Abstract
Given a costly to compute function \(m: {\mathbb {R}}^d\rightarrow {\mathbb {R}}\), which is part of a simulation model, and an \({\mathbb {R}}^d\)-valued random variable with known distribution, the problem of estimating a quantile \(q_{m(X),\alpha }\) is investigated. The presented approach has a nonparametric nature. Monte Carlo quantile estimates are obtained by estimating m through some estimate (surrogate) \(m_n\) and then by using an initial quantile estimate together with importance sampling to construct an importance sampling surrogate quantile estimate. A general error bound on the error of this quantile estimate is derived, which depends on the local error of the function estimate \(m_n\), and the convergence rates of the corresponding importance sampling surrogate quantile estimates are analyzed. The finite sample size behavior of the estimates is investigated by applying the estimates to simulated data.
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References
Arnold BC, Balakrishnan N, Nagaraja HN (1992) A first course in order statistics. Wiley, New York
Beirlant J, Györfi L (1998) On the asymptotic \({L}_2\)-error in partitioning regression estimation. J Stat Plan Inference 71:93–107
Bichon B, Eldred M, Swiler M, Mahadevan S, McFarland J (2008) Efficient global reliability analysis for nonlinear implicit performance functions. AIAA J 46:2459–2468
Bourinte J-M, Deheeger F, Lemaire M (2011) Assessing small failure probabilities by combined subset simulation and support vector machines. Struct Saf 33:343–353
Bucher C, Bourgund U (1990) A fast and efficient response surface approach for structural reliability problems. Struct Saf 7:57–66
Cannamela C, Garnier J, Iooss B (2008) Controlled stratification for quantile estimation. Ann Appl Stat 2(4):1554–1580
Das P-K, Zheng Y (2000) Cumulative formation of response surface and its use in reliability analysis. Probab Eng Mech 15:309–315
de Boor C (1978) A practical guide to splines. Springer, Berlin
Deheeger F, Lemaire M (2010) Support vector machines for efficient subset simulations: \(^2\)SMART method. In: Proceedings of the 10th international conference on applications of statistics and probability in civil engineering (ICASP10), Tokyo, Japan
Devroye L (1982) Necessary and sufficient conditions for the almost everywhere convergence of nearest neighbor regression function estimates. Zeitschrift für Wahrscheinlichkeitstheorie und verwandte Gebiete 61:467–481
Devroye L, Krzyżak A (1989) An equivalence theorem for \({L}_1\) convergence of the kernel regression estimate. J Stat Plan Inference 23:71–82
Devroye L, Wagner TJ (1980) Distribution-free consistency results in nonparametric discrimination and regression function estimation. Ann Stat 8:231–239
Devroye L, Györfi L, Krzyżak A, Lugosi G (1994) On the strong universal consistency of nearest neighbor regression function estimates. Ann Stat 22:1371–1385
Dvoretzky A, Kiefer J, Wolfowitz J (1956) Asymptotic minimax character of the sample distribution function and of the classical multinomial estimator. Ann Math Stat 27:642–669
Egloff D, Leippold M (2010) Quantile estimation with adaptive importance sampling. Ann Stat 38:1244–1278
Enss GC, Kohler M, Krzyżak A, Platz R (2016) Nonparametric quantile estimation based on surrogate models. IEEE Trans Inf Theory 62:5727–5739
Falk M (1985) Asymptotic normality of the kernel quantile estimator. Ann Stat 13:428–433
Glasserman P (2004) Monte Carlo methods in financial engineering. Springer, New York
Greblicki W, Pawlak M (1985) Fourier and Hermite series estimates of regression functions. Ann Inst Stat Math 37:443–454
Györfi L (1981) Recent results on nonparametric regression estimate and multiple classification. Probl Control Inf Theory 10:43–52
Györfi L, Kohler M, Krzyżak A, Walk H (2002) A distribution-free theory of nonparametric regression. Springer series in statistics. Springer, New York
Hurtado J (2004) Structural reliability: statistical learning perspectives, vol 17. Lecture notes in applied and computational mechanics. Springer, Berlin
Kaymaz I (2005) Application of Kriging method to structural reliability problems. Strut Saf 27:133–151
Kim S-H, Na S-W (1997) Response surface method using vector projected sampling points. Strut Saf 19:3–19
Kohler M (2000) Inequalities for uniform deviations of averages from expectations with applications to nonparametric regression. J Stat Plan Inference 89:1–23
Kohler M, Krzyżak A (2001) Nonparametric regression estimation using penalized least squares. IEEE Trans Inf Theory 47:3054–3058
Kohler M, Krzyżak A, Tent R, Walk H (2018) Nonparametric quantile estimation using importance sampling. Ann Inst Stat Math 70:439–465
Lugosi G, Zeger K (1995) Nonparametric estimation via empirical risk minimization. IEEE Trans Inf Theory 41:677–687
Massart P (1990) The tight constant in the Dvoretzky–Kiefer–Wolfowitz inequality. Ann Probab 18(3):1269–1283
Morio J (2012) Extreme quantile estimation with nonparametric adaptive importance sampling. Simul Model Pract Theory 27:76–89
Nadaraya EA (1964) On estimating regression. Theory Probab Appl 9:141–142
Nadaraya EA (1970) Remarks on nonparametric estimates for density functions and regression curves. Theory Probab Appl 15:134–137
Neddermeyer JC (2009) Computationally efficient nonparametric importance sampling. J Am Stat Assoc 104(486):788–802
Papadrakakis M, Lagaros N (2002) Reliability-based structural optimization using neural networks and Monte Carlo simulation. Comput Methods Appl Mech Eng 191:3491–3507
Pepelyshev A, Rafajłowicz E, Steland A (2014) Estimation of the quantile function using Bernstein–Durrmeyer polynomials. J Nonparametr Stat 26:1–20
Platz R, Enss GC (2015) Comparison of uncertainty in passive and active vibration isolation. In: Atamturktur H, Moaveni B, Papadimitriou C, Schoenherr T (eds) Model validation and uncertainty quantification, vol 3. Conference proceedings of the society for experimental mechanics series. Springer, Berlin
Rafajłowicz E (1987) Nonparametric orthogonal series estimators of regression: a class attaining the optimal convergence rate in L2. Stat Probab Lett 5:219–224
Sheather S, Marron S (1990) Kernel quantile estimators. J Am Stat Assoc 85:410–416
Stone CJ (1977) Consistent nonparametric regression. Ann Stat 5:595–645
Stone CJ (1982) Optimal global rates of convergence for nonparametric regression. Ann Stat 10:1040–1053
Wahba G (1990) Spline models for observational data. SIAM, Philadelphia
Watson GS (1964) Smooth regression analysis. Sankhya Ser A 26:359–372
Acknowledgements
The authors would like to thank the German Research Foundation (DFG) for funding this project within the Collaborative Research Centre 805 (Projektnummer 57157408 - SFB 805). Furthermore the authors would like to thank two anonymous referees and the Associate editor for many very useful comments which improved an early version of this article.
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Kohler, M., Tent, R. Nonparametric quantile estimation using surrogate models and importance sampling. Metrika 83, 141–169 (2020). https://doi.org/10.1007/s00184-019-00736-3
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DOI: https://doi.org/10.1007/s00184-019-00736-3