A Pooled Quantile Estimator for Parallel Simulations

Quantile is an important risk measure quantifying the stochastic system random behaviors. This paper studies a pooled quantile estimator, which is the sample quantile of detailed simulation outputs after directly pooling independent sample paths together. We derive the asymptotic representation of the pooled quantile estimator and further prove its normality. By comparing with the classical quantile estimator used in stochastic simulation, both theoretical and empirical studies demonstrate the advantages of the proposal under the context of parallel simulation

Monte Carlo and Quasi-Monte Carlo Density Estimation via Conditioning

Estimating the unknown density from which a given independent sample originates is more difficult than estimating the mean, in the sense that for the best popular density estimators, the mean integrated square error converges more slowly than at the canonical rate of $\mathcal{O}(1/n)$. When the sample is generated from a simulation model and we have control over how this is done, we can do better. We examine an approach in which conditional Monte Carlo permits one to obtain a smooth estimator of the cumulative distribution function, whose sample derivative is, under certain conditions, an unbiased estimator of the density at any point, and therefore converges at a faster rate than the usual density estimators. We can achieve an even faster rate by combining this with randomized quasi-Monte Carlo to generate the samples.Comment: Main manuscript: 30 pages, 6 figures, 5 tables. Supplement: 11 pages, 5 figures, 5 tables. We are very thankful to the anonymous referees, whose comments were considered in this submissio

A Multi-Level Simulation Optimization Approach for Quantile Functions

Quantile is a popular performance measure for a stochastic system to evaluate its variability and risk. To reduce the risk, selecting the actions that minimize the tail quantiles of some loss distributions is typically of interest for decision makers. When the loss distribution is observed via simulations, evaluating and optimizing its quantile functions can be challenging, especially when the simulations are expensive, as it may cost a large number of simulation runs to obtain accurate quantile estimators. In this work, we propose a multi-level metamodel (co-kriging) based algorithm to optimize quantile functions more efficiently. Utilizing non-decreasing properties of quantile functions, we first search on cheaper and informative lower quantiles which are more accurate and easier to optimize. The quantile level iteratively increases to the objective level while the search has a focus on the possible promising regions identified by the previous levels. This enables us to leverage the accurate information from the lower quantiles to find the optimums faster and improve algorithm efficiency