Math 181A
Spring 2017

                               ********** Announcements ************

 

·     It has been in the UCSD catalog for a couple of years at least, that the pre-requisite for this course is MATH 180A, and not any other statistics or probability oriented courses like ECON 120A; however, our WebReg system remains outdated (at the writing of this note) to imply that this might be a very recent change.  Students with ECON 120A but not MATH 180A generally will NOT obtain this instructor’s consent to enroll in the course.

[A related message: you are strongly encouraged to take MATH 180A even after ECON 120A.  We have recently given umbrella approval to petitions for graduating with 2 fewer units due to this partial duplicated credits for prob&stats majors.]

 

·     This course is intended to be taken as a sequence with MATH 181B for any student.  Applied math majors will be required to either take both or take another stand-alone course. Students of any major are discouraged from taking only 181A.

 

·     In place of syllabus: the core materials of this course are chapters 5, 6, 7 in the textbook (see below), and we may or may not get to the start of Chapter 9. Materials in Chapter 3 that was not covered in MATH 180A, will be covered as needed. Chapters 4 and 8 will be skipped.  The textbook will be followed pretty closely, with the addition of asymptotic theory of the maximum likelihood estimator (mostly from the Rice book below) as well as a bit more in depth on likelihood ratio test (uniformly most powerful).

·     Statistical programming language and environment R will be taught during TA sessions, and will be used in homework assignments.

 

·     I will be collecting feedback slips every 2 weeks on Fridays for students to tell me: 1) what you have learned in the past 2 weeks; 2) what you don’t understand about what we have discussed so far; 3) any other feedbacks in general.  Students who turn in their slips at the scheduled times will receive 2 bonus points towards their total scores for the quarter. Schedule: section A01 – week 2, section A02 – week 4, section A03 – week 6, section A04 – week 8.

 

·     Midterm will be Monday, May 8, in class. A cheat sheet is allowed (this is true for all exams). Bring sheets of paper (or a blue book) to write the exam on.

 

·     Wednesday 4/26 in place of lecture, there will be an extra TA session for all in the lecture room (CTR 105) from 1-1:50pm.

 

·     For practice (not required, instead the emphasis is on understanding the concepts and statistical thinking, and not ‘do as many problems as possible’) please use the textbook problems of the materials that we have covered in class.

 

·     See bottom of page for updated grading scheme.

 

·     Final week my office hour: Wed 6/14 1-3pm, APM 5856. (I do not have office hours on M Tu, but the TA’s do)

 

·     Be sure to bring your ID to the final exam.

 

 

 

Lecture:  MWF 1:00 - 1:50pm, CENTER 105 

 

Instructor: Ronghui (Lily) Xu

Office:  APM 5856

Phone:  534-6380
Email: rxu@ucsd.edu
 

Office Hours:

M:  4-5pm
F:  3-4pm

Or, by appointment

 

Teaching Assistants:  Jue (Marquis) Hou, Andrew Ying

Email:  j7hou@ucsd.edu, anying@ucsd.edu
Office Hours:  see announcements in TritonEd (TED)

 

 

Textbook:

Larsen and Marx, "An Introduction to Mathematical Statistics and Its Applications" (6^th edition)

Reference books:

1.    Rice, "Mathematical Statistics and Data Analysis";

2.    Wackerly et al., "Mathematical Statistics with Applications"

Additional reading (not required):

Efron, B. and Hinkley, D.V. (1978) Assessing the accuracy of the maximum likelihood estimator: observed versus expected Fisher information. Biometrika, 65, 457-487. (Introduction section only)

 

Lecture 1 slides (motivation and background mainly, not “required” material)

 

Weekly topics:

Week 1: estimation; method of moments estimator (MME), case study 5.2.2 (4th ed);

Week 2: maximum likelihood estimator (MLE), interval estimation;

Week 3: unbiasedness, efficiency, mean squared error (MSE); Cramer-Rao lower bound, Fisher information;

Week 4: convergence in probability, consistency and asymptotic normality of the MLE;

Week 5: confidence intervals based on the MLE, asbestos data;

Week 6: hypothesis testing paradigm, type I error, rejection region; one-sample normal with known variance; p-value; tests for Binomial including exact;

Week 7: type II error, sample size; tests for non-normal (and non-Binomial) data; duality of CI and hypothesis testing, Wald test;

Week 8: likelihood ratio test, Neyman-Pearson lemma, uniformly most powerful test (see Rice and Wackerly books for additional materials);

Week 9: distributions related to normal; one-sample t-test;

Week 10: inference about variance of one-sample normal (adjustable mortgage data); Bayesian inference (notes).

 

 

Homework: due each (following) week at TA sessions or in TA dropbox by end of that day

– be sure to append your R program codes at the back of your assignments, but you need to summarize the relevant results in the ‘main’ part, as opposed to have the grader look for them among the codes.  Good presentation is important for any work.

 

Week1 (due 4/13): 5.2.18, 23, 26 (only do the estimation part),

R simulation: for n=10, simulate a random sample of size n from N(μ, σ²), where μ = 1 and σ² = 2;  plot the histogram and superimpose by the N(1, 2) density function, and compute your estimate of μ and σ². Repeat for n=100 and 1000. Describe what you observe as well as what you expect.

Week 2 (due 4/20): 5.2.10, 14; 5.3.2, 8, 12, 17, 27;

R simulation: for n=10, simulate a random sample of size n from N(μ, 2), where μ = 1. Plot in different figures: 1) the likelihood function of μ, 2) the log-likelihood function; mark the maximum likelihood estimate in both plots.

Week 3 (due 4/27): 5.4.18 (add: show that the two estimators in the problem are unbiased), 19; 5.5.7, 2, 3

Week 4 (due 5/4): 5.7.1, 3(a), 4;

Construct (i.e. give an example of) a sequence of real functions g_n(x) converging to a function g(x), such that the corresponding sequence of maximizers of g_n(x) converges to that of g(x).

R simulation: for n=10, simulate a random sample of size n from N(μ, σ²), where μ = 1 and σ² = 2;  compute the same mean. Repeat the above simulation 500 times, plot the histogram of the 500 sample means. Now repeat the 500 simulations for n=1,000. Compare these two sets of results for different sample sizes, and discuss it in the context of consistency.

Week 5 (due 5/11): For X ~ B(n, p) [think of it as equivalent to a random sample of size n

from Bernoulli distribution with probability p], derive the 95% CI for p based on the MLE, and compare it to the one obtained in Section 5.3 of Larsen&Marx book. Derive also the observed Fisher information.

R simulation: in class we derived confidence intervals (CI) for the parameter λ of Poisson(λ). Now take sample size n=100, λ=1, and carry out 500 simulation runs. For each simulation, compute the MLE and the 95% CI based on the MLE, and see if the 95% CI contains the true λ=1. Report out of the 500 runs, how many times the 95% CI contains the true λ=1; explain whether and why your simulation result is as desired. (Hint, this is similar to the Larsen and Marx example 5.3.1 with Table 5.3.1 on page 296.)

Week 6 (due 5/18): 6.2.7, 1, 8; 6.3.3, 7, 9

Week 7 (due 5/25): 6.4.4, 7, 10, 16 (I think it’s “X>=5” instead of “k>=5” at the end of the problem), 21;

Discuss: based on the duality of CI and hypothesis testing,

1)        how testing against a one-sided alternative hypothesis corresponds to a one-sided CI (i.e. with one end of the CI at + or – infinity);

2)        derive the Wald test based on the MLE for the Poisson problem used in R simulation of Week 5 assignment.

Week 8 (due 6/1): 6.5.1, 2, 5;  do also:

1)        Show that for testing simple versus simple hypothesis H₀: μ = μ₀ against H₁: μ = μ₁ where μ₁ > μ₀, based on a random sample Y₁, …, Y_n from N(μ, σ²) where σ² is known, the likelihood ratio test rejects for large values of the sample mean Y_bar.

2)        For a random sample of size n=25 from Normal (μ, 1), under the null hypothesis: μ=0. Plot the power curves in the same figure of the three Z-tests (given in Theorem 6.2.1 of Larsen&Marx) for the following three alternative hypotheses: 1) μ>0; 2) μ<0; 3) μ≠0.  Assume a 0.05 significance level.  Comment on the relation of your plots with the UMP test.

Week 9 (due 6/8): 7.3.4, 5, 13; 7.4.7, 19; also the following:

1)        For a random sample of size n from Exponential(λ), and the hypotheses H₀: λ = λ ₀ versus H₁: λ ≠ λ ₀ (similar to problem 6.5.2 from week 8), derive a) the Wald test, b) the likelihood ratio test using its asymptotic distribution, both assuming a significance level α.

2)        Show that the one-sample t-test of Section 7.4 in Larsen&Marx is equivalent to the likelihood ratio test for a random sample from N(μ, σ²) and H₀: μ = μ₀ versus H₀: μ ≠ μ₀ with unknown σ².

Week 10 (not due): 7.5.9, 16;

 

 

For additional practice (not required, see top of the page) please use the textbook problems of the materials that we have covered in class.

 

Grading (updated): max(30% Homework + 30% Midterm, 35% Homework + 25% Midterm) + 40% Final

[1 lowest homework score will be dropped]