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Estimation of conditional distribution functions from data with additional errors applied to shape optimization

Author

Listed:
  • Matthias Hansmann

    (TU Darmstadt)

  • Benjamin M. Horn

    (TU Darmstadt)

  • Michael Kohler

    (TU Darmstadt)

  • Stefan Ulbrich

    (TU Darmstadt)

Abstract

We study the problem of estimating conditional distribution functions from data containing additional errors. The only assumption on these errors is that a weighted sum of the absolute errors tends to zero with probability one for sample size tending to infinity. We prove sufficient conditions on the weights (e.g. fulfilled by kernel weights) of a local averaging estimate of the codf, based on data with errors, which ensure strong pointwise consistency. We show that two of the three sufficient conditions on the weights and a weaker version of the third one are also necessary for the spc. We also give sufficient conditions on the weights, which ensure a certain rate of convergence. As an application we estimate the codf of the number of cycles until failure based on data from experimental fatigue tests and use it as objective function in a shape optimization of a component.

Suggested Citation

  • Matthias Hansmann & Benjamin M. Horn & Michael Kohler & Stefan Ulbrich, 2022. "Estimation of conditional distribution functions from data with additional errors applied to shape optimization," Metrika: International Journal for Theoretical and Applied Statistics, Springer, vol. 85(3), pages 323-343, April.
    • Handle: RePEc:spr:metrik:v:85:y:2022:i:3:d:10.1007_s00184-021-00831-4
    DOI: 10.1007/s00184-021-00831-4
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    References listed on IDEAS

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    1. Hall, Peter & Wolff, Rodney C. L. & Yao, Qiwei, 1999. "Methods for estimating a conditional distribution function," LSE Research Online Documents on Economics 6631, London School of Economics and Political Science, LSE Library.
    2. Matthias Hansmann & Michael Kohler & Harro Walk, 2019. "Correction to: On the strong universal consistency of local averaging regression estimates," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 71(5), pages 1265-1269, October.
    3. Algoet, Paul & Györfi, László, 1999. "Strong Universal Pointwise Consistency of Some Regression Function Estimates," Journal of Multivariate Analysis, Elsevier, vol. 71(1), pages 125-144, October.
    4. Hall, Peter & Yao, Qiwei, 2005. "Approximating conditional distribution functions using dimension reduction," LSE Research Online Documents on Economics 16333, London School of Economics and Political Science, LSE Library.
    5. Harro Walk, 2001. "Strong Universal Pointwise Consistency of Recursive Regression Estimates," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 53(4), pages 691-707, December.
    6. Cai, Zongwu, 2002. "Regression Quantiles For Time Series," Econometric Theory, Cambridge University Press, vol. 18(1), pages 169-192, February.
    7. Noël Veraverbeke & Irène Gijbels & Marek Omelka, 2014. "Preadjusted non-parametric estimation of a conditional distribution function," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 76(2), pages 399-438, March.
    8. Matthias Hansmann & Michael Kohler & Harro Walk, 2019. "On the strong universal consistency of local averaging regression estimates," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 71(5), pages 1233-1263, October.
    9. Dmytro Furer & Michael Kohler, 2015. "Smoothing spline regression estimation based on real and artificial data," Metrika: International Journal for Theoretical and Applied Statistics, Springer, vol. 78(6), pages 711-746, August.
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    11. Ann-Kathrin Bott & Michael Kohler, 2017. "Nonparametric estimation of a conditional density," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 69(1), pages 189-214, February.
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