical Inference: Level 6 and 7 Examination 2020 Page 1
STAT0008: STATISTICAL INFERENCE
2020
• Answer ALL questions.
• You may submit only one answer to each question.
• The relative weights attached to each question are as follows: Q1 (18), Q2 (26), Q3 (27),
Q4 (29).
• The numbers in square brackets indicate the relative weights attached to each part ques-
tion.
• Marks are awarded not only for a final answer but also for the clarity and coherence of
your solution.
• Information regarding some common probability distributions is provided at the begin-
ning of the examination paper.
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STAT0008 Statistical Inference: Level 6 and 7 Examination 2020 Page 2
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Continued
STAT0008 Statistical Inference: Level 6 and 7 Examination 2020 Page 3
Common Probability Distributions
Binomial Distribution
Notation: X ∼ Bin(n, p)
Probability mass function: P(X = x) =
(
n
x
)
px(1− p)n−x
with p ∈ [0, 1], n ∈ N
Support: x ∈ {0, 1, . . . , n}
Mean: E(X) = np
Variance: Var(X) = np(1− p)
Poisson Distribution
Notation: X ∼ Poi(λ)
Probability mass function: P(X = x) = λxe−λ
x!
with λ ∈ [0,∞)
Support: x ∈ {0, 1, . . .}
Mean: E(X) = λ
Variance: Var(X) = λ
Negative Binomial Distribution
Notation: X ∼ NegBin(k, p)
Probability mass function: P(X = x) =
(
x−1
k−1
)
pk(1− p)x−k
with p ∈ [0, 1], k ∈ N
Support: x ∈ {k, k + 1, . . .}
Mean: E(X) = k
p
Variance: Var(X) = k(1−p)
p2
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STAT0008 Statistical Inference: Level 6 and 7 Examination 2020 Page 4
(Continuous) Uniform Distribution
Notation: X ∼ U [a, b]
Probability density function: fX(x; a, b) =
1
b−a
with (a, b) ∈ R2, a < b
Support: x ∈ [a, b]
Mean: E(X) = 1
2
(a+ b)
Variance: Var(X) = 1
12
(b− a)2
Normal Distribution
Notation: X ∼ N (µ, σ2)
Probability density function: fX(x;µ, σ
2) = 1√
2piσ2
exp
{− 1
2σ2
(x− µ)2}
with µ ∈ R, σ ∈ (0,∞)
Support: x ∈ (−∞,∞)
Mean: E(X) = µ
Variance: Var(X) = σ2
Exponential Distribution
Notation: X ∼ Exp(λ)
Probability density function: fX(x;λ) = λe
−λx
with λ ∈ (0,∞)
Support: x ∈ (0,∞)
Mean: E(X) = 1
λ
Variance: Var(X) = 1
λ2
Gamma Distribution
Notation: X ∼ Gamma(α, β)
Probability density function: fX(x;α, β) =
βα
Γ(α)
xα−1e−βx
with α ∈ (0,∞), β ∈ (0,∞)
Support: x ∈ (0,∞)
Mean: E(X) = α
β
Variance: Var(X) = α
β2
Continued
STAT0008 Statistical Inference: Level 6 and 7 Examination 2020 Page 5
Q1. Suppose that X1, . . . , Xn are independent and identically distributed Poisson random
variables, such that Xi ∼ Poi(µ) where µ > 0 and i ∈ {1, . . . , n}.
(a) Determine the maximum likelihood estimator of µ, denoting this estimator µˆ.
[3]
(b) Determine the asymptotic (large sample) distribution of µˆ.
[4]
(c) Is µˆ weakly consistent for µ? Justify your answer.
[3]
Suppose that Y1, . . . , Yn are independent and identically distributed exponential random
variables, such that Yi ∼ Exp(λ) where λ > 0 and i ∈ {1, . . . , n}. The random variable
Z is defined as the minimum of {Y1, . . . , Yn}.
(d) Determine P(Z > z) for z > 0 and hence show that λZ is a pivotal quantity for λ.
[5]
(e) Using λZ, construct a two-sided, equal-tailed, 95% confidence interval for λ, based
on the sample Y1, . . . , Yn.
[3]
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STAT0008 Statistical Inference: Level 6 and 7 Examination 2020 Page 6
Q2. The random variables X1, . . . , Xn are independent and sampled from a distribution with
probability density function
f(x;ψ) =

1
ψ
exp
(
1
ψ
− 1
ψ
ex + x
)
x ≥ 0;
0 otherwise;
where ψ > 0 is an unknown parameter.
(a) Determine the maximum likelihood estimator of ψ and call this estimator ψˆ.
[3]
(b) Is ψˆ is the minimum variance bound unbiased estimator (MVBUE) of ψ? Justify
your answer.
[3]
(c) Determine the Crame´r-Rao lower bound for the variance of unbiased estimators of
1/ψ.
[3]
(d) Determine the median of the distribution of Xi, for any i ∈ {1, . . . , n}.
[6]
A sample of X1, . . . , Xn, denoted x1, . . . , xn, is taken where n = 60,
∑60
i=1 xi = 45.95 and∑60
i=1 e
xi = 219.73.
(e) Construct an approximate 95% confidence interval for ψ. You may find the following
quantiles of the standard N (0, 1) distribution helpful:
z0.025 = 1.96, z0.05 = 1.645, z0.1 = 1.28
where, for a given value ∈ (0, 1), z is the quantile such that P(Z > z) = for
Z ∼ N (0, 1).
[4]
A Bayesian prior distribution for ψ is defined as
pi(ψ) =
βα
Γ(α)
1
ψα+1
exp
(
−β
ψ
)
.
where α > 0 and β > 0 are known parameters.
(f) Derive the posterior distribution of ψ and hence determine the Bayes’ estimate of
ψ under quadratic error loss.
[7]
Continued
STAT0008 Statistical Inference: Level 6 and 7 Examination 2020 Page 7
Q3. Suppose that X1, . . . , Xn are independent and identically distributed Bin(1, p) random
variables. The parameter θ is such that θ = p2.
(a) Find a minimal sufficient statistic for θ.
[4]
(b) Determine the maximum likelihood estimator of θ. Is this estimator unbiased for θ?
Justify your answer.
[7]
The estimator U is defined
U =
{
1 if X1 = 1 and X2 = 1
0 otherwise
(c) Show that U is unbiased for θ.
[1]
(d) Using the Rao-Blackwell theorem, determine an unbiased estimator of θ, denoting
this estimator θ˜, such that Var(θ˜) ≤ Var(U).
[7]
(e) Is θ˜ the minimum variance unbiased estimator (MVUE) of θ? Justify your answer.
[4]
(f) Does the variance of θ˜ attain the Crame´r-Rao lower bound for the variance of unbi-
ased estimators of θ? Justify your answer.
[4]
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STAT0008 Statistical Inference: Level 6 and 7 Examination 2020 Page 8
Q4. The random variables X1, . . . , Xn are independent and identically distributed, where
Xi ∼ Exp(λ) for i ∈ {1, . . . , n} and λ > 0.
(a) Based on the sample, X1, . . . Xn, construct the most powerful test of size α of the
hypotheses
H0 : λ = λ0 against H1 : λ = λ1
where α ∈ (0, 1), λ0 > 0, λ1 > 0 and λ1 > λ0. You may use the following results
without proof:
If Z1, . . . , Zn are independent Exp(θ) random variables then
∑n
i=1 Zi ∼ Gamma(n, θ).
If n ∈ N and Y ∼ Gamma(n, θ) then 2θY ∼ χ22n.
If U ∼ χ2ν then P(U > χ2ν()) = for ∈ (0, 1).
[9]
(b) Based on the sample X1, . . . , Xn, construct a 100(1 − α)% confidence lower bound
for λ.
[3]
(c) Derive the power function for the test that you constructed in (a). Your answer may
be written in terms of a cumulative distribution function, F (.), where the explicit
form of F need not be evaluated.
[3]
(d) Is the test that you constructed in (a) uniformly most powerful and of size α for a
test of the hypotheses
H0 : λ ≤ λ0 against H1 : λ > λ0?
Justify your answer. [4]
Associated with X1, . . . , Xn is the set of random variables {Z1, . . . , Zn} such that
Zi =
{
0 if Xi < γ;
1 if Xi ≥ γ;
for i ∈ {1, . . . , n}, where γ > 0 is a known parameter.
(e) Write down the likelihood function, L(λ, z), based on an observed sample z =
(z1, . . . , zn)
> of Z1, . . . , Zn and hence determine the maximum likelihood estimate
of λ.
[4]
(f) State a condition that must be satisfied by X1, . . . , Xn to ensure that the maximum
likelihood estimate of λ, derived in (e), is finite.
[2]
(g) Determine the Crame´r-Rao lower bound for the variance of unbiased estimators of
λ, where λ is to be estimated using z1, . . . , zn.
[4]
End of Paper

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