MATH5905 Term One 2021 Assignment One Statistical Inference
Problem One
During the cold winter months, the principle of the Winterville School District must decide
whether to call o↵ the next day’s school due to the snow conditions. If the principle fails to call
o↵ school and there is snow, there are serious concerns for children and teachers not showing
up for school and high rates of accidents. However, if the principle does call o↵ school and then
it does snow, the students undertake online classes and nothing is lost. On the other hand, if
the principle does call of school and it does not snow, the students must make-up the lost day
later in the year since it’s fine to go outside and students are unlikely to be able to focus in
the online class.
The principle decides that the costs of failing to close the school when there is snow is twice
the costs of closing the school when there is no snow. Therefore, the principle assigns two units
of loss to the first outcome and one to the second. If the principle closes the school when there
is snow or keeps it open when there is no snow, then no loss is incurred.
There are two local experts that provide the principle with independent and identically dis-
tributed weather forecasts. If there is snow, each expert will independently forecast snow with
probability 3/4 and no snow with probability 1/4. However, if there is to be no snow then each
expert independently predicts snow with probability 1/2. The principle will listen to both
forecasts and then make his decisions based on the data X which is the number of experts
forecasting snow.
a) There are two possible actions in the action space A = {a0, a1} where action a0 is to
keep the school open and action a1 is to close the school. There are two states of nature
⇥ = {✓0, ✓1} where ✓0 = 0 represents “no snow” and ✓1 = 1 represents “snow”. Define
the appropriate loss function L(✓, 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 d0. The decision d strictly dominates d0 if
R(✓, d)  R(✓, d0) for all values of ✓ and R(✓, d) < (✓, d0) for at least one value ✓. Hence,
given a choice between d and d0 we would always prefer to use d. Any decision rules
which is strictly dominated by another decisions rule (as d0 is in the above) is said to be
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MATH5905 Term One 2021 Assignment One Statistical Inference
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 principle’s admissible decision rules.
g) Find the risk point of the minimax rule in set of randomized decision rules D and deter-
mine its minimax risk. Compare the minmax risk from the minimax decision rule in D
and D.
h) Define the minimax rule in the set D in terms of rules in D.
i) For which prior on {✓1, ✓2} is the minimax rule in the set D also a Bayes rule?
j) Prior to listening to the forecasts, the principle believes there will be snow with probability
1/2. Find the Bayes rule and the Bayes risk with respect to this prior.
k) For a small positive ✏ = 0.1, illustrate on the risk set the risk points of all rules which
are ✏-minimax. Write down all the vertices that define the region.
Problem Two
Suppose that X is a binomial random variable with parameters n and ✓ where the number of
trials n is assumed to be known. Calculate the Bayes rule (based on a single observation of X)
for estimating ✓ when the prior distribution is the uniform distribution on [0, 1] and the loss
function is given by
L(✓, d) =
1
✓(1 ✓)(✓ d)
2.
Show that the rule you obtained is minimax.
Hint: The Beta function is given by
B(x, y) =
Z 1
0
tx1(1 t)y1dt
and the following relation holds:
B(x+ 1, y)
B(x, y)
=
x
x+ y
.
Problem Three
Suppose X = (X1, . . . , Xn) are i.i.d. Exponential(✓) with density
f(x|✓) = ✓e✓x, x > 0, ✓ > 0
and let ✓ have a Gamma(↵,) prior distribution with density
⌧(✓) =
1
(↵)↵
✓↵1e✓/ , ↵, > 0, ✓ > 0.
a) Find the posterior distribution for h(✓|X). Use the notation s =Pni=1Xi.
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MATH5905 Term One 2021 Assignment One Statistical Inference
b) Hence or otherwise determine the Bayes estimator of ✓ with respect to the quadratic loss
function L(a, ✓) = (a ✓)2.
c) Suppose the following ten observations were observed:
0.12, 0.28, 0.43, 0.34, 0.47, 0.67, 0.82, 0.12, 0.30, 0.45
Two actions a0 (accept H0) and a1 (reject H0) are possible and the losses when using
these actions are given by:
L(✓, a0) =
(
0 if ✓ 2 ⇥0
2 if ✓ 2 ⇥1
and L(✓, a1) =
(
1 if ✓ 2 ⇥0
0 if ✓ 2 ⇥1
Using the loss function above and the parameters ↵ = 2 and = 1 for the prior, what
is your decision when testing H0 : ✓  2.5 versus H1 : ✓ > 2.5. You may use the pgamma
function in R or another numerical integration routine from your favorite programming
package to answer this question.
Problem Four
Determine the form of the Bayes decision rule in an estimation problem with a one-dimensional
parameter ✓ 2 R1 and loss function
L

✓, d

=
(

✓ d if d  ✓,

d ✓ if d > ✓,
where ↵ and are known positive constants.
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