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Random approximations to some measures of accuracy in nonparametric curve estimation

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  • Marron, James Stephen
  • Härdle, Wolfgang

Abstract

This paper deals with a quite general nonparametric statistical curve estimation setting. Special cases include estimation or probability density functions, regression functions, and hazard functions. The class of "fractional delta sequence estimators" is defined and treated here. This class includes the familiar kernel, orthogonal series, and histogram methods. It is seen that, under some mild assumptions, both the average square error and integrated square error provide reasonable (random) approximations to the mean integrated square error. This is important for two reasons. First, it provides theoretical backing to a practice that has been employed in several simulation studies. Second, it provides a vital tool for proving theorems about selecting smoothing parameters for several different nonparametric curve estimators.

Suggested Citation

  • Marron, James Stephen & Härdle, Wolfgang, 1986. "Random approximations to some measures of accuracy in nonparametric curve estimation," Journal of Multivariate Analysis, Elsevier, vol. 20(1), pages 91-113, October.
    • Handle: RePEc:eee:jmvana:v:20:y:1986:i:1:p:91-113
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    Cited by:

    1. Timmermans, Catherine & Delsol, Laurent & von Sachs, Rainer, 2013. "Using Bagidis in nonparametric functional data analysis: Predicting from curves with sharp local features," Journal of Multivariate Analysis, Elsevier, vol. 115(C), pages 421-444.
    2. Inoue, Atsushi & Jin, Lu & Rossi, Barbara, 2017. "Rolling window selection for out-of-sample forecasting with time-varying parameters," Journal of Econometrics, Elsevier, vol. 196(1), pages 55-67.
    3. Max Köhler & Anja Schindler & Stefan Sperlich, 2014. "A Review and Comparison of Bandwidth Selection Methods for Kernel Regression," International Statistical Review, International Statistical Institute, vol. 82(2), pages 243-274, August.
    4. Delsol, Laurent & Ferraty, Frédéric & Vieu, Philippe, 2011. "Structural test in regression on functional variables," Journal of Multivariate Analysis, Elsevier, vol. 102(3), pages 422-447, March.
    5. K. Benhenni & F. Ferraty & M. Rachdi & P. Vieu, 2007. "Local smoothing regression with functional data," Computational Statistics, Springer, vol. 22(3), pages 353-369, September.
    6. J. Vilar, 1995. "Kernel estimation of the regression function with random sampling times," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 4(1), pages 137-178, June.
    7. Ferreira García, María Eva & Núñez Antón, Vicente Alfredo & Rodríguez Poo, Juan M., 1999. "Two-Stage Nonparametric Regression for Longitudinal Data," BILTOKI 1134-8984, Universidad del País Vasco - Departamento de Economía Aplicada III (Econometría y Estadística).
    8. Estévez-Pérez, Graciela, 2002. "On convergence rates for quadratic errors in kernel hazard estimation," Statistics & Probability Letters, Elsevier, vol. 57(3), pages 231-241, April.
    9. BOUEZMARNI, Taoufik & ROMBOUTS, Jeroen V.K., 2007. "Nonparametric density estimation for multivariate bounded data," LIDAM Discussion Papers CORE 2007065, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    10. Aldo Goia & Philippe Vieu, 2015. "A partitioned Single Functional Index Model," Computational Statistics, Springer, vol. 30(3), pages 673-692, September.
    11. Dippon, J. & Fritz, P. & Kohler, M., 2002. "A statistical approach to case based reasoning, with application to breast cancer data," Computational Statistics & Data Analysis, Elsevier, vol. 40(3), pages 579-602, September.
    12. Małgorzata Łazȩcka & Jan Mielniczuk, 2023. "Squared error-based shrinkage estimators of discrete probabilities and their application to variable selection," Statistical Papers, Springer, vol. 64(1), pages 41-72, February.
    13. Chacón, José E. & Rodríguez-Casal, Alberto, 2010. "A note on the universal consistency of the kernel distribution function estimator," Statistics & Probability Letters, Elsevier, vol. 80(17-18), pages 1414-1419, September.
    14. Wu, Colin O., 1997. "A Cross-Validation Bandwidth Choice for Kernel Density Estimates with Selection Biased Data," Journal of Multivariate Analysis, Elsevier, vol. 61(1), pages 38-60, April.
    15. Zhenyu Jiang & Nengxiang Ling & Zudi Lu & Dag Tj⊘stheim & Qiang Zhang, 2020. "On bandwidth choice for spatial data density estimation," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 82(3), pages 817-840, July.
    16. Kozek, A. S. & Yin, J., 2004. "On Gauss quadrature and partial cross validation," Computational Statistics & Data Analysis, Elsevier, vol. 45(3), pages 431-448, April.
    17. Rossi, Barbara & Inoue, Atsushi & Jin, Lu, 2014. "Window Selection for Out-of-Sample Forecasting with Time-Varying Parameters," CEPR Discussion Papers 10168, C.E.P.R. Discussion Papers.
    18. Élie Youndjé & Martin Wells, 2008. "Optimal bandwidth selection for multivariate kernel deconvolution density estimation," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 17(1), pages 138-162, May.
    19. Matthew D. Baird, 2014. "Cross Validation Bandwidth Selection for Derivatives of Multidimensional Densities," Working Papers WR-1060, RAND Corporation.
    20. Sun, Liuquan, 1997. "Bandwidth choice for hazard rate estimators from left truncated and right censored data," Statistics & Probability Letters, Elsevier, vol. 36(2), pages 101-114, December.
    21. Timmermans, Catherine & Delsol, Laurent & von Sachs, Rainer, 2011. "Using Bagidis in nonparametric functional data analysis: predicting from curves with sharp local features," LIDAM Discussion Papers ISBA 2011020, Université catholique de Louvain, Institute of Statistics, Biostatistics and Actuarial Sciences (ISBA).
    22. Daren Cline, 1990. "Optimal kernel estimation of densities," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 42(2), pages 287-303, June.
    23. Salim Bouzebda & Amel Nezzal & Tarek Zari, 2022. "Uniform Consistency for Functional Conditional U -Statistics Using Delta-Sequences," Mathematics, MDPI, vol. 11(1), pages 1-39, December.
    24. Sancetta, Alessio, 2013. "Weak conditions for shrinking multivariate nonparametric density estimators," Journal of Multivariate Analysis, Elsevier, vol. 115(C), pages 285-300.

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