Construction of the Chi-square Goodness-of-Fit Test

Abstract

The goal of statistical tests is to use data to learn about the nature of the system from which it is collected. It is an important bridge between raw data and interpreting its meaning/drawing conclusions from it. One recent use of statistical tests is in COVID-19 tests. The results were analysed to determine whether the patient is tested positive or negative, and statistics is used to determine the false positive rate. In this paper we will focus on one of the most commonly used statistical tests, the Chi-square test, and why it works. We will first discuss the general procedure of hypothesis testing, followed by the construction of an appropriate estimator for statistical models before finally, finding the limiting distribution of the estimator. Using these properties, we will construct the Chi-square distribution and discuss its relevance to hypothesis testing. Python simulations using the Chi-square test will be generated to investigate the effect of multiple variables on its statistical power.