MATH10028 May 2020 ???
Theory of Statistical Inference
1. Suppose that Y1, . . . , Yn are independent and identically distribution Bernoulli(θ)
random variables, where θ ∈ (0, 1) is unknown.
(a) Find the maximum likelihood estimator θˆMLE for θ. [2 Marks]
(b) Find Bias(θˆMLE). [1 Mark]
(c) Find Var(θˆMLE). [1 Mark]
(d) Calculate the Fisher Information
I(θ) = E
{(∂l(θ;Y1, . . . , Yn)
∂θ
)2}
.
[4 Marks]
(e) Hence, or otherwise, deduce that the maximum likelihood estimator is a minimum
variance unbiased estimator for θ. [2 Marks]
[Hint: You may assume that the conditions of the Crame´r–Rao Lower Bound hold, and
you need not formally state the result.]
2. Suppose that the distribution of Y = (Y1, . . . , Yn) belongs to a parametric family
{fY(·|θ) : θ ∈ Θ}, where Θ ⊆ R. We are interested in testing H0 : θ ∈ Θ0 versus
H1 : θ ∈ Θ1. Suppose a hypothesis test rejects H0 if Y lies in a critical region C (otherwise
H0 is not rejected).
(a) (i) Define the power function of the test with critical region C. [2 Marks]
(ii) What is the size of the test with critical region C. [2 Marks]
(b) Suppose that Θ0 = {θ0}, Θ1 = {θ1} and that Y1, . . . , Yn are continuous random
variables. What is the critical region of the most powerful test of size α ∈ (0, 1) in
the the case? Clearly justify your answer. You may use any results from the course
without proof, provided they are clearly stated. [6 Marks]
(c) Now suppose that Y1, . . . , Yn are independent and identically distributed Exp(λ)
random variables, where λ > 0 is unknown.
(i) Find the most powerful size-α test of H0 : λ = λ0 against H1 : λ = λ1 where
0 < λ0 < λ1. [5 Marks]
(ii) Find the power function in this case. [2 Marks]
(iii) Deduce that the test you have constructed is a uniformly most powerful (UMP)
test of size α for testing H0 : λ = λ0 against H1 : λ > λ0. [3 Marks]
[Hint: you may use that fact that 2λ
∑n
j=1 Yj ∼ χ22n without proof]
3. Suppose that the distribution of Y = (Y1, . . . , Yn) belongs to a parametric family
{fY(·|θ) : θ ∈ Θ}, where Θ ⊆ R.
(a) (i) What does is mean to say that the statistic S = S(Y) is sufficient for θ. [2
Marks]
(ii) What does it mean to say that the estimator θˆ = θˆ(Y) is unbiased for θ. [2
Marks]
(iii) State (without proof) the factorisation criteria that can be used to determine
whether a statistic is sufficient. [2 Marks]
(b) Now suppose that Y1, . . . , Yn are independent and identically distributed U([−
√
θ,
√
θ])
random variables, where θ > 0 is an unknown parameter.
(i) Find a two-dimensional statistic S = (S1, S2) which is sufficient for θ. [4
Marks]
(ii) Show that the estimator θˆ = 3Y 21 is unbiased for θ. [2 Marks]
(iii) Find another unbiased estimator for θ, which is a function of the statistic S
you found in part (b)(i), and has lower variance than 3Y 21 . Clearly justify your
answer. You may use any results from the course without proof, provided they
are clearly stated. [8 Marks]

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