MATH5905 Term One 2022 Assignment One Statistical Inference
University of New South Wales
School of Mathematics and Statistics
MATH5905 Statistical Inference
Term One 2022
Assignment One
Given: Friday 25 February 2022 Due date: Sunday 13 March 2022
Instructions: This assignment is to be completed collaboratively by a group of at most 3
students. The same mark will be awarded to each student within the group, unless I have good
reasons to believe that a group member did not contribute appropriately. This assignment must
be submitted no later than 11:55 pm on Sunday, 13 March 2022. The first page of the submitted
PDF should be this page. Only one of the group members should submit the PDF file
on Moodle, with the names of the other students in the group clearly indicated in the document.
I/We declare that this assessment item is my/our own work, except where acknowledged, and
has not been submitted for academic credit elsewhere. I/We acknowledge that the assessor of
this item may, for the purpose of assessing this item reproduce this assessment item and provide
a copy to another member of the University; and/or communicate a copy of this assessment
item to a plagiarism checking service (which may then retain a copy of the assessment item on
its database for the purpose of future plagiarism checking). I/We certify that I/We have read
and understood the University Rules in respect of Student Academic Misconduct.
Name Student No. Signature Date
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MATH5905 Term One 2022 Assignment One Statistical Inference
Problem One
Consider a random vector with two components X and Y. Denote the cumulative distribution
function (cdf) as FX,Y (x, y), and the marginal cdf as FX(x) and FY (y), respectively.
i) Show, using first principles, that
FX(x) + FY (y) − 1 ≤ FX,Y (x, y) ≤
p
FX(x)FY (y)
always holds.
ii) Suppose that the X and Y are components of continuous random vector with a density
fX,Y (x, y) = cxy, 0 < y < x, 0 < x < 2 (and zero else). Here c is a normalizing constant.
a) Show that c =12.
b) Find the marginal density fX(x) and FX(x).
c) Find the marginal density fY (y) and FY (y).
d) Find the conditional density fY |X(y|x).
e) Find the conditional expected value a(x) = E(Y |X = x).
Make sure that you show your working and do not forget to always specify the support of
the respective distribution.
Problem Two
At a critical stage, a fund manager has to make a decision about investing or not investing in
certain company stock. He intends to apply a statistical decision theory approach to work out
the appropriate decision based on the potential long-term profitability of the investment. He
uses two independent advisory teams with teams of experts and each team should provide him
with an opinion about the profitability. Data X represents the number of teams recommending
investing in the stock (due, of course, to their belief in its profitability).
If the investment is not made and the stock is not profitable, or when the investment is
made and the stock turns out profitable, nothing is lost. In the manager’s judgement, if the
stock turns out to be not profitable and decision is made to invest in it, the loss is three time
higher than the cost of not investing when the stock turns out profitable.
The two independent expert teams have a history of forecasting the profitability as follows. If
a stock is profitable, each team will independently forecast profitability with probability 4/5
(and no profitability with 1/5). On the other hand, if the stock is not profitable, then each
team predicts profitability with probability 1/2. The fund manager will listen to both teams
and then make his decisions based on the data X.
a) There are two possible actions in the action space A = {a0, a1} where action a0 is to
invest and action a1 is not to invest. There are two states of nature Θ = {θ0, θ1} where
θ0 = 0 represents “profitable stock” and θ1 = 1 represents “stock not profitable”. Define
the appropriate loss function L(θ, a) for this problem.
b) Compute the probability mass function (pmf) for X under both states of nature.
c) The complete list of all the non-randomized decisions rules D based on x is given by:
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MATH5905 Term One 2022 Assignment One Statistical Inference
d1 d2 d3 d4 d5 d6 d7 d8
x = 0 a0 a1 a0 a1 a0 a1 a0 a1
x = 1 a0 a0 a1 a1 a0 a0 a1 a1
x = 2 a0 a0 a0 a0 a1 a1 a1 a1
For the set of non-randomized decision rules D compute the corresponding risk points.
d) Find the minimax rule(s) among the non-randomized rules in D.
e) Sketch the risk set of all randomized rules D generated by the set of rules in D. You
might want to use R (or your favorite programming language) to make this sketch more
precise.
f) Suppose there are two decisions rules d and d
′
. The decision d strictly dominates d
′
if
R(θ, d) ≤ R(θ, d′
) for all values of θ and R(θ, d) < (θ, d′
) for at least one value θ. Hence,
given a choice between d and d
′ we would always prefer to use d. Any decision rules
which is strictly dominated by another decisions rule (as d
′
is in the above) is said to be
inadmissible. Correspondingly, if a decision rule d is not strictly dominated by any other
decision rule then it is admissible. Show on the risk plot the set of randomized decisions
rules that correspond to the fund manager’s admissible decision rules.
g) Find the risk point of the minimax rule in the set of randomized decision rules D and
determine its minimax risk. Compare the two minimax risks of the minimax decision
rule in D and in D. Comment.
h) Define the minimax rule in the set D in terms of rules in D.
i) For which prior on {θ1, θ2} is the minimax rule in the set D also a Bayes rule?
j) Prior to listening to the two teams, the fund manager believes that the stock will be
profitable with probability 1/2. Find the Bayes rule and the Bayes risk with respect to
his prior.
k) For a small positive ϵ = 0.1, illustrate on the risk set the risk points of all rules which
are ϵ-minimax.
Problem Three
In a Bayesian estimation problem, we sample n i.i.d. observations X = (X1, X2, . . . , Xn)
from a population with conditional distribution of each single observation being the geometric
distribution
fX1|Θ(x|θ) = θ
x
(1 − θ), x = 0, 1, 2, . . . ; 0 < θ < 1.
The parameter θ is considered as random in the interval Θ = (0, 1).
i) If the prior on Θ is given by τ (θ) = 3θ
2
, 0 < θ < 1, show that the posterior distribution
h(θ|X = (x1, x2, . . . , xn)) is in the Beta family. Hence determine the Bayes estimator of θ with
respect to quadratic loss.
Hint: For α > 0 and β > 0 the beta function B(α, β) = R 1
0
x
α−1
(1 − x)
β−1dx satisfies
B(α, β) = Γ(α)Γ(β)
Γ(α+β) where Γ(α) = R ∞
0
exp(−x)x
α−1dx. A Beta (α, β) distributed random
variable X has a density f(x) = 1
B(α,β)
x
α−1
(1 − x)
β−1
, 0 < x < 1, with E(X) = α/(α + β).
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MATH5905 Term One 2022 Assignment One Statistical Inference
ii) Five observations form this distribution were observed: 2, 4, 7, 2, 3. Using zero-one loss,
what is your decision when testing H0 : θ ≤ 0.80 against H1 : θ > 0.80. (You may use
the integrate function in R or another numerical integration routine from your favourite
programming package to answer the question.)
Problem Four
Let X1, X2, . . . , Xn be i.i.d. uniform in (0, θ) and let the prior on θ be the Pareto prior given by
τ (θ) = βαβ
θ
−(β+1), θ > α. (Here α > 0 and β > 0 are assumed to be known constants). Show
that the Bayes estimator with respect to quadratic loss is given by ˆθBayes = max(α, x(n)
)
n+β
n+β−1
.
Justify all steps in the derivation.
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