Download STAT 1632 Intermediate Mathematical Statistics STAT 2640

STAT 1632 Intermediate Mathematical Statistics
STAT 2640 Intermediate Mathematical Statistics
Spring 2017
Tuesday-Thursdays 9:30-10:45AM
Posvar 1811
Instructor Sungkyu Jung
Department of Statistics, University of Pittsburgh
Email: [email protected] Office: Posvar 1806 Phone: 412-624-9033
Office Hours: By appointment. You can always talk to me after class.
TA/Grader Bowen Yi
Email: [email protected]
Office Hours: TBA
Course description: This course is the second half of a two semester sequence in mathematical statistics. It is for upper-level undergraduates and graduate statistics majors and interested students from other
fields. Topics to be covered include point and interval estimation, hypotheses testing, sufficiency, the information inequality, exponential families, finite and large-sample efficiency. The two major statistical paradigms,
Frequentist and Bayesian approaches, will be presented and contrasted.
Prerequisite:
Textbook:
STAT 1631 (STAT 2630) or its equivalents.
DeGroot and Schervish, Probability and Statistics, 4th Ed. Pearson.
Reference:
• Student Solutions Manual for Probability and Statistics, DeGroot and Schervish, Pearson.
• Miller and Miller, John E. Freunds Mathematical Statistics with Applications, 8th Ed. Pearson.
• Hogg and Craig, Introduction to Mathematical Statistics, 5th Ed. Prentice Hall.
Course Requirements and Grading:
• Homework 25%
• Midterm exam 30% - Tuesday, February 28 (in class).
• Final exam 30% - Thursday, April 20 (in class).
• Mini-lecture 15%
Course website: This term we will be using Piazza for class discussion. The system is highly catered to
getting you help fast and efficiently from classmates, the TA, and myself. Rather than emailing questions to
the teaching staff, I encourage you to post your questions on Piazza. If you have any problems or feedback for
the developers, email [email protected].
Find our class page at: https://piazza.com/pitt/spring2017/stat1632/home
Tentative course schedule:
Dates
Jan. 5
Jan. 10
Jan. 12
Jan. 17
Jan. 19
Jan. 24, 26
Jan. 31, Feb. 2
Feb. 7, 9, 14
Feb. 16
Feb. 21
Feb. 23
Feb. 28
Mar. 2
Mar. 6–10
Mar. 14, 16
Mar. 21
Mar. 23, 28
Mar. 30, Apr. 4
Apr. 6
Apr. 11
Apr. 13
Apr. 18, 20
Topic
Introduction
Review of preliminary probability theory
Simulations
Decision theory fundamentals
Bayes estimators
Maximum likelihood estimators
Properties of M.L.E. (and mini-lectures)
Sufficient statistics and Rao-Blackwell theorem
Normal sampling theory
Confidence interval
Review (and mini-lectures)
Midterm exam
Midterm exam review
No class. Spring Break
Unbiasedness, efficiency, Fisher information and
asymptotic normality of M.L.E.
Fundamentals of hypothesis testing
Hypotheses testing and Neyman-Pearson lemma
(and mini-lectures)
Composite hypotheses and uniformly most powerful
tests
Bayesian analysis
Mini-lectures
No class. Easter holiday.
Review and final exam
Textbook
sections
7.1
1.1–6.3
6.1–6.3, 8.1
4.8, 7.4, 7.9, 9.8
7.2-7.4
7.5
7.6
7.7–7.9
8.1–8.4
8.5
Homework due
HW 1 (Jan. 19)
HW 2 (Feb. 2)
HW 3 (Feb. 16)
HW 4 (Feb. 23)
7.1–8.5
8.7–8.8
9.1
9.1–9.3
HW 5 (Mar. 23)
9.3–9.4
HW 6 (Mar. 30)
8.6, 9.8
9.5–9.7
HW 7 (Apr. 11)
HW 8 (Apr. 18)
Homework assignments:
• Homework assignments consist of problems from the text and supplemental problems. You must do all
homework problems on your own, although you may discuss the problems with other students. Assignments must be turned in on time (due at the beginning of the lecture) for credit.
• No late homework will be accepted. No electronic copy will be accepted. Missed homework will receive
a grade of zero. Show all work neatly on letter-sized papers. Clearly label each problem. Homework
pages must be stapled together. A homework violating any of the above will receive a grade of zero. All
homework grades will be counted for final grade.
Mini-lectures:
• Mini-lectures are required to all students. A total of 10 mini-lectures will be given by graduate students
and undergraduate students. Students will have ten to fifteen minutes to present their work to class.
Use of powerpoint or beamer slides is strongly recommended. The slides (or a report) will also be made
available to all students enrolled in the class.
• In general, the presentation consists of a theory and (numerical) examples. Consult with your instructor
for preparation.
• List of topics:
1.1. Variance stabilizing transformation. (Textbook section 7.3, p.365.)
1.2. Bayes estimators for conjugate prior families: a) Poisson-gamma, b) exponential-gamma, c) Normal
mean-normal, and d) normal precision-gamma (Textbook section 7.3, pp. 397-402.)
1.3. Bayes estimators for the maximum. (See textbook section 7.3 and Assignment 2.9.)
1.4. Bayesian sensitivity analysis for coin fairness involving an extended binomial-beta family. Students
who choose this part must be familiar with R or Matlab. (See textbook p. 387 and Assignment
2.10.)
1.5. Normal sampling theory: Does it really work? Does it work for non-normal distributions? (See
Textbook section 8.2–8.4. See Assignment 4.12.)
1.6. The meaning of “approximate” confidence intervals. (See Textbook section 8.5, p. 487 and Assignment 4.13)
1.7. Simulation for consistency, (asymptotic) unbiasedness, and asymptotic distribution of an M.L.E.
Students who choose this part must be familiar with R or Matlab. (Related textbook sections are
7.6, 8.7, 8.8, 12.1–12.3. See Assignment 5.11.)
1.8. Empirically comparing two estimators: bias, variance and asymptotic distributions. Students who
choose this part must be familiar with R or Matlab. (Related textbook sections are 7.6, 8.7, 8.8,
12.1–12.3. See Assignment 5.12.)
1.9. One-sample t-test: the level, power function and p-value. (Textbook section 9.5, pp. 577-583.)
1.10. One-sample t-test as a likelihood ratio test (Textbook section 9.5, pp. 583-585.)
1.11. Two-sample t-test: the level, power function and p-value. (Textbook section 9.6, pp. 587-591.)
1.12. Two-sample t-test for unequal variances. (Textbook section 9.6, pp. 593-595)
1.13. Power comparison among the t-test and an alternative, for normal and non-normal data. (See
Textbook sections 9.5 and 10.8 and Assignment 7.8.)
1.14. Comparing two ad-hoc tests for testing the location of Cauchy distributions. (See Assignment
7.9. For background on Cauchy distribution, read https://en.wikipedia.org/wiki/Cauchy_
distribution, Textbook Example 7.6.3, Exercise 12 in Section 9.1)
1.15. F-test for comparing variances of two normal distributions: the level, power function and p-value.
(Textbook section 9.7, pp. 597-602.)
1.16. F-test as a likelihood ratio test (Textbook section 9.7, pp. 602-603.)
University Policies:
• Academic Integrity
Students in this course will be expected to comply with the University of Pittsburgh’s Policy on Academic
Integrity. Any student suspected of violating this obligation for any reason during the semester will be
required to participate in the procedural process, initiated at the instructor level, as outlined in the
University Guidelines on Academic Integrity. This may include, but is not limited to, the confiscation
of the examination of any individual suspected of violating University Policy. Furthermore, no student
may bring any unauthorized materials to an exam, including dictionaries and programmable calculators.
• Disability Services
If you have a disability for which you are or may be requesting an accommodation, you are encouraged
to contact both your instructor and Disability Resources and Services (DRS), 140 William Pitt Union,
(412) 648-7890, [email protected], (412) 228-5347 for P3 ASL users, as early as possible in the term.
DRS will verify your disability and determine reasonable accommodations for this course.
• Copyright Notice
Course materials may be protected by copyright. United States copyright law, 17 USC section 101, et
seq., in addition to University policy and procedures, prohibit unauthorized duplication or retransmission
of course materials. See Library of Congress Copyright Office and the University Copyright Policy.
Assignment 1 | Review of preliminary probability theory
Textbook sections: 1.1 - 6.3
1.1. Let A1 , . . . , An ∈ F be events of a probability model (Ω, F, P ).
P
(a) Show that P (∪ni=1 Ai ) ≤ ni=1 P (Ai ).
P
(b) Show that P (∩ni=1 Ai ) ≥ 1 − ni=1 P (Aci ).
1.2. The cumulative distribution function of Exponential(λ) is
1 − eλx , x > 0;
F (x) =
0,
x ≤ 0.
(a) Verify that the quantile function for Exponential(λ) is Q(p) = − log(1−p)
, p ∈ (0, 1).
λ
(b) What is the median of Exponential(λ)? Is it the same as the mean of the distribution?
1.3. Using the normal table (p.861 of the textbook) or a statistical software to evaluate the values of the
quantile function for the standard normal. Specifically, let Φ−1 denote the quantile function for the
standard normal (there is no closed form for this function).
(a) Evaluate Φ−1 (p) for p = 0.1, 0.5, 0.975.
(b) Sketch the graph of Φ, and mark the coordinates (Φ−1 (p), p) for p = 0.1, 0.5, 0.975.
1.4. Read Textbook section 4.7. Show that if X and Y are random variables for which the necessary moments
exist, then
(a) E(Y ) = E(E(Y |X)), and
(b) Var(Y ) = E(Var(Y |X)) + Var(E(Y |X)).
1.5. Textbook section 6.2, Exercise 1. (on p-convergence)
1.6. Textbook section 6.2, Exercise 9. (on p-convergence)
1.7. Textbook section 6.3, Exercise 10. (on CLT and Chebyshev)
1.8. Textbook section 6.3, Exercise 13. (on delta method)
1.9. Textbook section 6.3, Exercise 14. (a) (on delta method)
Assignment 2 | Bayesian analysis I
Textbook sections: 7.1–7.4.
2.1. Read textbook section 7.3 and read the Wikipedia article at
https://en.wikipedia.org/wiki/Conjugate_prior. Compute the posterior distributions of some conjugate prior families, including a) Poisson-gamma, b) exponential-gamma, c) Normal mean-normal, and
d) normal precision-gamma family.
2.2. Textbook section 7.2, Exercise 2. (on posterior probability computation)
2.3. Textbook section 7.2, Exercise 10. (on posterior probability computation)
2.4. Textbook section 7.3, Exercise 3. (on Bernoulli-Beta family)
2.5. Textbook section 7.3, Exercise 14. (on Neg.Binomial-Beta family)
2.6. Textbook section 7.3, Exercise 18. (on Pareto-Uniform family)
2.7. Textbook section 7.3, Exercise 21. (on use of improper prior)
2.8. Textbook section 7.4, Exercise 4. (asymptotics of Bernoulli-Beta posterior mean )
2.9. (Do not submit; for mini-lectures) Read the short article “Bayesian inference of a uniform distribution” by
Thomas P. Minka. (http://research.microsoft.com/en-us/um/people/minka/papers/minka-uniform.pdf).
2.10. (Do not submit; for mini-lectures) Suppose you are about to flip a coin that you believe is fair. If p
denotes the probability of flipping a head, then your “best guess at p is .5. Moreover, you believe that it
is highly likely that the coin is close to fair, which you quantify by P (.44 < p < .56) = .9. Consider the
following two priors for p:
P 1 : p ∼ Beta(100, 100)
P 2 : p distributed according to the mixture prior g(p) = .9fB (p; 500, 500) + .1fB (p; 1, 1),
where fB (p; a, b) is the beta density with parameters a and b.
(a) Simulate 1000 values from each prior density P1 and P2. By summarizing the simulated samples,
show that both priors match the given prior beliefs about the coin flipping probability p.
(b) Suppose you flip the coin 100 times and obtain 45 heads. Simulate 1000 values from the posteriors
from priors P1 and P2, and compute 90% probability intervals. [A 90% probability interval is given
by [Q(0.05), Q(0.95)], where Q is the quantile function of the (empirical) distribution.]
(c) Suppose that you only observe 30 heads out of 100 flips. Again simulate 1000 values from the two
posteriors and compute 90% probability intervals.
(d) Obtain the Bayes estimates under the squared and absolute loss functions, for each of the cases (b)
and (c), under each of the prior P1 and P2.
(e) Looking at your results from (b) and (c), comment on the sensitivity of the inference with respect
to the choice of prior density in each case.
Assignment 3 | Maximum likelihood estimation
Textbook sections: 7.5–7.9
3.1. Let X1 , . . . , Xn represent a random sample from each of the populations having the following probability
density functions:
(a) f (x; θ) = θx e−θ /x!, x = 0, 1, 2, . . . ., 0 ≤ θ < ∞, zero elsewhere, where f (0; 0) = 1.
(b) f (x; θ) = θxθ−1 , 0 < x < 1, 0 < θ < ∞, zero elsewhere.
(c) f (x; θ) = (1/θ)e−x/θ , 0 < x < ∞, 0 < θ < ∞, zero elsewhere.
(d) f (x; θ) = 21 e−|x−θ| , −∞ < x < ∞, −∞ < θ < ∞, zero elsewhere.
(e) f (x; θ) = e−(x−θ) , θ ≤ x < ∞, −∞ < θ < ∞, zero elsewhere.
(f) f (x; θ) = (θ2 − θ1 )−1 , θ1 ≤ x ≤ θ2 , −∞ < θ1 < θ2 < ∞, zero elsewhere.
In each case, find the maximum likelihood estimator (m.l.e.) Θ̂ of θ.
3.2. (Do not submit) Read textbook example 7.5.6.
3.3. (Do not submit) Read textbook example and 7.5.8; Textbook section 7.5, Exercise 8.
3.4. (Invariance property of m.l.e.) Let X1 , . . . , Xn be a random sample from the distribution with pdf
f (x) = (1/θ)e−x/θ , x > 0. Let η = 1/θ > 0.
(a) Compute the m.l.e. η̂ of η defined by
η̂ = arg max
η>0
n
Y
ηe−ηxi .
i=1
(b) Obtain the m.l.e. θ̂ of θ (this is done in Exercise 3.1), then compare 1/θ̂ with the answer of part
(a).
3.5. Textbook section 7.6, Exercise 4. (on mle of a transformed parameter)
3.6. Textbook section 7.6, Exercise 6. (on mle of a transformed parameter)
3.7. Textbook section 7.6, Exercise 14. [Hint: Read Textbook sections 5.2 and 5.5.] (on mles under Neg.Bin
and Bin)
3.8. Textbook section 7.6, Exercise 15. (on mles for non i.i.d. sample)
3.9. For each of the sampling situations described in Exercise 3.1, find a sufficient statistic (or non-trivial
jointly sufficient statistics).
3.10. Textbook section 7.8, Exercise 4. (on sufficient statistics)
3.11. Let X1 , . . . , Xn represent a random sample from Bernoulli(p).
(a) Is X̄n a minimal sufficient statistic?
(b) Write p̂B for the Bayes estimator of p with respect to the squared error loss under a certain specified
Beta prior distribution. Is p̂B a minimal sufficient statistic?
3.12. (Extra credit) Textbook section 7.8, Exercise 12. (on computing the median and its mle)
Q
3.13. Textbook section 7.9, Exercise 6. [Hint: i) T (X1n ) = ni=1 Xi is a sufficient statistic; ii) Use Theorem
7.9.1. (Rao-Blackwell)]
3.14. Textbook section 7.9, Exercise 13. (on performing Rao-Blackwellization)
Assignment 4 | Normal sampling theory and confidence intervals
Textbook sections: 8.1–8.5
4.1. Suppose X1 , . . . , X20 are a random sample of size 20 from N (100, 100) population. What are the distributions of the following quantities?
1
20 (X1
(a) sample mean: X̄ =
19 2
100 S ,
X̄−100
√ ;
10/ 20
(b) a scaled sample variance:
(c) standardized mean:
(d) studentized mean:
+ . . . + X20 );
where S 2 =
1
19
P20
i=1 (Xi
− X̄)2 ;
X̄−100
√ .
S/ 20
4.2. Textbook section 8.3, Exercise 6. (on the sampling dist’n of S 2 )
4.3. Textbook section 8.3, Exercise 8. (on the CLT for average iid χ2 )
4.4. Textbook section 8.3, Exercise 9. (on independence of S 2 and X̄)
4.5. Textbook section 8.4, Exercise 2. (on a transformation for t-dist’n)
4.6. Textbook section 8.4, Exercise 3. (on the definition of t-dist’n)
4.7. Textbook section 8.5, Exercise 1. (on the definition of C.I. and normal sampling theory)
4.8. Textbook section 8.5, Exercise 4. (on the sample size calculation and normal sampling theory)
4.9. Textbook section 8.5, Exercise 6. (on pivotal quantities)
4.10. Suppose X1 , . . . , Xn form a random sample from Bernoulli(p).
(a) Show that for the m.l.e. p̂ = X̄n , V (X1 , . . . , Xn , p) = X̄n − p is not a pivot (pivotal quantity).
√
(b) By the central limit theorem, n(X̄n −p) converges in distribution (as n → ∞) to some distribution.
What is the limiting distribution?
√
(c) Let α(x) = 2 arcsin(x1/2 ). Using the delta method, show that the limiting distribution of n[α(X̄n )−
α(p)] is the standard normal.
(d) Use the result in (c), show that an approximate coefficient γ confidence interval for arcsin(p1/2 ) is
the interval with endpoints
1 + γ −1/2
arcsin(X̄n1/2 ) ± Φ−1 (
)n
,
2
where Φ is the cumulative distribution function of the standard normal distribution.
4.11. (Do not submit) Read textbook example 8.5.11.
4.12. (Do not submit; for mini-lectures) In this exercise, we empirically confirm the theoretical sampling
distributions of statistics from normal population. Set µ = σ = 1 and obtain a random sample of
size n = 25 from N (µ, σ 2 ). The sampling distribution of any statistic T (X1n ) can be approximated by
the repeatedly computing T (X1n ) for different random samples. Denote these values by T1 , . . . , TN (for
N = 1000).
(a) Take T (X1n ) = X̄. Confirm the fact, X̄ ∼ N (1, 1/25), by
i. providing a histogram of {Tj : j = 1, . . . , N }, overlaid with the theoretical density function,
ii. and providing a QQ-plot.
p
(b) Repeat (a) for T (X1n ) = (X̄ − 1)/ Sn2 /n.
(c) Provide a graphical evidence of the fact that X̄ and Sn are independent.
(d) Now suppose that the model is Poisson(2). Re-do (a)-(c) and approve or disapprove the claims
therein.
4.13. (Do not submit; for mini-lectures)
(a) Read textbook page 487. Re-create Figure 8.5 using your own random sample and varying the
sample size from n = 10 to n = 50.
(b) Read textbook example 8.5.10. Create a figure showing one hundred observed confidence intervals
for the estimation of Poisson mean parameter. Vary the sample size from n = 10 to n = 50.
Comment on the quality of the confidence interval.
Assignment 5 | Theory of point estimation
Textbook sections: 8.7–8.8
5.1. Textbook section 8.7, Exercise 1. (on mle for transformed parameter and unbiasedness)
5.2. Read textbook example 8.7.6. Suppose X1 , . . . , Xn form a random sample from N (µ, σ 2 ). Let σ̂02 and σ̂12
be the two estimators of σ 2 , which are defined as follows:
n
σ̂02
1X
=
(Xi − X̄n )2 ,
n
i=1
n
σ̂12
1 X
=
(Xi − X̄n )2 .
n−1
i=1
(a) Which estimator is unbiased?
(b) Show that the M.S.E. of σ̂02 is smaller than the M.S.E. of σ̂12 for all possible values of µ and σ 2 .
5.3. Textbook section 8.7, Exercise 8. [Hint: Use the uniqueness of power series representation.]
5.4. Textbook section 8.7, Exercise 11. (on unbiasedness, variance and MSE)
5.5. Suppose X = (X1 , . . . , Xn ) form a random sample from a distribution with pdf or pmf f (x; θ), for
−∞ < x < ∞, −∞ < θ < ∞. Assume that δ(X) is an unbiased estimator of θ.
(a) Let T = r(X) be a sufficient statistic. Argue that δ0 (X) = Eθ (δ(X)|T ) is a function of T , and does
not depend on the value of θ. [Hint: This is straightforward due to the definition of a sufficient
statistic.]
(b) Show that δ0 (X) is also an unbiased estimator of θ. [Hint: Use the result in Assignment 1.4.]
(c) Show that Varθ (δ0 ) ≤ Varθ (δ) for every value of θ. [Hint: Use the result in Assignment 1.4.]
5.6. Let X1 , . . . , Xn represent a random sample from each of the populations having the following probability
density functions:
(a) f (x; θ) = θx e−θ /x!, x = 0, 1, 2, . . . ., 0 ≤ θ < ∞, zero elsewhere, where f (0; 0) = 1.
(b) f (x; θ) = θx (1 − θ)1−x , x = 0, 1, 0 < θ < 1, zero elsewhere.
(c) f (x; θ) = (1/θ)e−x/θ , 0 < x < ∞, 0 < θ < ∞, zero elsewhere.
In each case, compute the Fisher information in the random sample (X1 , . . . , Xn ).
5.7. For each of the distributions in Assignment 5.6, show that the m.l.e. θ̂ of θ is the uniformly minimum
variance unbiased estimator (UMVUE). [That is, show that the m.l.e. is unbiased and efficient.]
√
5.8. For each of the distributions in Assignment 5.6, evaluate the asymptotic distribution of n(θ̂ − θ).
5.9. Textbook section 8.8, Exercise 14. [Note: Do not attempt to actually determine the m.l.e. (cf. Textbook
example 7.6.4). Use Theorem 8.8.5. ]
5.10. Textbook section 8.8, Exercise 4+10. (on computing information)
5.11. (Do not submit; for mini-lectures) Consider estimating the population variancePfrom a random sample
of size n. Assume that the model is N (µ, σ 2 ), and take the estimator σ̂n2 = 1/n ni=1 (Xi − X̄n )2 .
(a) Create a computer program to generate random values of σ̂n2 .
(b) Use simulations to “empirically” show that σ̂n2 is biased.
(c) Use simulations to empirically demonstrate that σ̂n2 is consistent, thus asymptotically unbiased, and
asymptotically normally distributed.
Repeat (a-c), but assume that the model is Exponential(θ).
5.12. (Do not submit; for mini-lectures) Suppose a random sample X1 , . . . , Xn from a Uniform distribution on
(0, θ), where θ > 0 is unknown.
(a) Compute the m.l.e. of θ and the method-of-moment estimator of θ.
(b) Use simulations to approximate the sampling distributions of θ̂nMLE and θ̂nMME .
(c) Based on (b), estimate the bias, variance and MSE of each estimator.
(d) Use simulations to reveal the limiting distributions of estimators.
Assignment 6 | Testing hypothesis I
Textbook sections: 9.1–9.2
6.1. Textbook section 9.1, Exercise 1. (on power and size calculation)
6.2. Textbook section 9.1, Exercise 2. (on power and size calculation)
6.3. Textbook section 9.1, Exercise 8 + 10. (on power, size and p-value calculation)
6.4. Textbook section 9.1, Exercise 12. (on power, size and p-value calculation)
6.5. Textbook section 9.1, Exercise 16. (on the duality between CI and test)
6.6. A random sample of size n, {X1 , . . . , Xn } from an exponential population with mean θ, is to be used to
test H0 : θ = θ0 versus H1 : θ 6= θ0 for a given value of θ0 .
n nx̄
− +n
(a) Show that the expression for likelihood ratio statistic is λ = θx̄0 e θ0 .
(b) Show that the critical region of the likelihood ratio test can be written as
x̄e−x̄/θ0 ≤ K.
(c) Without referring to Wilks’ theorem (Theorem 9.1.4), show that −2 log(Λ) is approximately distributed as χ2 (1) for large n under H0 .
Hints: Use
• the second order Taylor expansion log(1 + x) x − x2 /2;
• and the central limit theorem for X̄;
• and note that
n
X̄
− nX̄ +n
e θ0
−2 log(Λ) = −2 log
θ0
X̄
X̄
−1 +1 −
−1
= −2n log
θ0
θ0
(d) Using Wilks’ theorem (or part (c)), show that for large n the critical region −2 log(λ) > 3.841 has
size 0.05, approximately.
√0 (e) Use the central limit theorem to show that for large n a critical region θx̄−θ
> 1.96 has size 0.05,
0/ n
approximately.
(f) Discuss the relationship between two tests in (d) and (e).
6.7. Textbook section 9.2, Exercise 2. [Hint: Use Theorem 9.2.1.] (on minimizing a linear combination of
error probs)
6.8. Textbook section 9.2, Exercise 4. [Hint: Use Neyman-Pearson Lemma.]
6.9. Textbook section 9.2, Exercise 12. (on applying n-p lemma)
Assignment 7 | Testing hypothesis II
Textbook sections: 9.3–9.7
7.1. (Do not submit) Textbook section 9.3, Exercises 1, 2, 3, 4, 8, 9. (on MLR for UMP)
7.2. (Do not submit) The following facts may be useful in determining the critical region of a test.
P
(a) If Xi , i = 1, . . . , n, are i.i.d. Poisson(λ), then ni=1 Xi ∼Poisson(nλ).
P
(b) If Xi , i = 1, . . . , n, are i.i.d. N (0, 1), then ni=1 Xi2 ∼ χ2 (n).
(c) If Xi , i = 1, . . . , n, are i.i.d. Uniform(0,θ), then max{X1 , . . . , Xn }/θ ∼ Beta(1, n). Hint: Using the
fact that if Xi , i = 1, . . . , n, are i.i.d. with pdf f (x), x ∈ X , then Y = max{X1 , . . . , Xn } has pdf
g(x) = nf (x)(P (X ≤ x)n−1 , x ∈ X .
7.3. Let X1 , . . . , Xn represent a random sample from each of the following populations:
(a) Poisson distribution with unknown mean θ (θ > 0).
(b) The normal distribution with known mean 0 and unknown variance θ
(c) (Extra credit) The uniform distribution on [0, θ], where θ > 0 is unknown.
In each case, suppose it is desired to test the following hypotheses:
H0 : θ ≤ 2,
H1 : θ > 2.
Find expressions for a statistic T (= r(X) = r(X1 , . . . , Xn )) and a constant c satisfying that i) the joint
distribution of X has a monotone likelihood ratio in T , and ii) the uniformly most-powerful level α0 test
is of the form “reject H0 if T ≥ c.”
7.4. Textbook section 9.3, Exercise 18. (on MLR to UMP and sample size calculation)
7.5. (Do not submit) Textbook section 9.5, Exercises 12, 14. (on t-test)
7.6. (Do not submit) Textbook section 9.6, Exercise 6. (on two-sample t-test)
7.7. (Do not submit) Textbook section 9.7, Exercises 8, 10. (on F-test)
7.8. (Do not submit; for mini-lectures) Read textbook section 10.8, pp. 678–680. The test discussed in p.
679, Section 10.8, is called a “sign test”. Suppose X1 , . . . , Xn form a random sample from an unknown
distribution, and consider testing that the null hypothesis “median = 0,” against the alternative “median
is greater than 0”.
(a) Understand why the sign test works.
(b) We will “empirically” compare the powers of the one-sample t-test and the sign test. Do the
following for various values of the sample size n = 10 to 50. In each situation, is there a uniformly
most-powerful test?
i. With
tests,
ii. With
tests,
the assumption that the underlying distribution is N (µ, σ 2 ), compare the powers of two
for various values of µ.
the assumption that the underlying distribution is Exp(µ), compare the powers of two
for various values of µ. (Is µ the median?)
7.9. (Do not submit; for mini-lectures) Suppose X1 , . . . , Xn form a random sample from the Cauchy distribution with location parameter θ, and scale parameter 1. To test H0 : θ = 0 vs. H1 : θ > 0, one may
wish to use any of the two statistics: the sample mean X̄n and the sample median X̃n . Keep
(a) For each statistic, T = X̄n or X̃n , and a fixed constant c, write a computer program to approximate
Pθ (T > c), and empirically check that Pθ (T > c) is an increasing function of θ.
(b) Argue that the two test statistics are acceptable choices.
(c) Suggest an approximate value of c1 such that the test that rejects H0 when X̄n > c1 has size
α = 0.05.
(d) Suggest an approximate value of c2 such that the test that rejects H0 when X̃n > c2 has size
α = 0.05.
(e) Compare the powers of the two tests, for various values of θ.
[Note that for the Cauchy sample, all of the “standard tasks” such as computing the m.l.e., deriving
the LRT, and applying Neyman-Pearson lemma are analytically prohibitive and require numerical
computation. We do not pursue these here. For a special case of these tasks, see Exercise 17,
Textbook section 9.3.]
Assignment 8 | Bayesian analysis II
Textbook sections: 8.6, 9.8
Note: We may entirely skip these exercises, if the class progress is not fast enough.
8.1. Textbook section 8.6, Exercise 5.
8.2. Textbook section 8.6, Exercise 6.
8.3. Textbook section 8.6, Exercise 8.
8.4. Textbook section 8.6, Exercise 12.
8.5. Textbook section 8.6, Exercise 16. [Hint: The average of all 30 observations is x̄30 = 1.442 and s230 =
2.671.]
8.6. Textbook section 9.8, Exercise 2.
8.7. Textbook section 9.8, Exercise 4.