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Egloff, Daniel; Leippold, Markus

Quantile estimation with adaptive importance sampling. (English) Zbl 1183.62141

Ann. Stat. 38, No. 2, 1244-1278 (2010).
Summary: We introduce new quantile estimators with adaptive importance sampling. The adaptive estimators are based on weighted samples that are neither independent nor identically distributed. Using a new law of iterated logarithm for martingales, we prove the convergence of the adaptive quantile estimators for general distributions with non-unique quantiles thereby extending the work of D. Feldman and H. G. Tucker [Ann. Math. Stat. 37, 451–457 (1966; Zbl 0152.36907)]. We illustrate the algorithm with an example from credit portfolio risk analysis.

Cited in 10 Documents

MSC:

62L20 Stochastic approximation
60F15 Strong limit theorems
62P05 Applications of statistics to actuarial sciences and financial mathematics
65C05 Monte Carlo methods
65C60 Computational problems in statistics (MSC2010)
62G05 Nonparametric estimation

Keywords:

quantile estimation; law of iterated logarithm; adaptive importance sampling; stochastic approximation; Robbins-Monro

Citations:

Zbl 0152.36907
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Full Text: DOI arXiv

References:

[1] Andrieu, C., Moulines, E. and Priouret, P. (2005). Stability of stochastic approximation under verifiable conditions. SIAM J. Control Optim. 44 283-312. · Zbl 1083.62073 · doi:10.1137/S0363012902417267
[2] Arouna, B. (2004). Adaptive Monte Carlo method, a variance reduction technique. Monte Carlo Methods Appl. 10 1-24. · Zbl 1063.65003 · doi:10.1515/156939604323091180
[3] Arouna, B. (2004). Robbins-Monro algorithms and variance reduction in finance. J. Computational Finance 7 35-62.
[4] Avramidis, A. N. and Wilson, J. R. (1998). Correlation-induction techniques for estimating quantiles in simulation experiments. Oper. Res. 46 574-591. · Zbl 1009.62598 · doi:10.1287/opre.46.4.574
[5] Benveniste, A., Metiver, M. and Priouret, P. (1990). Adaptive Algorithms and Stochastic Approximations . Springer, Berlin. · Zbl 0752.93073
[6] Chen, H. F. (2002). Stochastic Approximation and Its Applications. Nonconvex Optimization and Its Applications 64 . Kluwer, Dordrecht. · Zbl 1008.62071
[7] Chen, H. F., Guo, L. and Gao, A. J. (1988). Convergence and robustness of the Robbins-Monro algorithm truncated at randomly varying bounds. Stochastic Process. Appl. 27 217-231. · Zbl 0632.62082 · doi:10.1016/0304-4149(87)90039-1
[8] Delyon, B. (1996). General results on the convergence of stochastic algorithms. IEEE Transactions on Automatic Control 41 1245-1256. · Zbl 0867.93075 · doi:10.1109/9.536495
[9] Delyon, B. (2000). Stochastic approximation with decreasing gain: Convergence and asymptotic theory. Technical report, Publication Interne 952, IRISA.
[10] Delyon, B., Lavielle, M. and Mouliens, E. (1999). Convergence of a stochastic approximation version of the EM algorithm. Ann. Statist. 27 94-128. · Zbl 0932.62094 · doi:10.1214/aos/1018031103
[11] Egloff, D., Leippold, M. and Vanini, P. (2007). A simple model of credit contagion. Journal of Banking and Finance 8 2475-2492.
[12] Feldman, D. and Tucker, H. G. (1966). Estimation of non-unique quantiles. Ann. Math. Statist. 37 451-457. · Zbl 0152.36907 · doi:10.1214/aoms/1177699527
[13] Glasserman, P. and Li, J. (2005). Importance sampling for portfolio credit risk. Management Sci. 51 1643-1656. · Zbl 1232.91621 · doi:10.1287/mnsc.1050.0415
[14] Glasserman, P. and Wang, Y. (1997). Counterexamples in importance sampling for large deviation probabilities. Ann. Appl. Probab. 7 731-746. · Zbl 0892.60043 · doi:10.1214/aoap/1034801251
[15] Hall, P. and Heyde, C. C. (1980). Martingale Limit Theory and its Applications . Academic Press, New York. · Zbl 0462.60045
[16] Hesterberg, T. and Nelson, B. L. (1998). Control variates for probability and quantile estimation. Management Sci. 44 1295-1312. · Zbl 1103.90374 · doi:10.1287/mnsc.44.9.1295
[17] Heyde, C. C. (1977). On central limit and iterated logarithm supplements to the martingale convergence theorem. J. Appl. Probab. 14 758-775. JSTOR: · Zbl 0385.60033 · doi:10.2307/3213349
[18] Hsu, J. C. and Nelson, B. L. (1990). Control variates for quantile estimation. Management Sci. 36 835-851. JSTOR: · Zbl 0711.62024 · doi:10.1287/mnsc.36.7.835
[19] Jin, X., Fu, M. C. and Xiong, X. (2003). Probabilistic error bounds for simulation quantile estimators. Management Sci. 14 230-246. · Zbl 1232.91349 · doi:10.1287/mnsc.49.2.230.12743
[20] Jost, J. (2005). Riemannian Geometry and Geometric Analysis . Springer, Berlin. · Zbl 1083.53001
[21] Kiefer, J. and Wolfowitz, J. (1952). Stochastic estimation of the maximum of a regression function. Ann. Math. Statist. 23 462-466. · Zbl 0049.36505 · doi:10.1214/aoms/1177729391
[22] Kobayashi, S. and Nomizu, K. (1996). Foundations of Differential Geometry . I, II. Wiley, Chichester. · Zbl 0119.37502
[23] Kushner, H. J. and Clark, D. S. (1978). Stochastic Approximation Methods for Constrained and Unconstrained Systems . Springer, New York. · Zbl 0381.60004
[24] Kushner, H. J. and Yin, G. G. (1997). Stochastic Approximation Algorithms and Applications . Springer, New York. · Zbl 0914.60006
[25] Ljung, L., Pflug, G. and Walk, H. (1992) Stochastic Approximation and Optimization of Random Systems . Birkhaeuser, Basel. · Zbl 0747.62090
[26] Merino, S. and Nyfeler, M. (2004). Applying importance sampling for estimating coherent credit risk contributions. Quantitative Finance 4 199-207. · Zbl 1095.91018
[27] Polyak, B. and Juditsky, A. (1992). Acceleration of stochastic approximation by averaging. SIAM J. Control Optim. 30 838-855. · Zbl 0762.62022 · doi:10.1137/0330046
[28] Rao, C. R. (1945). Information and the accuracy atainable in the estimation of statistical parameters. Bull. Calcutta Math. Soc. 37 81-91. · Zbl 0063.06420
[29] Resnick, S. (1987). Extreme Values, Regular Variation, and Point Processes . Springer, New York. · Zbl 0633.60001
[30] Robbins, H. and Monro, S. (1951). A stochastic approximation method. Ann. Math. Statist. 22 400-407. · Zbl 0054.05901 · doi:10.1214/aoms/1177729586
[31] Smirnov, N. (1952). Limit distribution for the terms of a variational series. Amer. Math. Soc. Translation 1952 64. · Zbl 0046.11701
[32] Tierney, L. (1983). A space-efficient recursive procedure for estimating a quantile of an unknown distribution. SIAM J. Sci. Statist. Comput. 4 706-711. · Zbl 0524.65099 · doi:10.1137/0904048
[33] van der Vaart, A. W. (1998). Asymptotic Statistics . Cambridge Univ. Press, Cambridge. · Zbl 0910.62001 · doi:10.1017/CBO9780511802256
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