| The main motivation of this work is given by the implementation of the antithetic principle in Markov chain Monte Carlo algorithms. In the past, researchers typically have used antithetic variates in pairs. In the present paper it is argued that the negative coupling of more than two antithetic variates can significantly increase the variance reductions compared to the paired case. Central to the generating methods we investigate in Chapter 2, are the qualitative and quantitative characterizations of negative dependence, negative association, negative dependence through stochastic ordering, respectively, extreme antithesis.; The emerging area of perfect simulation provides a perfect setting for implementing the K-process parallel antithetic coupling because this class of methods is designed to deliver genuine independent draws from the target density. The theoretical results we obtain to justify antithetic backward coupling simultaneously answer questions related to the joint stationarity of a coupled chain. We show that for an entire class of forward MCMC chains having the so-called attractive transition kernels, the K-process coupling produces variance reductions. We compare three methods for constructing K-tuples of negative associated random variables.; In Chapter 4 we generalize Hall's (1989) antithetic resampling scheme to more than two parallel bootstrap samples. Similar to the Monte Carlo implementation we increase the efficiency when we draw more than two parallel antithetic bootstrap samples.; In Chapter 5 we introduce a different way to implement stratification which results in shorter running times for perfect sampling algorithms. We discuss the multistage coupling from the past and we use a random walk example to illustrate its efficiency.; For a fixed K and as the number of iterations tend to infinity, the distributional support of the K-tuple of antithetic variates generated with the iterative Latin hypercube method is an intriguing antihype fractal. In Chapter 6 we investigate its geometrical properties using the theory of self-similar sets. |
