Variance Reduction In Random Sampling And Convergence Diagnosis Of MCMC

Monte Carlo method is a widely used statistical simulation method,which is gradually applied by people along with the rapid development of the computer.Through scientific and reasonable statistical modeling,the problems such as integral or other complex research objects which were difficult to be calculated manually by traditional methods are converted into the process of simulation calculation generating random numbers or data with certain characteristics,so as to solve mathematical problems we encounter.This method essentially simplifies the problem and eventually can obtain an approximate solution within the error range.In the first part of this paper,we introduce the basic idea of Monte Carlo method and the variance reduction of random sampling,which includes antithetic variable method,control variable method,important sampling method and stratified sampling method.Furthermore,we put forward uneven stratified sampling method and important uneven stratified sampling method.After introducing the specific implementation steps of the new methods,the excellent performances of the new methods in variance reduction are verified by some numerical simulation results.In the second part,we focus on the convergence monitoring method of Markov chain Monte Carlo(MCMC),including Yu-Myklan method,Gelman-Rubin method and Geweke method.We mainly improve the Yu-Myklan method and the Geweke method for faster monitoring the convergence of Markov chain.In the numerical simulation,through the convergence diagnostic of Gibbs samples from a bivariate normal distribution and hybrid Gibbs samples from the posterior distribution of parameters of the competing risk model,we compare and analyze the performances of the original methods and the improved methods in convergence monitoring.The results show that the improved two methods are easier to detect convergence,and the more accurate Bayesian estimation of the parameters can be obtained by the improved Geweke method.