Course 3 Mathematical Statistics

Mathematical Statistics is a branch of statistics that emphasizes the theoretical aspects of statistical methods. Unlike applied statistics, which focuses on using statistical tools in real-world problems, mathematical statistics seeks to understand the properties of these methods, their underlying assumptions, and their limitations. This section provides a solid foundation in the mathematical framework that supports statistical inference, estimation, and hypothesis testing.

We will begin by exploring the concepts of estimation theory, focusing on the properties of estimators, such as unbiasedness, consistency, and efficiency. These properties are crucial for understanding how well an estimator performs in different contexts and how we can evaluate and improve the precision of our statistical models. Building on this, we will delve into hypothesis testing and the formulation of tests based on statistical decision rules, including the concepts of Type I and Type II errors, power, and significance levels.

In addition to traditional estimation and hypothesis testing, this section will introduce decision theory, which provides a systematic approach to making optimal choices in uncertain environments. Decision theory helps to formalize the selection process between competing hypotheses or models by considering the costs and benefits associated with each decision and is particularly important in situations where uncertainty or risk is present. We will also cover advanced topics such as likelihood theory, the Cramer-Rao lower bound, and the asymptotic properties of estimators.

Through this section, you will develop a deeper understanding of the mathematical foundations of statistical methods, allowing you to apply these techniques and critically assess their assumptions and limitations. This knowledge will be essential as you progress to more advanced areas of statistical theory and its application to real-world problems.

Prerequisite: Course 1 Probability Concept and Course 2 Elementary Statistics (familiarity with Calculus and Linear Algebra would be beneficial).