TY - JOUR

T1 - Bias-corrected Estimation of the Density of a Conditional Expectation in Nested Simulation Problems

AU - Yang, Ran

AU - Kent, David

AU - Apley, Daniel W.

AU - Staum, Jeremy

AU - Ruppert, David

N1 - Funding Information:
The authors gratefully acknowledge the support from NSF grant AST-1814840. Authors’ addresses: R. Yang, D. W. Apley (corresponding author), and J. Staum, Department of Industrial Engineering & Management Sciences, Northwestern University, 2145 Sheridan Road, Evanston, IL, 60208; emails: ranyang2011@ u.northwestern.edu, {apley, j-staum}@northwestern.edu; D. Kent, Department of Statistics and Data Science, Cornell University, Ithaca, NY, 14853; email: [email protected]; D. Ruppert, Department of Statistics and Data Science and School of Operations Research and Information Engineering, Cornell University, Ithaca, NY, 14853; email: [email protected]. Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. Copyrights for components of this work owned by others than ACM must be honored. Abstracting with credit is permitted. To copy otherwise, or republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. Request permissions from [email protected]. © 2021 Association for Computing Machinery. 1049-3301/2021/07-ART22 $15.00 https://doi.org/10.1145/3462201
Publisher Copyright:
© 2021 ACM.

PY - 2021/10

Y1 - 2021/10

N2 - Many two-level nested simulation applications involve the conditional expectation of some response variable, where the expected response is the quantity of interest, and the expectation is with respect to the inner-level random variables, conditioned on the outer-level random variables. The latter typically represent random risk factors, and risk can be quantified by estimating the probability density function (pdf) or cumulative distribution function (cdf) of the conditional expectation. Much prior work has considered a naïve estimator that uses the empirical distribution of the sample averages across the inner-level replicates. This results in a biased estimator, because the distribution of the sample averages is over-dispersed relative to the distribution of the conditional expectation when the number of inner-level replicates is finite. Whereas most prior work has focused on allocating the numbers of outer- and inner-level replicates to balance the bias/variance tradeoff, we develop a bias-corrected pdf estimator. Our approach is based on the concept of density deconvolution, which is widely used to estimate densities with noisy observations but has not previously been considered for nested simulation problems. For a fixed computational budget, the bias-corrected deconvolution estimator allows more outer-level and fewer inner-level replicates to be used, which substantially improves the efficiency of the nested simulation.

AB - Many two-level nested simulation applications involve the conditional expectation of some response variable, where the expected response is the quantity of interest, and the expectation is with respect to the inner-level random variables, conditioned on the outer-level random variables. The latter typically represent random risk factors, and risk can be quantified by estimating the probability density function (pdf) or cumulative distribution function (cdf) of the conditional expectation. Much prior work has considered a naïve estimator that uses the empirical distribution of the sample averages across the inner-level replicates. This results in a biased estimator, because the distribution of the sample averages is over-dispersed relative to the distribution of the conditional expectation when the number of inner-level replicates is finite. Whereas most prior work has focused on allocating the numbers of outer- and inner-level replicates to balance the bias/variance tradeoff, we develop a bias-corrected pdf estimator. Our approach is based on the concept of density deconvolution, which is widely used to estimate densities with noisy observations but has not previously been considered for nested simulation problems. For a fixed computational budget, the bias-corrected deconvolution estimator allows more outer-level and fewer inner-level replicates to be used, which substantially improves the efficiency of the nested simulation.

KW - Deconvolution

KW - Stein's unbiased risk estimation

KW - estimating a conditional variance

KW - quadratic programming

KW - rate of convergence

KW - shape constraints

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U2 - 10.1145/3462201

DO - 10.1145/3462201

M3 - Article

AN - SCOPUS:85113156842

SN - 1049-3301

VL - 31

JO - ACM Transactions on Modeling and Computer Simulation

JF - ACM Transactions on Modeling and Computer Simulation

IS - 4

M1 - 22

ER -