Confidence Intervals for Quantiles Using Sectioning When Applying Variance-Reduction Techniques

Abstract

We develop confidence intervals (CIs) for quantiles when applying variance-reduction techniques (VRTs) and sectioning. Similar to batching, sectioning partitions the independent and identically distributed (i.i.d.) outputs into nonoverlapping batches and computes a quantile estimator from each batch. But rather than centering the CI at the average of the quantile estimators across the batches, as in batching, sectioning centers the CI at the overall quantile estimator based on all the outputs. A similar modification is made to the sample variance, which is used to determine the width of the CI. We establish the asymptotic validity of the sectioning CI for importance sampling and control variates, and the proofs rely on first showing that the corresponding quantile estimators satisfy a Bahadur representation, which we have done in prior work. Here, we present some numerical results. ACM Reference Format:Marvin K. Nakayama. 2014. Confidence intervals for quantiles using sectioning when applying variancereduction techniques. ACM Trans. Model. Comput.

Distribution‐free Approximate Methods for Constructing Confidence Intervals for Quantiles

“…We discussed in the previous section that the proposed method is not range preserving. Neither is Chen and Hall (), Shao and Wu () or Nakayama (). For the latter two cases, we can see from and , respectively, that the margin of error computation does not guarantee a range‐preserving interval.…”

“…Recent advances on this topic have been in computer science and finance, particularly for constructing confidence intervals for value‐at‐risk estimates (Glasserman et al , ; Jin et al , ; Chu & Nakayama, ). Applications in these areas have resulted in the development of two general approaches: batching (Schmeiser, ; Glasserman, ; Muñoz, ) and sectioning (Asmussen & Glynn, ; Nakayama, ). These methods are based on functions of order statistics generated from dividing the sample into smaller groups.…”

“…These methods are based on functions of order statistics generated from dividing the sample into smaller groups. A good survey is provided by Nakayama (). In these approaches, one uses the percentiles of either a standard normal or a t distribution whose DF depends on the batch sizes used.…”

Quantile Estimation with Latin Hypercube Sampling

“…Sectioning (Section III.5a of Asmussen and Glynn (2007) for SRS, Nakayama (2014) for IS and CV) addresses this issue by replacing the batching point estimatorξ p,b,m in the batching CI with the overall quantile estimator ξ p,n , based on the CDF estimator F n from all n outputs. The SRS…”

“…To address this issue, we can instead apply sectioning, which was originally proposed in Section III.5a of Asmussen and Glynn (2007) for SRS and extended to certain variance-reduction techniques (VRTs) by Nakayama (2014). Similar to batching, sectioning replaces the batching point estimator with the overall point estimator throughout the batching CI.…”