Estimating the relative frequency of a binomial distribution using classical and Bayesian approaches in R

The aim of the bachelor thesis is to investigate different methods of estimating the relative frequency of binomial distribution in Bayesian and Frequentist statistics. In the introduction, emphasis is placed on the differences between the approaches to statistics that distinguish methods used. Frequentist statistics is represented by the maximum likelihood method, which derives the maximum likelihood estimate and its asymptotic convergence to a normal distribution. Bayesian methods are represented by the Beta-Binomial model, whose posterior distribution is analytically derived. The Bayesian part is further extended by the introduction of three simulation programs that can be used to simulate the posterior distributions without knowledge of the analytical solution. The practical part compares point and interval estimates for different sample sizes and population relative frequencies. The results show that for point estimates, the Beta-Binomial is the most appropriate model with informed apriori distribution. In the case of borderline relative frequencies or small sample sizes, the maximum likelihood method is unsuitable primarily because of the asymptotic assumption. In the case of interval estimation, it has been shown to be the same model with a non-informative apriori distribution to be the best. Again, it can be seen that the intervals produced by the frequentist method are more likely to be incorrect in small-samples and borderline relative frequencies. The main contribution of this bachelor thesis is highlighting the practical advantages of Bayesian methods in estimation with limited data, where apriori parameter information can partially compensate for the lack of data.