Importance sampling methods with multiple sampling distributions

The complexity of integrands in modern scientific, industrial and financial problems increases rapidly with the development of data collection technologies. Monte Carlo method is widely used for complicated integration. In Monte Carlo integration, it is a natural and flexible method to consider multiple simulation mechanisms instead of one to address different aspects of the integrand. New methods are needed to combine the multiple mechanisms efficiently. Monte Carlo integration methods are reviewed, with focus on importance sampling methods (IS) and sequential Monte Carlo methods (SMC). The former is commonly used for low-dimension problems. The latter is a variation of IS, which has been developed to be a new branch itself in the recent two decades, and promising for high- dimension problems with sequential nature. For IS, techniques for combining multiple proposal distributions have been well developed, including Owen and Zhou (2000) and Tan (2004). Important implementation issues are needed to be resolved, including the allocation of sample budgets and the selection of proposals. A two-stage procedure is proposed to optimize the sample allocation, and although little theoretical investigation has been done for such a two-stage procedure in literatures, its optimality among current approaches is theoretically justified. The choice of the first stage sample size is also discussed through investigating the high order performance of estimators. About the construction of proposals, suggestions are given to approximate the perfect case. For SMC, only the plain vanilla combination of multiple proposals has been used in literatures. A novel SMC filtering scheme is proposed to combine the multiple proposals through the control variates approach in Tan (2004). Control variates are used in both resampling and estimation. The new algorithm is shown to be asymptotically more efficient than the direct use of multiple proposals and control variates. The guidance for selecting multiple proposals and control variates is also given. Numerical studies of the AR(1) model observed with noise and the stochastic volatility model with AR(1) dynamics show that the new algorithm can significantly improve over the bootstrap filter and auxiliary particle filter.Ph. D.Includes bibliographical referencesby Wentao L