Minimax Framework and Hilbert Space (Under Construction)

Hypothesis Testing

Introduction

Hypothesis testing is an fundamental tool in statistics for building a complete and scientific procedure to identify our beliefs about the world and evaluate the errors in our conjectures. Typically, the core of a hypothesis testing procedure consists of two statements:

\[\begin{aligned} H_0 &: \text{A belief about something};\\ H_1 &: \text{Another belief}. \end{aligned}\]

$H _{0}$ is called the null hypothesis, while $H _{1}$ is called the alternative hypothesis. We require that $H _{0}$ and $H _{1}$ are at least mutually exclusive to precisely convey the testing analysis. For example, good $H _{0}$ and $H _{1}$ can be “There will be no rain tomorrow” and “There will be rain tomorrow”, or “The vaccine is not effective” and “The vaccine is effective”. In contrast, “Tomorrow will be sunny sometime” and “Tomorrow will be rainy sometime” are bad $H _{0}$ and $H _{1}$, since both statements can be true at the same time and therefore it is hard to distinguish them.