Marvin Nakayama - New Jersey Institute of Technology

Title: Efficient Simulation of Risk and its Error: Confidence Intervals for Quantiles When Using Variance-Reduction Techniques. Abstract: The p-quantile of a continuous random variable is the constant for which exactly p of the mass of its distribution lies to the left of the quantile; e.g., the median is the 0.5-quantile. Quantiles are widely used to assess risk. For example, a project manager may want to determine a time T such that the project has a 95% chance of completing by T, which is the 0.95-quantile. In finance, where a quantile is known as a value-at-risk, analysts frequently measure risk with the 0.99-quantile of a portfolio’s loss. For complex stochastic models, analytically computing a quantile often is not possible, so simulation is employed. In addition to providing a point estimate for a quantile, we also want to measure the simulation estimate's error, and this is typically done by giving a confidence interval (CI) for the quantile. Indeed, the U.S. Nuclear Regulatory Commission requires that licensees of nuclear power plants demonstrate compliance using a “95/95 criterion,” which entails ensuring (with 95% confidence) that a 0.95-quantile lies below a mandated limit. In this talk we present some methods for constructing CIs for a quantile estimated via simulation. Unfortunately, crude Monte Carlo often produces wide CIs, so analysts often apply variance-reduction techniques (VRTs) in simulations to decrease the error. We first discuss forming a CI using a finite difference, and the second approach applies a procedure known as sectioning, which is closely related to batching. The asymptotic validity of both CIs follows from a so-called Bahadur representation, which shows that a quantile estimator can be approximated by a linear transformation of a probability estimator. We have established Bahadur representations for a broad class of VRTs, including antithetic variates, control variates, replicated Latin hypercube sampling, and importance sampling. We present some empirical results comparing the different CIs. This work is supported by NSF grants CMMI-0926949, CMMI-1200065, and DMS-1331010.