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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.

References

Adler, R. J., Blanchet, J. H. and Liu, J. C. (2009). Efficient Monte Carlo for large excursions of Gaussian random fields. Preprint.Google Scholar
Asmussen, S. and Glynn, P. W. (2007). Stochastic Simulation: Algorithms and Analysis. Springer, New York.CrossRefGoogle Scholar
Asmussen, S. and Kroese, D. P. (2006). Improved algorithms for rare event simulation with heavy tails. Adv. Appl. Prob. 38, 545–558.CrossRefGoogle Scholar
Asmussen, S., Binswanger, K. and Højgaard, B. (2000). Rare events simulation for heavy-tailed distributions. Bernoulli 6, 303–322.Google Scholar
Babu, G. J. (1986). A note on bootstrapping the variance of sample quantile. Ann. Inst. Statist. Math. 38, 439–443.CrossRefGoogle Scholar
Bahadur, R. R. (1966). A note on quantiles in large samples. Ann. Math. Statist. 37, 577–580.CrossRefGoogle Scholar
Blanchet, J. H. (2009). Efficient importance sampling for binary contingency tables. Ann. Appl. Prob. 19, 949–982.Google Scholar
Blanchet, J. and Glynn, P. (2008). Effcient rare-event simulation for the maximum of heavy-tailed random walks. Ann. Appl. Prob. 18, 1351–1378.Google Scholar
Blanchet, J. H. and Liu, J. (2008). State-dependent importance sampling for regularly varying random walks. Adv. Appl. Prob. 40, 1104–1128.Google Scholar
Blanchet, J. and Liu, J. (2010). Efficient importance sampling in ruin problems for multidimensional regularly varying random walks. J. Appl. Prob. 47, 301–322.CrossRefGoogle Scholar
Blanchet, J., Glynn, P. and Liu, J. C. (2007). Effcient rare event simulation for multiserver queues. Preprint.Google Scholar
Blanchet, J., Glynn, P. and Liu, J. C. (2007). Fluid heuristics, Lyapunov bounds and efficient importance sampling for a heavy-tailed G/G/1 queue. Queueing Systems 57, 99–113.Google Scholar
Bucklew, J. A. (2004). Introduction to Rare Event Simulation. Springer, New York.CrossRefGoogle Scholar
Chu, F. and Nakayama, M. K. (2010). Confidence intervals for quantiles and value-at-risk when applying importance sampling. In Proc. of the 2010 Winter Simulation Conference, IEEE, pp. 2751–2761.Google Scholar
David, H. A. and Nagaraja, H. N. (2003). Order Statistics, 3rd edn. Wiley-Interscience, Hoboken, NJ.CrossRefGoogle Scholar
De la Peña, V. H., Lai, T. L. and Shao, Q.-M. (2009). Self-Normalized Processes. Springer, Berlin.Google Scholar
Dupuis, P. and Wang, H. (2005). Dynamic importance sampling for uniformly recurrent Markov chains. Ann. Appl. Prob. 15, 1–38.Google Scholar
Dupuis, P., Leder, K. and Wang, H. (2007). Importance sampling for sums of random variables with regularly varying tails. ACM Trans. Model. Comput. Simul. 17, 21 pp.CrossRefGoogle Scholar
Dupuis, P., Sezer, A. L. and Wang, H. (2007). Dynamic importance sampling for queueing networks. Ann. Appl. Prob. 17, 1306–1346.CrossRefGoogle Scholar
Durrett, R. (2010). Probability: Theory and Examples, 4th edn. Cambridge University Press.CrossRefGoogle Scholar
Falk, M. (1986). On the estimation of the quantile density function. Statist. Prob. Lett. 4, 69–73. (Correction: 4 (1986), 217.)Google Scholar
Ghosh, M., Parr, W. C., Singh, K. and Babu, G. J. (1984). A note on bootstrapping the sample median. Ann. Statist. 12, 1130–1135.Google Scholar
Glasserman, P. and Li, J. (2005). Importance sampling for portfolio credit risk. Manag. Sci. 51, 1643–1656.Google Scholar
Glasserman, P., Heidelgerger, P. and Shahabuddin, P. (1999). Importance sampling and stratification for value-at-risk. In Computational Finance 1999 (Proc. 6th Internat. Conf. Comput. Finance), MIT Press, Cambridge, MA, pp. 7–24.Google Scholar
Glasserman, P., Heidelberger, P. and Shahabuddin, P. (2000). Variance reduction techniques for estimating value-at-risk. Manag, Sci. 46, 1349–1364.Google Scholar
Glasserman, P., Heidelberger, P. and Shahabuddin, P. (2002). Portfolio value-at-risk with heavy-tailed risk factors. Math. Finance 12, 239–269.CrossRefGoogle Scholar
Glynn, P. W. (1996). Importance sampling for Monte Carlo estimation of quantiles. In Mathematical Methods in Stochastic Simulation and Experimental Design: Proc. 2nd St. Petersburg Workshop on Simulation, Publishing House of Saint Petersburg, pp. 180–185.Google Scholar
Hall, P. and Martin, M. A. (1988). Exact convergence rate of bootstrap quantile variance estimator. Prob. Theory Relat. Fields 80, 261–268.CrossRefGoogle Scholar
Hult, H. and Svensson, J. (2009). Efficient calculation of risk measures by importance sampling – the heavy tailed case. Preprint.Google Scholar
Hult, H. and Svensson, J. (2009). Efficient calculation of risk measures by importance sampling – the heavy tailed case. Preprint.Google Scholar
Juneja, S. and Shahabuddin, P. (2002). Simulating heavy tailed processes using delayed hazard rate twisting. ACM Trans. Model. Comput. Simul. 12, 94–118.CrossRefGoogle Scholar
Kish, L. (1965). Survey Sampling. John Wiley, New York.Google Scholar
Komlós, J., Major, P. and Tusnády, G. (1975). An approximation of partial sums of independent RV's and the sample DF. I. Z. Wahrscheinlichkeitsth. 32, 111–131.CrossRefGoogle Scholar
Komlós, J., Major, P. and Tusnády, G. (1975). Weak convergence and embedding. In Limit Theorems of Probability Theory (Colloq. Math. Soc. Jànos Bolyai 11), North-Holland, Amsterdam, pp. 149–165.Google Scholar
Komlós, J., Major, P. and Tusnády, G. (1976). An approximation of partial sums of independent RV's, and the sample DF. II. Z. Wahrscheinlichkeitsth. 34, 33–58.Google Scholar
Lohr, S. L. (1999). Sampling: Design and Analysis. Duxbury Press, Pacific Grove, CA.Google Scholar
Maritz, J. S. and Jarrett, R. G. (1978). A note on estimating the variance of the sample median. J. Amer. Statist. Assoc. 73, 194–196.CrossRefGoogle Scholar
McKean, J. W. and Schrader, R. M. (1984). A comparison of methods for studentizing the sample median. Commun. Statist. Simul. Comput. 13, 751–773.Google Scholar
Sadowsky, J. S. (1996). On Monte Carlo estimation of large deviations probabilities. Ann. Appl. Prob. 6, 399–422.CrossRefGoogle Scholar
Sheather, S. J. (1986). A finite sample estimate of the variance of the sample median. Statist. Prob. Lett. 4, 337–342.Google Scholar
Siegmund, D. (1976). Importance sampling in the Monte Carlo study of sequential tests. Ann. Statist. 4, 673–684.Google Scholar
Sun, L. and Hong, L. J. (2009). A general framework of importance sampling for value-at-risk and conditional value-at-risk. In Proc. 2009 Winter Simulation Conference, pp. 415–422.Google Scholar
Sun, L. and Hong, L. J. (2010). Asymptotic representations for importance-sampling estimators of value-at-risk and conditional value-at-risk. Operat. Res. Lett. 38, 246–251.CrossRefGoogle Scholar
Wang, R.-H., Lin, S.-K. and Fuh, C.-D. (2009). An importance sampling method to evaluate value-at-risk for asset with Jump risk. Asia-Pacific J. Financial Studies 38, 745–772.Google Scholar