STAT 4102 Exam 2 Name:
Spring 2020
Time Limit: Due at noon, April 12 Student ID:
On my honor, I have neither given nor received unauthorized
aid on this examination.
Signature: Date:
Instructions:
• This exam contains 6 pages and 5 problems.
• You may use any resource available to you, but you cannot plagiarize directly.
• Unless otherwise noted, you must show your work to receive credit.
• If a printer is available, print out the exam, write down your answers neatly on the
exam paper, and submit a copy of your exam through Canvas. If there is no printer,
write down your answers neatly on sheets of paper.
STAT 4102 Exam 2 - Page 2 of 6 Spring 2020
1. Let Y1, Y2, . . . , Yn denote a random sample from a population with mean µ ∈ (−∞,∞) and
variance σ2 ∈ (0,∞). Let Y¯n = n−1
∑n
i=1 Yi. Recall that, by the law of large numbers, Y¯n is a
consistent estimator of µ.
(a) (10 points) Prove that Un = nn+1 Y¯n is a consistent estimator of µ.
(b) (5 points) Prove that Vn = Y¯n − nn+1 is not a consistent estimator of µ.
(c) (5 points) Suppose that, for each i, P(|Yi − µ| > 3) = 0.025. Is Y2 a consistent estimator
of µ? Prove what you assert.
STAT 4102 Exam 2 - Page 3 of 6 Spring 2020
2. Let Y1, Y2, . . . , Yn be iid Uniform(a, b) random variables, where a < b. Recall that this implies
that E(Yi) = (a+ b)/2, and that V (Yi) = (b− a)2/12.
(a) (5 points) Prove that
E(Y 2i ) =
(a+ b)2
4
+
(b− a)2
12
.
(b) (5 points) Let µ′1 = E(Yi), and let µ′2 = E(Y 2i ). Use what you showed in (a) to prove that
a+ b = 2µ′1, b− a = 2
√
3 [µ′2 − (µ′1)2].
(c) (5 points) Use results in (b) to express a and b as functions of µ′1 and µ′2.
(d) (5 points) Let y1, y2, . . . , yn be observed values of Y1, Y2, . . . , Yn. Suppose that
m′1 = n
−1
n∑
i=1
yi = 1.05, m
′
2 = n
−1
n∑
i=1
y2i = 2.14.
Based on these observed sample moments, find estimates of a and b using the method of
moments.
STAT 4102 Exam 2 - Page 4 of 6 Spring 2020
3. Let Y1, Y2, . . . , Yn a random sample from a Uniform(−θ, 0) distribution, where θ > 0. This
implies that the marginal pdf of Yi is
f(yi|θ) =
{
1/θ, −θ ≤ yi ≤ 0
0, elsewhere .
(a) (10 points) Write down the likelihood function of the sample, L(y1, y2, . . . , yn|θ).
(b) (10 points) Find the MLE of θ. (Hint: Refer to Example 9.16. Let y1, y2, . . . , yn be
observed values of Y1, Y2, . . . , Yn. Then yi < 0 for each i, and L(y1, y2, . . . , yn|θ) ̸= 0 if and
only if θ ≥ max{−y1,−y2, . . . ,−yn}.)
STAT 4102 Exam 2 - Page 5 of 6 Spring 2020
4. Let Y1, Y2, . . . , Yn be iid random variables from a Pareto distribution with parameters α > 0
and β = 2. The marginal pdf of Yi is
f(yi|α) =
{
α2αy
−(α+1)
i , yi ≥ 2
0, elsewhere
.
(a) (10 points) Prove that U = Πni=1Yi is sufficient for α.
(b) (10 points) Find the MLE of α. (You can assume that the log likelihood function is
maximized when its derivative is zero.)
STAT 4102 Exam 2 - Page 6 of 6 Spring 2020
5. Let Y1, Y2, . . . , Yn denote a random sample from a Bernoulli distribution with mean θ. Note
that this implies that E(Yi) = θ, and V (Yi) = θ(1−θ). Consider the following three estimators
for θ.
Sn = Y1 − Y2 + Y3, Tn = Y1
2
+
Y2 + Y3 + · · ·+ Yn
2(n− 1) , Un = n
−1
n∑
i=1
Yi.
It’s clear that all three estimators are unbiased.
(a) (10 points) Find the efficiency of Un relative to Sn and Tn, respectively.
(b) (10 points) Use Slutsky’s Theorem (Theorem 9.3) to show that √n(Un−θ)/
√
Un(1− Un)
converges to a standard normal distribution as n→∞.