A Comparison of Confidence Intervals of Negative Binomial Parameter p by Maximum Likelihood, Bayesian and Markov Chain Monte Carlo Methods

Abstract

The objective of this research is to estimate the confidence interval of population parameter (p) or probability of success in each experiment based on negative binominal distribution. The interval estimation is evaluated by Maximum Likelihood (ML), Beyesian, and Markov Chain Monte Carlo (MCMC) methods. The performance of these methods is considered by Confidence Coefficients (CC) and Average Width (AW), When the CC Value are greater than the fixed confidence interval, the AW of the confidence interval will focus the performance of these methods. The data is generated from negative binomial distribution as follows : the population parameter as small (0.2), medium (0.5), and large (0.8), the sample sizes and parameter r or called number of success as small sample sizes (n=30) r=10, 15, 20, medium sample sizes (n=50) r=10, 20, 30 and large sample size (n=70) r=15, 30, 45 and the 90%, 95% and 99% confidence interval. The results show that the ML method exhibits poor interval estimation in most all cases, but the Bayesian method performs very satisfactorily in most all cases especially when true parameters are midium or largevalues for all sample sizes and parameter r. For the small true parameters, the MCMC method is a good performance in most cases. However, the confidence coefficient and average width of MCMC and Baysian are equal in some case, so MCMC are reasonable working as good as Bayesian.