Comparison of two Markov chain Monte Carlo (MCMC) methods

As the world advances, statisticians/mathematicians are being involved into more and more complex surveys for the development of society and human beings. Consequently, these complex survey data requires complicated and high-dimensional models for final analysis. We need sophisticated and efficient statistical/mathematical tools for estimation and prediction of these models. Frequently, we simulate samples from these complicated models to estimate parameters because direct estimation is sometimes not efficient. Markov chain Monte Carlo (MCMC) methods have been developed for simulation and efficient estimation. In these times of availability of highspeed computing facilities, MCMC methods are popular tools to generate samples from these complex and high-dimensional distributions. There are some MCMC methods available for application, for instance, Gibbs sampler, Metropolis-Hastings algorithm, etc. In this paper we compare the performance of two MCMC methods, namely the Hybrid (HY) algorithm and the Random Walk Metropolis-Hastings (M-H) algorithm, by employing two logistic regression models and a multivariate normal distribution. We get data used in one of the examples from a Prostate Cancer Study presented in a book by D.W. Hosmer and S. Lemeshow. The Bayesian approach is followed to fit the two logistic regression models. Evaluation of performance is based on convergence of Markov chains, efficiency of estimation and Monte Carlo error variances of the two methods.