MATH5905 Term One 2023 Assignment One Statistical Inference

Problem One
i) It is known that for independent Poisson distributed random variables X1 ∼ Poisson(λ1)
and X2 ∼ Poisson(λ2) it holds that
X1 +X2 ∼ Poisson(λ1 + λ2).
Show that ifXi ∼ Poisson(λi), i = 1, 2, . . . , k are independent then the conditional distribution
of X1 given X1 + X2 + . . . Xk is Binomial and determine the parameters of this Binomial
distribution.
ii) Suppose that the X and Y are components of continuous random vector with a density
fX,Y (x, y) = cxy
2, 0 < x < y, 0 < y < 2 (and zero else). Here c is a normalizing constant.
a) Show that c = 516 .
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
Let X and Y be independent uniformly distributed in (0, 1) random variables. Further, let
U(X,Y ) = X + Y, V (X,Y ) = Y −X be a transformation.
a) Sketch the support S(X,Y ) of the random vector X,Y in R
2.
b) Sketch the support S(U,V ) of the random vector (U, V ) in R
2.
c) Determine the Jacobian of the transformation
d) Determine the density of the random vector (U, V )
Justify completely each step.
Problem Three
You are going to the races and want to decide whether or not to bet on the horse Thunderbolt.
You want to apply decision theory to make a decision. You use the information from two
independent horse-racing experts. Data X represents the number of experts recommending
you to bet on Thunderbolt (due, of course, to their belief that this horse will win the race).
If you decide not to bet and Thunderbolt does not win, or when you bet and Thunderbolt
wins the race, nothing is lost. If Thunderbolt does not win and you have decided to bet on
him, your subjective judgment is that your loss would be four times higher than the cost of not
betting but the Thunderbolt does win (as you will have missed other opportunities to invest
your money).
You have investigated the history of correct winning bets for the two horse-racing experts and
it is as follows. When Thunderbolt had been a winner, both experts have correctly predicted
his win with probability 5/6 (and a loss with a probability 1/6). When Thunderbolt has not
won a race, both experts had a prediction of 3/5 for him to win. You listen to both experts
and make your decision based on the data X.
a) There are two possible actions in the action space A = {a0, a1} where action a0 is to bet
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MATH5905 Term One 2023 Assignment One Statistical Inference
and action a1 is not to bet. There are two states of nature Θ = {θ0, θ1} where θ0 = 0
represents “Thunderbolt winning” and θ1 = 1 represents “Thunderbolt not winning”.
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:
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 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 experts, you believe that Thunderbolt will win the race with
probability 1/2. Find the Bayes rule and the Bayes risk with respect to your 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 Four
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) = θx(1− θ), x = 0, 1, 2, . . . ; 0 < θ < 1.
The parameter θ is considered as random in the interval Θ = (0, 1).
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MATH5905 Term One 2023 Assignment One Statistical Inference
i) If θ is interpreted as a probability of success in a single trial in a success-failure scheme, give
an interpretation of the conditional distribution of the random variable X1 given Θ = θ.
ii) If the prior on Θ is given by τ(θ) = 20θ3(1 − θ), 0 < θ < 1, show that the posterior
distribution h(θ|X = (x1, x2, . . . , xn)) is also 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) =
∫ 1
0 x
α−1(1 − x)β−1dx satisfies
B(α, β) = Γ(α)Γ(β)Γ(α+β) where Γ(α) =
∫∞
0 exp(−x)xα−1dx. A Beta (α, β) distributed random vari-
able X has a density f(x) = 1B(α,β)x
α−1(1− x)β−1, 0 < x < 1, with E(X) = α/(α+ β).
iii) Seven observations form this distribution were observed:3, 4, 7, 3, 6, 5, 2. 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.)