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MATH5905, T1 2020 Assignment One Statistical Inference

Department of Statistics

Term 1, 2020

MATH5905 – Statistical Inference

Assignment 1

Instructions: This assignment must be submitted no later than the beginning of the lecture at 6 pm

on Friday, 13th March 2020. Please, declare on the first page that the assignment is your own work,

except where acknowledged. State also that you have read and understood the University Rules in

respect to Student Academic Misconduct. You need to submit the PDF file on Moodle Turnitin by

6 pm on Friday 13th March 2020. You will also need to bring a printed copy of your PDF file to the

lecture by 6 pm on Friday 13th March 2020 as well.

Maximal number of pages: 8 pages

Problem 1

Consider a decision problem with parameter space Θ = {θ1, θ2} and a set of non randomized decisions

D = {di

, 1 ≤ i ≤ 7} with risk points {R(θ1, di), R(θ2, di)} as follows:

i 1 2 3 4 5 6 7

R(θ1, di) 1 3 7 13 15 13 10

R(θ2, di) 15 11 11 5 11 17 14

a) Find the minimax rule(s) amongst the non-randomized rules in D.

b) Plot the risk set of all randomized rules D generated by the set of rules in D.

c) Find the risk point of the minimax rule in D and determine its minimax risk.

d) Define the minimax rule in the set D in terms of rules in D.

e) For which prior on {θ1, θ2} is the minimax rule in the set D also a Bayes rule?

f) Determine the Bayes rule and the Bayes risk for the prior

2

3

,

1

3

on {θ1, θ2}.

g) For a small positive = 2, illustrate on the risk set the risk points of all rules which are -minimax.

Problem 2

Suppose X = (X1, . . . , Xn) are i.i.d. Poisson(θ) with density

f(x | θ) = e

−θ

θ

x

x!

, x ∈ {0, 1, 2, . . . }, θ > 0

and let θ have a Gamma(α, β) distribution with density

τ (θ) = 1

Γ(α)β

α

θ

α−1

e

−θ/β, α, β > 0, θ > 0.

a) Find the posterior distribution for θ.

b) Hence or otherwise determine the Bayes estimator of θ with respect to the quadratic loss function

L(a, θ) = (a − θ)

2

.

c) Suppose the following nine observations were observed:

0, 1, 0, 0, 2, 3, 1, 1, 0.

MATH5905, T1 2020 Assignment One Statistical Inference

Using a zero-one loss with the parameters α = 2 and β = 1 for the prior, what is your decision

when testing H0 : θ ≤ 1 versus H1 : θ > 1. (You may use the integrate function in R or another

numerical integration routine from your favourite programming package to answer the question.)

Problem 3

Determine the form of the Bayes decision rule in an estimation problem with a one-dimensional parameter

θ ∈ R

1 and loss function

L

θ, d

=

(

a

θ − d

if d ≤ θ,

b

d − θ

if d > θ,

where a and b are known positive constants.

Problem 4

Suppose you are responsible for a deciding whether the light rail project in Sydney was to continue

to be developed during a critial stage in its development. You must decide whether the project will

continue or the project will be abandoned. The parameter θ ∈ (0, 1) measures the financial viability of

the project and the project is deemed profitable when θ > 0.5. You have some data x available to you

that provides information about the parameter θ. If θ < 0.5 then the cost to the taxpayer of continuing

the project is 0.5−θ (in units of $ million) whereas if θ > 0.5 then the costs is zero since the project will

be privatized if profitable. Furthermore, if θ > 0.5 then the cost of abandoning the project is θ − 0.5

due to the contractual arrangement for purchasing the machinery whereas if θ < 0.5 then it is zero.

Therefore, two actions will be available to you:

a0 : continue the project and a1 : abandon it.

The prototype light-rail has been subjected to independent trials each with a probability θ of success.

The data x then consists of the total number of trials required for the first successful result to be

obtained. That is, one realization from a geometric distribution has been obtained. Two important

parties are involved in the decision making. The Minister for Transport who has a prior density

τ1(θ) = 30θ

2

(1 − θ)

2

,

and the Prime Minister who has prior density

τ2(θ) = 4θ

3

.

a) Derive the Bayesian decision rule in terms of the posterior mean of θ given x.

b) Determine the values of x for which there is the most serious ministerial disagreement between

the two parties.