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Minimum quadratic distance density estimation using nonparametric mixtures

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  • Chee, Chew-Seng
  • Wang, Yong
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    Abstract

    Quadratic loss is predominantly used in the literature as the performance measure for nonparametric density estimation, while nonparametric mixture models have been studied and estimated almost exclusively via the maximum likelihood approach. In this paper, we relate both for estimating a nonparametric density function. Specifically, we consider nonparametric estimation of a mixing distribution by minimizing the quadratic distance between the empirical and the mixture distribution, both being smoothed by kernel functions, a technique known as double smoothing. Experimental studies show that the new mixture-based density estimators outperform the popular kernel-based density estimators in terms of mean integrated squared error for practically all the distributions that we studied, thanks to the substantial bias reduction provided by nonparametric mixture models and double smoothing.

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    File URL: http://www.sciencedirect.com/science/article/pii/S0167947312002447
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    Bibliographic Info

    Article provided by Elsevier in its journal Computational Statistics & Data Analysis.

    Volume (Year): 57 (2013)
    Issue (Month): 1 ()
    Pages: 1-16

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    Handle: RePEc:eee:csdana:v:57:y:2013:i:1:p:1-16

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    Web page: http://www.elsevier.com/locate/csda

    Related research

    Keywords: Bandwidth selection; Double smoothing; Kernel-based density estimator; Minimum distance estimation; Nonparametric mixture; Quadratic loss;

    References

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    1. Seo, Byungtae & Lindsay, Bruce G., 2010. "A computational strategy for doubly smoothed MLE exemplified in the normal mixture model," Computational Statistics & Data Analysis, Elsevier, vol. 54(8), pages 1930-1941, August.
    2. Yong Wang, 2007. "On fast computation of the non-parametric maximum likelihood estimate of a mixing distribution," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 69(2), pages 185-198.
    3. Ayanendranath Basu & Bruce Lindsay, 1994. "Minimum disparity estimation for continuous models: Efficiency, distributions and robustness," Annals of the Institute of Statistical Mathematics, Springer, vol. 46(4), pages 683-705, December.
    4. Cao, Ricardo & Cuevas, Antonio & Fraiman, Ricardo, 1995. "Minimum distance density-based estimation," Computational Statistics & Data Analysis, Elsevier, vol. 20(6), pages 611-631, December.
    5. Jones, M.C. & Henderson, D.A., 2009. "Maximum likelihood kernel density estimation: On the potential of convolution sieves," Computational Statistics & Data Analysis, Elsevier, vol. 53(10), pages 3726-3733, August.
    6. Wellner, Jon & Balabdaoui, Fadoua, 2010. "Estimation of a k-monotone density: characterizations, consistency and minimax lower bounds," Economics Papers from University Paris Dauphine 123456789/4650, Paris Dauphine University.
    7. Fadoua Balabdaoui & Jon A. Wellner, 2010. "Estimation of a "k"-monotone density: characterizations, consistency and minimax lower bounds," Statistica Neerlandica, Netherlands Society for Statistics and Operations Research, vol. 64(1), pages 45-70.
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    Cited by:
    1. Chee, Chew-Seng & Wang, Yong, 2014. "Least squares estimation of a k-monotone density function," Computational Statistics & Data Analysis, Elsevier, vol. 74(C), pages 209-216.

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