連續型分布之有效率馬可夫鏈蒙地卡羅抽樣法

Abstract

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In statistics, Markov chain Monte Carlo (MCMC) is a classical sampling algorithm from a probability distribution, especially for estimating the expectation of real-valued function f. In this study, we focus on developing new algorithms to generates samples on a continuous state space rather than the independent and identically distributed (i.i.d.) sampling. Chen et al. (2012) derived the optimal transition matrices for finite discrete state space. It is shown that the MCMC based on their optimal transition is more efficient than the independent and identically distributed (i.i.d.) sampling in terms of the asymptotic variance. Motivated by the performance of the MCMC sampling for discrete state space, we propose two MCMC algorithms for continuous state space in chapter 2 with discussion and theoretic justification. There were many different comparison criteria to evaluate the performance between different MCMC based algorithms, we choose two different criteria to test for our proposed algorithm. These two criteria are the variance of some real function f and the maximum spacing defined below is a similar concept to the worst-case analysis: (max)┬(1<i≤N)⁡〖{x_((i))-x_((i-1)) 〗,x_((1) ),1-x_((N))}, where x_((i)) is the order statistics of x_i. Based on these two criteria, simulation comparisons of two proposed MCMC algorithms with the i.i.d. sampling are presented in chapter 3. In the end, we have our conclusion remarks in chapter 4.