STAT 4101 Final Exam, Spring 2021
First name: Last name:
Hawkid: @uiowa.edu
Please read the following instructions before you start the exam.
• The exam is from May 11th 10:00 AM to noon.
• There is imposed an x% penalty on any submission that is x minutes late.
• There are 5 problems. The total is 10 × 5 = 50 points.
• Show all work for each problem below. If you only present a correct answer without stating how you get it, no credit will be given.
• Calculator ready solutions are not sufficient to obtain full credits. A numer- ical answer is needed if a problem asks for it.
• You must work independently and cannot discuss with anyone else expect for the instructor.
• It is prohibited to share the exam content or the solution with others. See code of academic honesty: https://clas.uiowa.edu/students/handbook/academic-fraud-honor-code
1
1. (10 pts) Multiple choices. Select the most accurate answer.
• Which one is the Type II error?
(a) Reject H0 when H0 is true.
(b) Reject H0 when H1 is true.
(c) Fail to reject H0 when H1 is true.
• Suppose Y1 is sufficient for θ and Y2 is unbiased for θ. Which of the following statistics is also unbiased and its variance is no larger than the variance of Y2?
(a) Y1.
(b) E(Y1|Y2).
(c) E(Y2|Y1).
• Suppose the whole sequence X1,…,Xn,… and X are random variables; a is a constant. Which of
the following statements is incorrect?
(a) If Xn converges to X in probability, then Xn converges to X in distribution. (b) If Xn converges to a in distribution, then Xn converges to a in probability. (c) If Xn converges to X in distribution, then Xn converges to X in probability.
• Which of the following statements is incorrect?
(a) If X is an efficient estimator for θ, then X is also unbiased for θ. (b) If X is an efficient estimator for θ, then X is also an MVUE for θ. (c) If X is an efficient estimator for θ, then X is also consistent for θ.
• Suppose a distribution with pdf f(x;θ) belongs to the regular exponential class and has the monotone decreasing likelihood ratio property in the statistics Y = ni=1 K(Xi). To test
H0 :θ≥θ0 vsH1 :θ<θ0.
which of the following tests could be the uniformly most powerful? (a) Y >k.
(b) Y =k.
(c) Y