Statistics

Many statistical models try to explain how one variable relates to others. In this course, how to analyze multiple variables simultaneously, multivariate analysis. Both how they depend upon other variables, but also how they depend upon each other. With the tremendous amount of data available nowadays, e.g. in genetics, it is often the case that the number of variables is far greater than the number of observations. This demands special techniques that are learned in this course. Course content includes matrices and multivariate normal distribution, singular value decomposition and its geometric interpretation, principal component analysis including its functional formulation, factor analysis, cluster analysis, prediction theory including prediction with high-dimensional predictors, penalized regression and prediction, sparse matrices, linear discriminant analysis, and large-scale inference.  

This course introduces the fundamentals of regression modeling, providing essential knowledge for students pursuing advanced study in statistics or careers as professional statisticians. Topics include parameter estimation in linear models, hypothesis testing for model comparison, model selection techniques for predictive purposes, detection of assumption violations, and identification of influential observations.

This course develops and interprets the mathematical foundations of statistics including theories of stochastic variables, variable transformations, sample distribution, estimation, and hypothesis testing. 

Students explore calculus and probability theory necessary to analyze data and draw statistical inferences about the population. Thus, there is a major focus on deriving statistical estimators which are functions of data and a focus on studying their statistical properties. The course covers point and interval estimation methods which are widely used in academia and industry. After establishing statistical procedures to obtain inference about the population, we apply them to real problems by using Excel. If time permits, we will talk about linear regression models and Bayesian methods, which are both essential in quantitative finance, actuarial science, medical science, etc. Please note that the lectures are in-depth regarding mathematical proofs of theorems in the textbook; students should expect the class to be theoretical and rigorous. 

Prerequisites: An undergraduate level understanding of calculus and probability. Students should have a solid understanding of integral with one variable (calculus with two variables will be very helpful for advanced topics) and integral by parts.  

This course covers the properties and applications of maximum likelihood estimation (MLE), including its consistency, asymptotic normality, and efficiency, and applies these concepts to real-world statistical problems. Students analyze hypothesis testing frameworks, covering the Neyman-Pearson lemma, likelihood ratio tests, and their implementation for single-parameter and multi-parameter models and study the principles of sufficiency and completeness in statistical inference and use the factorization theorem to identify sufficient statistics for various distributions. Students also explore confidence interval construction methods, focusing on pivotal quantities, and evaluate their properties such as coverage probability and efficiency and we apply advanced inferential techniques to solve problems involving exponential families, sequential analysis, and decision-theoretic approaches, linking theory to practice. 

Prerequisites: Mathematical Statistics I, Linear Algebra, Calculus 

This course forms part of the third-year major in Applied Statistics. It is an introduction to the study of Operational Research (OR) and explores fundamental quantitative techniques in the OR armamentarium with a strong focus on computer-based application. The course is intended for students in the applied statistics stream but may be taken as an elective by students in the mathematical statistics stream. Topics covered include linear and non-linear programming where students will learn to find optimal solutions by characterizing problems in terms of objectives, decision variables and constraints, decision making under uncertainty through decision trees, decision rules and scenario planning, Queueing Theory simulation through modelling the operation of real-world systems as they evolve over time. Course entry requirements: STA2030S or STA2005S; STA3030F is recommended.

This course introduces Bayesian data analysis using the WinBUGS software package and R. Topics include the Bayesian paradigm, hypothesis testing, point and interval estimates, graphical models, simulation and Bayesian inference, diagnosing MCMC, model checking and selection, ANOVA, regression, GLMs, hierarchical models and time series. Classical and Bayesian methods and interpretations are compared.

At the end of the course, the student can analyze structural and dynamic aspects of populations. Particularly, the student is able: to measure population size, variation and distribution in time and space; to measure and analyze population structural characteristics; and to measure and analyze demographic dynamics components.

Course contents include: Population studies; sources of demographic data; population composition and variation; relations between demographic dynamics and structures; elements of demographic analysis: crude and age-specific rates, standardization methods; age, period and cohort; Lexis diagram, managing data and processes on the diagram; period and cohort analysis; life tables and life tables conceived as a stationary population; mortality by cause; fertility and reproduction measures; the demographic transition and forecasts (elements); and migration in population analysis.

This course is part of the Laurea Magistrale degree program and is intended for advanced level students. Enrollment is by permission of the instructor. At the end of the module, students: know the most relevant univariate and multivariate analytical techniques and tools; can perform statistical analyses of cross-sector and longitudinal data and deliver suitable graphic representations, making results comprehensible to organizations’ decision makers.

The course content is divided as follows:

1. Introduction to data: Data basics; Sampling principles; Experiments and observational studies
2. Summarizing data: Examining numerical data; Considering categorical data
3. Probability: Defining probability; Conditional probability; Bayes theorem
4. Random variables: Discrete and continuous; Expectation
5. Distributions of random variables: Normal; Geometric; Bernoulli
6. Foundations for inference: Point estimates; Confidence intervals; Hypothesis testing

This course develops foundational skills in statistical reasoning and data analysis, with an emphasis on parameter estimation and model interpretation. Throughout the semester, students will learn how to estimate and interpret key statistical parameters that describe both individual variables and relationships between variables.