STAT 3445Q
Name:
Spring 2020
Midterm exam 2
3/27 - 3/28 noon
Time Limit: 36 hours, EST 3/27 0:00am - 3/28 noon 11:59am
• This exam contains 6 pages (including this cover page) and 5 questions.
• Total of points is 100.
• Justify your answers whenever possible to ensure full credit, not just the final answer
you arrive at. If you’re using any theorem for any step, please write out
which theorem you’re using. You will receive most of the credit for well-argued
derivations.
• This exam is open book, open note. You can use all available materials including
textbook, notes, previous homework or exams.
• Never google the question and copy the answer directly. I may set some trap by giving
you a question that you may find answer online, but actually the answer is wrong. If I
find that you copy that wrong answer, your total score for this exam will be 0.
• Never discuss with other students. Your answer should be totaly your independent
work and thinking. You can use materials mentioned above to help you learn and
think, but no discussion with other students. If I find two students have identical
answer for any question, both of them will have grade 0 for this whole exam.
There are some questions asking you to illustrate use your own words, so it’s very easy
to say if you’re copying others or not.
• Try your best to make your answers organized and neat.
Grade Table (for instructor use only)
Question Points Bonus Points Score
1
30
0
2
10
0
3
15
0
4
20
0
5
25
0
Total:
100
0
STAT 3445Q
Midterm exam 2 - Page 2 of 6
3/27 - 3/28 noon
1. Let X1 , . . . , Xn be i.i.d. Gamma(α, 2), where α > 0 is unknown. The PDF of Xi ’s is
given by
(
1
α−1
exp(− x2 ), x > 0,
αx
f (x) = Γ(α)2
0, elsewhere.
(a) (10 points) Find the MOM estimator for α, denoted as α
b. Carefully argue that α
b
is an unbiased and consistent estimator for α.
(b) (10 points) Is X̄ consistent for α? If not, which parameter is it consistent for?
(c) (10 points) Now define another estimator for α as: (X1 + X2 )/4 + 1/n. Prove that
this estimator is not consistent for α. (hint: use the definition of consistency to
prove.)
STAT 3445Q
Midterm exam 2 - Page 3 of 6
3/27 - 3/28 noon
2. (10 points) Suppose X1 , . . . , Xn is an i.i.d. sample from a distribution with pdf:
α(α + 1)xα−1 (1 − x),
0 < x < 1,
where α is unknown parameter. Show that 2X̄/(1 − X̄) is consistent for α.
STAT 3445Q
Midterm exam 2 - Page 4 of 6
3/27 - 3/28 noon
3. Suppose X1 , . . . , Xn is an i.i.d. sample from a uniformly distributed population between
θ and 2θ, where θ > 0 is unknown. The population distribution PDF is
(
1
, θ < x < 2θ,
f (x) = θ
0 , elsewhere.
Denote X(1) = min(X1 , . . . , Xn ), X(n) = max(X1 , . . . , Xn ).
(a) (5 points) Show U = (X(1) + X(n) , X(n) ) is jointly sufficient for θ.
(b) (5 points) We know that “sufficiency” is defined based on the so called “information” concept. Can you explain what we mean by saying U defined above is sufficient
for θ using your own words?
(c) (5 points) If I define a statistics T = (X1 , . . . , Xn ), which is just a n dimensional
vector consisting of original sample. Is T sufficient for θ? Is T minimal sufficient
for θ? Please give the answer and explain the reason.
STAT 3445Q
Midterm exam 2 - Page 5 of 6
3/27 - 3/28 noon
4. Suppose X1 , . . . , Xn is an i.i.d. sample of size n from a P oisson(λ) population, where
λ > 0 is an unknown parameter. The probability mass function of the population
distribution is
( x −λ
λ e
, x = 0, 1, 2, . . . ,
x!
P (X = x) =
0, otherwise.
P
(a) (5 points) Show U = ni=1 Xi is sufficient for λ.
(b) (10 points) Suppose we want to estimate P (Xi = 1) = λe−λ . Define a estimator
θ̂ = I(X1 = 1) this way: I is still an indicator function, and the resulting θ̂ will just
be a binary random variable that only takes values 0 or 1. The probability to have
value 1 will be the probability that I(X1 = 1), which is equivalent to X1 = 1.
It’s easy to show that θ̂ is unbiased for λe−λ . Starting from this naive unbiased
estimator
P θ̂, use the definition of Rao-Blackwell Theorem and sufficient statistic
U = ni=1 Xi to get an improved unbiased estimator for λe−λ .
(hint: from the Rao-Blackwell theorem, you need to work on a conditional expectation of θ̂. Note that even with condition, the possible values for θ̂ are still 0 and
1. So the conditional distribution of θ̂ is still binary, but with a probability that
you need to calculate. You can follow the example on the Note 12 where we have
a Bernoulli population distribution.)
(c) (5 points) Explain what the “improvement” means in part (b). And is this improved unbiased estimator MVUE for λe−λ ? And why?
STAT 3445Q
Midterm exam 2 - Page 6 of 6
3/27 - 3/28 noon
5. On Note 16, we discussed about the confidence interval for large sample. This is built
on central limit theorem (CLT).
(a) (5 points) CLT is talking about the asymptotic property of sample mean X̄ under
large sample. But we also include the CI for population proportion p in Note 16,
which is constructed using sample proportion p̂. why CLT also works for sample
proportion?
(b) (15 points) The two-sided CI for two population proportion difference p1 − p2 is
already shown in Note 16 without detailed derivation. Please derive this conclusion using pivot method by yourself. (I derived the CI for one population mean
in the lecture video. You may use that as a template.) To formally state the
problem, please see below: There are two population distributions: Binary(p1 ) and
Binary(p2 ), where p1 and p2 are unknown parameters, showing the population proportions. Two set of samples X1 , . . . , Xn1 and Y1 , . . . , Yn2 are two sets of sample
from the two populations, respectively. Construct the two-sided CI with confidence
level 1 − α using pivot method and explain the meaning of this CI:
s
s
X̄ − Ȳ + zα/2 X̄(1 − X̄) + Ȳ (1 − Ȳ ) , X̄ − Ȳ − zα/2 X̄(1 − X̄) + Ȳ (1 − Ȳ ) ,
n1
n2
n1
n2
where zα/2 is the lower tail quantile.
(c) (5 points) If now you have the value for X̄ = x̄, and Ȳ = ȳ, where x̄ and ȳ are two
constants. After plug in the CI above, you’ll get:
s
s
x̄ − ȳ + zα/2 x̄(1 − x̄) + ȳ(1 − ȳ) , x̄ − ȳ − zα/2 x̄(1 − x̄) + ȳ(1 − ȳ) .
n1
n2
n1
n2
Can you still use the same explanation in part (b) to explain this CI after plug in?
Why?
(Blank page)
...

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