An Estimating Parameter of the Mean in Normal Distribution by Maximum Likelihood, Bayes, and Markov Chain Monte Carlo Methods
Authors: Autcha Araveeporn
This paper is to compare the parameter estimation of the mean in normal distribution by Maximum Likelihood (ML), Bayes, and Markov Chain Monte Carlo (MCMC) methods. The ML estimator is estimated by the average of data, the Bayes method is considered from the prior distribution to estimate Bayes estimator, and MCMC estimator is approximated by Gibbs sampling from posterior distribution. These methods are also to estimate a parameter then the hypothesis testing is used to check a robustness of the estimators. Data are simulated from normal distribution with the true parameter of mean 2, and variance 4, 9, and 16 when the sample sizes is set as 10, 20, 30, and 50. From the results, it can be seen that the estimation of MLE, and MCMC are perceivably different from the true parameter when the sample size is 10 and 20 with variance 16. Furthermore, the Bayes estimator is estimated from the prior distribution when mean is 1, and variance is 12 which showed the significant difference in mean with variance 9 at the sample size 10 and 20.
Digital Object Identifier (DOI): doi.org/10.5281/zenodo.1126349Procedia APA BibTeX Chicago EndNote Harvard JSON MLA RIS XML ISO 690 PDF Downloads 933
 Rohatgi, V. K. and Saleh, E. (2001) An Introduction to Probability and Statistics. John Wiely & Sons, New York.
 Gilk, W., Richardson, S. and Spiegelhalter, D. (1996) Markov Chain Monte Carlo in Practice. Chapman & Hall, London.
 Carlin, B. P. and Louis, T. A. (2009) Bayesian Methods for Data analysis. Florida: CRC Press Taylor & Francis Group
 Metropolis, N., Rosenbluth, A., Rosenbluth, M., Teller, A. and Teller, E. (1953) Equations of State Calculations by Fast Computing Machine. Journal of Chemical Physics, 21, 1087-1092.
 Geman, S. and Geman, D. (1984) Stochastic Relaxation, Gibbs Distribution and the Bayesian Restoration of Images. IEEE Transactions on Pattern Analysis and Machine Intelligence, 6 721-741.
 Gelfand, A., Hills, S., Racine-Poon, A. and Smith, A. (1990) Illustration of Bayesian inference in normal data models using Gibbs sampling. Journal of the American Statistical Association, 85, 972-985.
 Ntzoufran, I. (2009) Bayesian Modeling Using WinBUGS. John Wiely & Sons, New Jersey.
 R Development Core Team. (2004) R: A Language and Environment for Statistical Computing. R Foundation for Statistical Computing, Vienna, Austria.