MSc: Advanced Statistics

Advanced Statistics

• Course name: Advanced Statistics
• Code discipline: DS-03
• Subject area:

Short Description

This course in advanced statistics with a view toward applications in data sciences. It is intended for masters students who are looking to expand their knowledge of theoretical methods used in modern research in data sciences. The course presents some of the key probabilistic methods and results that may form an essential mathematical toolbox for a data scientist. This course places particular emphasis on random vectors, random matrices, and random projections. It teaches basic theoretical skills for the analysis of these objects, which include concentration inequalities, covering and packing arguments, decoupling and symmetrization tricks, chaining and comparison techniques for stochastic processes, combinatorial reasoning based on the VC dimension, and a lot more.

Prerequisites

Prerequisite subjects

• CSE329 - Empirical Methods

Course Topics

Course Sections and Topics

Section	Topics within the section
Sampling Distributions Associated with the Normal Population	Kolmogorov-Smirnov test Size of samples, Kolmogorov-Smirnov, Fisher exact Logistic regression

Intended Learning Outcomes (ILOs)

What is the main purpose of this course?

The main purpose of this course is to present the fundamentals of inferential statistics to the future software engineers and data scientists, on one side providing the scientific fundamentals of the disciplines, and on the other anchoring the theoretical concepts on practices coming from the world of software development and engineering. The course covers the statistical analysis of data with limited assumptions on the distribution, with reference to testing hypotheses, measuring correlations, building samples, and performing regressions.

ILOs defined at three levels

Level 1: What concepts should a student know/remember/explain?

By the end of the course, the students should be able to ...

• Remember the fundamentals of inferential statistics
• Remember the specifics and purpose of different hypothesis tests
• Distinguish between parametric and non parametric tests

Level 2: What basic practical skills should a student be able to perform?

By the end of the course, the students should be able to ...

• the basic concepts of inferential statistics
• the fundamental laws in statistics
• the concept of null and alternative hypotheses
• the hypotheses test procedure

Level 3: What complex comprehensive skills should a student be able to apply in real-life scenarios?

By the end of the course, the students should be able to ...

• To understand the problems related to analyse statistically data not distributed normally
• To know the more recent computationally-intensive techniques that can help to describe samples and to infer properties of populations in absence of normality
• To identify situations when the data is on nominal scales so alternative techniques should be use, and act accordingly.
• To be able to run experiment to evaluate hypotheses for situation of scarce data, distributed non normally, on different kinds of scales.

Grading

Course grading range

Grade	Range	Description of performance
A. Excellent	85-100	-
B. Good	65-84	-
C. Satisfactory	51-64	-
D. Poor	0-50	-

Course activities and grading breakdown

Activity Type	Percentage of the overall course grade
Quiz during each lecture (weekly evaluations)	15
Labs classes (weekly evaluations)	15
Midterm	20
Final exam	50

There are two constraints for passing the course:

1. You must attend all labs.
2. You must submit all lab reports.
3. You must have at least 50% on the Final Exam.

Recommendations for students on how to succeed in the course

• Watch the video lecture and read the lecture notes before coming to the onsite lectures and to the labs.
• Attend the onsite lectures
• Ask questions and provide answers to the questions during the onsite lectures.
• Attend all of the labs and submit all of the lab reports.
• Prepare seriously for the midterm exam.
• Prepare seriously for the final exam.

Resources, literature and reference materials

Open access resources

• The lecture notes and the video lectures provided via Moodle are sufficient for passing this course with grade A.

Teaching Methodology: Methods, techniques, & activities

Activities and Teaching Methods

Activities within each section

Learning Activities	Section 1
Testing (written or computer based)	1
Discussions	1

Formative Assessment and Course Activities

Ongoing performance assessment

Section 1

Activity Type	Content	Is Graded?
Question	Let X1,X2, ...,X10 be a random sample from a distribution whose probability density function is ${\textstyle f(x)=(1\quad if\;0<x<1}$ , otherwise 0). Based on the observed values 0.62, 0.36, 0.23, 0.76, 0.65, 0.09, 0.55, 0.26, 0.38, 0.24, test the hypothesis H0 : X UNIF(0, 1) against H1 : X UNIF(0, 1) at a significance level = 0.1.	1
Question	If X1,X2, ...,Xn is a random sample from a distribution with density function ${\textstyle f(x)=((1-\theta )x^{\theta }\;if\;0<x<1}$ , otherwise 0), what is the maximum likelihood estimator of ${\textstyle \theta }$ ?	1
Question	Let X1,X2, ...,Xn be a random sample of size n from a distribution with a probability density function ${\textstyle f(x)=((1-\theta )x^{\theta }\;if\;0<x<1,}$ otherwise 0), where ${\textstyle 0<\theta }$ is a parameter. Using the maximum likelihood method find an estimator for the parameter ${\textstyle \theta }$ .	1
Question	Suppose you are told that the likelihood of ${\textstyle \theta }$ at ${\textstyle \theta =2}$ is given by 1/4. Is this the probability that ${\textstyle \theta =2}$ ? Explain why or why not.	1
Question	If X1,X2, ...,Xn is a random sample from a distribution with density function${\textstyle f(x)=({\frac {1}{\theta }}\;if\;0<x<1,}$ otherwise 0), then what is the maximum likelihood estimator of ${\textstyle \theta }$ ?	0
Question	Let X1,X2, ...,Xn be a random sample from a normal population with mean ${\textstyle \mu }$ and variance ${\textstyle \sigma ^{2}}$ . What are the maximum likelihood estimators of ${\textstyle \mu }$ and ${\textstyle \sigma ^{2}}$ ?	0
Question	Suppose that you have the following data points: 0.36, 0.32, 0.10, 0.13, 0.45, 0.11, 0.12, 0.09; compute Dn to determine if they come from the uniform distribution [0,0.5].	0
Question	The data on the heights of 12 infants are given below: 18.2, 21.4, 22.6, 17.4, 17.6, 16.7, 17.1, 21.4, 20.1, 17.9, 16.8, 23.1. Test the hypothesis that the data came from some normal population at a significance level = 0.1.	0

Final assessment

Section 1

1. Providing full example of two sequences (in case of computational overhead, you can approximate at the first decimal digit). Compute their:

Covariance. Pearson’s correlation coefficient. Spearman’s Rank Correlation Coefficient. Kendall’s tau Correlation coefficient.

1. What is an empirical distribution?
2. Present, prove, and discuss the evaluation of the asymptotic confidence interval for the empirical distribution, detailing the role of the binomial.
3. Prove, under the simplified hypotheses, the distribution free property of Dn.
4. Write the Shannon Theorem and discuss its implications.
5. Discuss how we could proceed to compute the confidence interval of the Kendall Tau correlation coefficient of the population.
6. Suppose that you have the following datapoints: 0.4, 2, 0.6, 2.4, 2.2, 3.6, 3.8, 4; compute Dn to determine if they come from the uniform distribution [0,4].
7. Prove that ${\textstyle {\tilde {F}}_{n}}$ is a consistent and unbiased estimator of F.

The retake exam

Section 1