With the advent of powerful computing and the availability of massive sets of data, statistics has become a valuable tool in numerous applications. This course covers the basics of advanced mathematical statistics, both classical and modern.

Review of basic probability and statistics
Parametric models
Estimation: criteria and construction of estimators, maximum likelihood, asymptotics
Hypothesis testing, multiple hypotheses testing
Non parametric models, empirical distribution function, jackknife and bootstrap, non-parametric testing, density and regression estimation
Classification: discriminant analysis, support vector machines, multivariate statistics, EM algorithm and simulation including Monte Carlo Markov Chain
The level and detailed content of the course will be determined along with the students according to their background and interests.

Prerequisite: At least one good course in probability, and some basic statistics.
Instructors: Larry Goldstein and Yosi Rinott

Office Hours: Larry Goldstein, Tuesdays 14:00 – 15:00 and Fridays 15:45-16:45 (before the exercise sessions), room 610.

### Annoucements

During the first week of the course we will review probability theory, as covered in the first 5 chapters of our text.

An additional question, problem and lecture session will be available on Wednesday July 29th from 15:00-16:30 in Aula B3 for students who desire additional exposure to the material that is to be reviewed in week 1.

An additional class will be held in Aula B3 on Wednesday August 19th at 3:00.

### Problem Sessions and Exercises

Problem session coverage:

Tuesday, July 28th: Chapter 2
Friday, July 31st: Chapter 2:11,14,15,16,20, problems from Chapters 3 and 4 as time permits.
Tuesday, August 4th: Chapters 3 and 4.

### Exercises:

Chapter 2: 4,7,8,11,13a,14,15,16,20,21
Prove that αΓ(α)=Γ(α+1) (three words).
Prove that the standard normal density integrates to 1 (two words).
Prove or disprove: If N is an non-negative integer valued random variable, p is any number in [0,1], and X and Y are the number of heads and tails in
N independent tosses of a p-coin respectively, then X and Y are independent if and only if N is Poisson.
Find the variance of the Poisson and Gamma distributions.
Show that the N(μ,σ^2) distribution has mean μ and variance σ^2

Chapter 3: 1,3,5,7 (ignore hint, use Fubini’s theorem instead), 13 (you may use 17),14,16,17,20,21,23,24
Chapter 4: 1,2,3,4a,5,6. Also, prove the arithmetic/geometric mean inequality using Jensen’s inequality.
Chapter 5: 1,2,4,6,11,12,13,15,16
Day1Quiz: problems 1-6

Exercise Set 1
Midterm scores out of 90 possible points: 7,30,37,43,43,44,45,45,65,77,87,90,90,90,90 Median = 45, Average = 59
Exercise Set 2
Exercise Set 3
Exercise Set 4
Exercise Set 5
Exercise Set 6

### Structure and Evaluation

Lectures, Monday through Friday mornings, 8:30-9:30, Problem Sessions Tuesday 15:00-16:30 and Friday 16:45-18:15
Midterm, 30% Tuesday August 11th, 14:30-16:30, closed book.
Final Exam, 45% Wednesday August 26th, 15:00-17:00, closed book.
Course participation, 25%
Course Text: All of Statistics: A concise course in Statistical Inference, by Larry Wasserman.