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Estimation of integrated squared density derivatives from a contaminated sample

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  • A. Delaigle
  • I. Gijbels

Abstract

Summary. We propose a kernel estimator of integrated squared density derivatives, from a sample that has been contaminated by random noise. We derive asymptotic expressions for the bias and the variance of the estimator and show that the squared bias term dominates the variance term. This coincides with results that are available for non‐contaminated observations. We then discuss the selection of the bandwidth parameter when estimating integrated squared density derivatives based on contaminated data. We propose a data‐driven bandwidth selection procedure of the plug‐in type and investigate its finite sample performance via a simulation study.

Suggested Citation

  • A. Delaigle & I. Gijbels, 2002. "Estimation of integrated squared density derivatives from a contaminated sample," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 64(4), pages 869-886, October.
    • Handle: RePEc:bla:jorssb:v:64:y:2002:i:4:p:869-886
    DOI: 10.1111/1467-9868.00366
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    Cited by:

    1. Peter Hall & Tapabrata Maiti, 2008. "Non‐parametric inference for clustered binary and count data when only summary information is available," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 70(4), pages 725-738, September.
    2. J. Beirlant & G. Claeskens & C. Croux & H. Degryse & H. Dewachter & G. Dhaene & J. Dhaene & I. Gijbels & M. Goovaerts & M. Hubert & F. Roodhooft & W. Schouten & M. Willekens, 2005. "Managing Uncertainty: Financial, Actuarial and Statistical Modeling," Review of Business and Economic Literature, KU Leuven, Faculty of Economics and Business (FEB), Review of Business and Economic Literature, vol. 0(1), pages 23-48.
    3. Mia Hubert & Irène Gijbels & Dina Vanpaemel, 2013. "Reducing the mean squared error of quantile-based estimators by smoothing," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 22(3), pages 448-465, September.
    4. Aurore Delaigle & Peter Hall, 2016. "Methodology for non-parametric deconvolution when the error distribution is unknown," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 78(1), pages 231-252, January.
    5. Bissantz, Nicolai & Birke, Melanie, 2009. "Asymptotic normality and confidence intervals for inverse regression models with convolution-type operators," Journal of Multivariate Analysis, Elsevier, vol. 100(10), pages 2364-2375, November.
    6. Delaigle, A. & Gijbels, I., 2004. "Practical bandwidth selection in deconvolution kernel density estimation," Computational Statistics & Data Analysis, Elsevier, vol. 45(2), pages 249-267, March.
    7. Bissantz, Nicolai & Birke, Melanie, 2008. "Asymptotic normality and confidence intervals for inverse regression models with convolution-type operators," Technical Reports 2008,17, Technische Universität Dortmund, Sonderforschungsbereich 475: Komplexitätsreduktion in multivariaten Datenstrukturen.
    8. Parmeter, Christopher F., 2008. "The effect of measurement error on the estimated shape of the world distribution of income," Economics Letters, Elsevier, vol. 100(3), pages 373-376, September.
    9. Mark A. van de Wiel & Kyung In Kim, 2007. "Estimating the False Discovery Rate Using Nonparametric Deconvolution," Biometrics, The International Biometric Society, vol. 63(3), pages 806-815, September.
    10. Tiee-Jian Wu & Chih-Yuan Hsu & Huang-Yu Chen & Hui-Chun Yu, 2014. "Root $$n$$ n estimates of vectors of integrated density partial derivative functionals," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 66(5), pages 865-895, October.
    11. William Horrace & Christopher Parmeter, 2011. "Semiparametric deconvolution with unknown error variance," Journal of Productivity Analysis, Springer, vol. 35(2), pages 129-141, April.
    12. Delaigle, A. & Gijbels, I., 2006. "Data-driven boundary estimation in deconvolution problems," Computational Statistics & Data Analysis, Elsevier, vol. 50(8), pages 1965-1994, April.
    13. Birke, Melanie & Bissantz, Nicolai & Holzmann, Hajo, 2008. "Confidence bands for inverse regression models with application to gel electrophoresis," Technical Reports 2008,16, Technische Universität Dortmund, Sonderforschungsbereich 475: Komplexitätsreduktion in multivariaten Datenstrukturen.
    14. Stéphane Bonhomme & Jean-Marc Robin, 2010. "Generalized Non-Parametric Deconvolution with an Application to Earnings Dynamics," The Review of Economic Studies, Review of Economic Studies Ltd, vol. 77(2), pages 491-533.
    15. Yousri Slaoui, 2021. "Data-driven Deconvolution Recursive Kernel Density Estimators Defined by Stochastic Approximation Method," Sankhya A: The Indian Journal of Statistics, Springer;Indian Statistical Institute, vol. 83(1), pages 312-352, February.
    16. Bissantz, Nicolai & Dümbgen, Lutz & Holzmann, Hajo & Munk, Axel, 2007. "Nonparametric confidence bands in deconvolution density estimation," Technical Reports 2007,03, Technische Universität Dortmund, Sonderforschungsbereich 475: Komplexitätsreduktion in multivariaten Datenstrukturen.
    17. Staudenmayer, John & Ruppert, David & Buonaccorsi, John P., 2008. "Density Estimation in the Presence of Heteroscedastic Measurement Error," Journal of the American Statistical Association, American Statistical Association, vol. 103, pages 726-736, June.
    18. Holzmann, Hajo & Bissantz, Nicolai & Munk, Axel, 2007. "Density testing in a contaminated sample," Journal of Multivariate Analysis, Elsevier, vol. 98(1), pages 57-75, January.
    19. Yatracos, Yannis G., 2018. "PLUG-IN L2-UPPER ERROR BOUNDS IN DECONVOLUTION, FOR A MIXING DENSITY ESTIMATE IN Rd AND FOR ITS DERIVATIVES," IRTG 1792 Discussion Papers 2018-061, Humboldt University of Berlin, International Research Training Group 1792 "High Dimensional Nonstationary Time Series".
    20. Cornelis J. Potgieter, 2020. "Density deconvolution for generalized skew-symmetric distributions," Journal of Statistical Distributions and Applications, Springer, vol. 7(1), pages 1-20, December.
    21. D. Ioannides & Eric Matzner-Løber, 2009. "Regression quantiles with errors-in-variables," Journal of Nonparametric Statistics, Taylor & Francis Journals, vol. 21(8), pages 1003-1015.

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