The Generalized Splitting Method: Applications to Combinatorial Counting and Static Rare-event Probability Estimation

We describe a new Monte Carlo algorithm for the consistent and unbiased estimation of multidimensional integrals and the efficient sampling from multidimensional densities. The algorithm is inspired by the classical splitting method and can be applied to general static simulation models. We provide examples from rare-event probability estimation, counting, and sampling, demonstrating that the proposed method can outperform existing Markov chain sampling methods in terms of convergence speed and accuracy. The second part of the thesis presents a new adaptive kernel density estimator based on linear diffusion processes. The proposed estimator builds on existing ideas for adaptive smoothing by incorporating information from a pilot density estimate. In addition, we propose a new plug- in bandwidth selection method that is free from the arbitrary normal reference rules used by existing methods. We present simulation examples in which the proposed approach outperforms existing methods in terms of accuracy and reliability.