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The Convergence Rate and Asymptotic Distribution of the Bootstrap Quantile Variance Estimator for Importance Sampling

Part of: Probabilistic methods, simulation and stochastic differential equations Parametric inference

Published online by Cambridge University Press:  04 January 2016

Jingchen Liu  and
Xuan Yang
Jingchen Liu*
Affiliation:
Columbia University
Xuan Yang*
Affiliation:
Columbia University
*
∗ Postal address: Department of Statistics, Columbia University, 1255 Amsterdam Avenue, New York, NY 10027, USA.
∗ Postal address: Department of Statistics, Columbia University, 1255 Amsterdam Avenue, New York, NY 10027, USA.
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Abstract

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Importance sampling is a widely used variance reduction technique to compute sample quantiles such as value at risk. The variance of the weighted sample quantile estimator is usually a difficult quantity to compute. In this paper we present the exact convergence rate and asymptotic distributions of the bootstrap variance estimators for quantiles of weighted empirical distributions. Under regularity conditions, we show that the bootstrap variance estimator is asymptotically normal and has relative standard deviation of order O(n−1/4).

Keywords

Weighted quantilebootstrapvariance estimatorimportance sampling

MSC classification

Primary: 62F40: Bootstrap, jackknife and other resampling methods
Secondary: 65C05: Monte Carlo methods
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 44 , Issue 3 , September 2012 , pp. 815 - 841
Copyright
© Applied Probability Trust 

Footnotes

This research was supported in part by NSF grants CMMI-1069064 and SES-1123698, and the Institute of Education Sciences under grant R305D100017.

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