Mathematical Statistics

URL study guide

https://tue.osiris-student.nl/onderwijscatalogus/extern/cursus?cursuscode=2WS30&collegejaar=2025&taal=en

Description

In this course you will learn about the foundations of Mathematical Statistics. Statistics concerns what can be learned from data. Applied statistics comprises a body of methods for data collection and analysis across the whole range of science, and in areas such as engineering, medicine, business, and law - wherever variable data must be summarized, or used to test or confirm theories, or to inform decisions. Theoretical statistics underpins this by providing a framework for understanding the properties and scope of methods used in applications. In this course the main emphasis in given to statistical inference, and in particular to the theoretical underpinnings of statistical inference. This does not mean we will not be concerned with applied matters, and throughout the course the student will be exposed to several applied situations, where inferences about real data must be done.


Topics:
 

• Statistics and estimators: point estimators, maximum likelihood, the method of moments, Bayesian methods
• Evaluation of the performance of estimators: mean squared error, bias-variance decomposition, uniformly minimum variance unbiased estimation, the Cramer-Rao lower bound, Rao-Blackwell’s theorem
• Sufficiency and Completeness, the Lehmann-Scheffé’s theorem
• Asymptotic notions: consistency, MSE consistency, asymptotic unbiasedness
• Maximum likelihood estimation: Invariance property, consistency of the MLE for certain classes of models (finite classes, and Wald consistency conditions), asymptotic normality
• Confidence regions and intervals: interpretation, pivotal quantity method, approximate pivots and asymptotic normalityHypothesis testing: simple vs. composite hypothesis, type I and II errors, power function and p-value, relation between hyp. Testing and confidence intervals. Tests for the normal and binomial distribution
• Optimality of hypothesis testing: most powerful tests and the Neyman-Pearson lemma, uniformly most powerful tests, unbiased tests, generalized likelihood ratio tests and Wilk’s theorem
• Non-parametric methods: the empirical distribution function and the Glivenko-Cantelli lemma, GoF testing based on the ECDF, density estimation by histograms

Objectives

After successfully completing this course you should be able to carry out standard statistic analyses (both on paper and by means of statistic software) and be able to derive simple alternatives to these analyses. You should be able to adequately document calculations that form the bases of such analysis. Although a significant emphasis will be given to the development of statistical inference tools, by the end of the course the students should be able to apply this knowledge in practical settings.

Method of Assessment

Written examination