Monte Carlo Simulation and Derivation of Chi-Square Statistics

Abstract

Computer simulation has become an important tool in teaching statistics. Teaching using computer simulation would enhance the understanding of the concept using visual illustrations. This paper describes how to use simulation in R- programming language to perform a chi-square test. We try to show the distribution of most commonly used chi-square statistics we often found in statistical methods in both derivation and simulation. In statistical methods in such cases as test of independency, test of goodness of fit, test of significance, log likelihood ratio test, significance test and model selection we use chi-square statistic. The approach of the paper will enhance the students’ and researchers’ ability to understand simulation and sampling distribution. The paper contains an expository discussion of chi-square statistic, its derivation and distribution and its derivatives such as t-distribution and F-distribution. We consider two chi-squares, the empirical chi-square statistic and the theoretical chi-square distribution. The empirical distribution of chi-square statistic agrees closely with the theoretical chi- square distribution for large simulations, only the empirical distribution near to zero has lower density compared to the theoretical one for one degree of freedom. This is because the theoretical chi-square distribution at 1 degree of freedom has infinite density near to zero, but for any number of simulation the empirical distribution has finite density near to zero. Chi- square itself turns to normal distribution as the degree of freedom is large.