We present 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.Comment: Published in at http://dx.doi.org/10.1214/10-AOS799 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org

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Since the beginning of electronic computing, people have been interested in carrying out random experiments on a computer. Such Monte Carlo techniques are now an essential ingredient in many quantitative investigations. Why is the Monte Carlo method (MCM) so important today? This article explores the reasons why the MCM has evolved from a 'last resort' solution to a leading methodology that permeates much of contemporary science, finance, and engineering.

Simulation from the truncated multivariate normal distribution in high dimensions is a recurrent problem in statistical computing and is typically only feasible by using approximate Markov chain Monte Carlo sampling. We propose a minimax tilting method for exact independently and identically distributed data simulation from the truncated multivariate normal distribution. The new methodology provides both a method for simulation and an efficient estimator to hitherto intractable Gaussian integrals. We prove that the estimator has a rare vanishing relative error asymptotic property. Numerical experiments suggest that the scheme proposed is accurate in a wide range of set-ups for which competing estimation schemes fail. We give an application to exact independently and identically distributed data simulation from the Bayesian posterior of the probit regression model.

The cross-entropy method is a versatile heuristic tool for solving difficult estimation and optimization problems, based on Kullback-Leibler (or cross-entropy) minimization. As an optimization method it unifies many existing populationbased optimization heuristics. In this chapter we show how the cross-entropy method can be applied to a diverse range of combinatorial, continuous, and noisy optimization problems.

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.

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We propose a novel simulation-based method that exploits a generalized splitting (GS) algorithm to estimate the reliability of a graph (or network), defined here as the probability that a given set of nodes are connected, when each link of the graph is failed with a given (small) probability. For large graphs, in general, computing the exact reliability is an intractable problem and estimating it by standard Monte Carlo methods poses serious difficulties, because the unreliability (one minus the reliability) is often a rare-event probability. We show that the proposed GS algorithm can accurately estimate extremely small unreliabilities and we exhibit large examples where it performs much better than existing approaches. It is also flexible enough to dispense with the frequently made assumption of independent edge failures.