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Reducing the mean squared error of quantile-based estimators by smoothing

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  • Mia Hubert

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  • Irène Gijbels
  • Dina Vanpaemel
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    Abstract

    Many univariate robust estimators are based on quantiles. As already theoretically pointed out by Fernholz (in J. Stat. Plan. Inference 57(1), 29–38, 1997 ), smoothing the empirical distribution function with an appropriate kernel and bandwidth can reduce the variance and mean squared error (MSE) of some quantile-based estimators in small data sets. In this paper we apply this idea on several robust estimators of location, scale and skewness. We propose a robust bandwidth selection and bias reduction procedure. We show that the use of this smoothing method indeed leads to smaller MSEs, also at contaminated data sets. In particular, we obtain better performances for the medcouple which is a robust measure of skewness that can be used for outlier detection in skewed distributions. Copyright Sociedad de Estadística e Investigación Operativa 2013

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    File URL: http://hdl.handle.net/10.1007/s11749-012-0293-3
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    Bibliographic Info

    Article provided by Springer in its journal TEST.

    Volume (Year): 22 (2013)
    Issue (Month): 3 (September)
    Pages: 448-465

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    Handle: RePEc:spr:testjl:v:22:y:2013:i:3:p:448-465

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    Related research

    Keywords: Kernel smoothing; Robust bandwidth selection; Quantile based estimators; Medcouple; Adjusted boxplot; 62G35; 62G30; 62F35;

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    1. Jin Zhang & Xueren Wang, 2009. "Robust normal reference bandwidth for kernel density estimation," Statistica Neerlandica, Netherlands Society for Statistics and Operations Research, vol. 63(1), pages 13-23.
    2. Tamine, Julien & Čížek, Pavel & Härdle, Wolfgang, 2002. "Smoothed L-estimation of regression function," SFB 373 Discussion Papers 2002,88, Humboldt University of Berlin, Interdisciplinary Research Project 373: Quantification and Simulation of Economic Processes.
    3. 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.
    4. Hubert, M. & Vandervieren, E., 2008. "An adjusted boxplot for skewed distributions," Computational Statistics & Data Analysis, Elsevier, vol. 52(12), pages 5186-5201, August.
    5. Mia Hubert & Stephan Van der Veeken, 2010. "Robust classification for skewed data," Advances in Data Analysis and Classification, Springer, vol. 4(4), pages 239-254, December.
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