Dirk P. Kroese

Kernel density estimation via diffusion

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.

Application of the Cross-Entropy Method to the Buffer Allocation Problem in a Simulation-Based Environment

The buffer allocation problem (BAP) is a well-known difficult problem in the design of production lines. We present a stochastic algorithm for solving the BAP, based on the cross-entropy method, a new paradigm for stochastic optimization. The algorithm involves the following iterative steps: (a) the generation of buffer allocations according to a certain random mechanism, followed by (b) the modification of this mechanism on the basis of cross-entropy minimization. Through various numerical experiments we demonstrate the efficiency of the proposed algorithm and show that the method can quickly generate (near-)optimal buffer allocations for fairly large production lines.

Improved algorithms for rare event simulation with heavy tails

The estimation of P(S n >u) by simulation, where S n is the sum of independent, identically distributed random varibles Y 1 ,…,Y n , is of importance in many applications. We propose two simulation estimators based upon the identity P(S n >u)=nP(S n >u, M n =Y n ), where M n =max(Y 1 ,…,Y n ). One estimator uses importance sampling (for Y n only), and the other uses conditional Monte Carlo conditioning upon Y 1 ,…,Y n−1. Properties of the relative error of the estimators are derived and a numerical study given in terms of the M/G/1 queue in which n is replaced by an independent geometric random variable N. The conclusion is that the new estimators compare extremely favorably with previous ones. In particular, the conditional Monte Carlo estimator is the first heavy-tailed example of an estimator with bounded relative error. Further improvements are obtained in the random-N case, by incorporating control variates and stratification techniques into the new estimation procedures.

Convergence properties of the cross-entropy method for discrete optimization

We present new theoretical convergence results on the cross-entropy (CE) method for discrete optimization. We show that a popular implementation of the method converges, and finds an optimal solution with probability arbitrarily close to 1. We also give conditions under which an optimal solution is generated eventually with probability 1.

The Cross-Entropy Method for Network Reliability Estimation

Consider a network of unreliable links, modelling for example a communication network. Estimating the reliability of the network—expressed as the probability that certain nodes in the network are connected—is a computationally difficult task. In this paper we study how the Cross-Entropy method can be used to obtain more efficient network reliability estimation procedures. Three techniques of estimation are considered: Crude Monte Carlo and the more sophisticated Permutation Monte Carlo and Merge Process. We show that the Cross-Entropy method yields a speed-up over all three techniques.

Efficient Monte Carlo simulation via the generalized splitting method

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.

Global likelihood optimization via the cross-entropy method with an application to mixture models

Global likelihood maximization is an important aspect of many statistical analyses. Often the likelihood function is highly multiextremal. This presents a significant challenge to standard search procedures, which often settle too quickly into an inferior local maximum. We present a new approach based on the cross-entropy (CE) method, and illustrate its use for the analysis of mixture models.

A comparison of RESTART implementations

The RESTART method is a widely applicable simulation technique for the estimation of rare event probabilities. The method is based on the idea to restart the simulation in certain system states, in order to generate more occurrences of the rare event. One of the main questions for any RESTART implementation is how and when to restart the simulation, in order to achieve the most accurate results for a fixed simulation effort. We investigate and compare, both theoretically and empirically, different implementations of the RESTART method. We find that the original RESTART implementation, in which each path is split into a fixed number of copies, may not be the most efficient one. It is generally better to fix the total simulation effort for each stage of the simulation. Furthermore, given this effort, the best strategy is to restart an equal number of times from each state, rather than to restart each time from a randomly chosen state.

Efficient estimation of large portfolio loss probabilities in t-copula models

We consider the problem of accurately measuring the credit risk of a portfolio consisting of loans, bonds and other financial assets. One particular performance measure of interest is the probability of large portfolio losses over a fixed time horizon. We revisit the so-called t-copula that generalizes the popular normal copula to allow for extremal dependence among defaults. By utilizing the asymptotic description of how the rare event occurs, we derive two simple simulation algorithms based on conditional Monte Carlo to estimate the probability that the portfolio incurs large losses under the t-copula. We further show that the less efficient estimator exhibits bounded relative error. An extensive simulation study demonstrates that both estimators outperform existing algorithms. We then discuss a generalization of the t-copula model that allows the multivariate defaults to have an asymmetric distribution. Lastly, we show how the estimators proposed for the t-copula can be modified to estimate the portfolio risk under the skew t-copula model.

Network Reliability Optimization via the Cross-Entropy Method

Consider a network of unreliable links, each of which comes with a certain price, and reliability. Given a fixed budget, which links should be purchased in order to maximize the systems reliability? We introduce a new approach, based on the cross-entropy method, which can deal effectively with the constraints, and noise introduced when estimating the reliabilities via simulation, in this difficult combinatorial optimization problem. Numerical results demonstrate the effectiveness of the proposed technique