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Weighted likelihood estimation of multivariate location and scatter

Author

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  • Claudio Agostinelli

    (University of Trento)

  • Luca Greco

    (University of Sannio)

Abstract

A novel approach to obtain weighted likelihood estimates of multivariate location and scatter is discussed. A weighting scheme is proposed that is based on the univariate distribution of the Mahalanobis distances rather than the multivariate distribution of the data at the assumed model. This strategy allows to avoid the curse of dimensionality affecting multivariate non-parametric density estimation, that is involved in the construction of the weights through the Pearson residuals. Asymptotic properties of the proposed weighted likelihood estimator are also discussed. Then, weighted likelihood-based outlier detection rules and robust dimensionality reduction techniques are developed. The effectiveness of the methodology is illustrated through some numerical studies and real data examples.

Suggested Citation

  • Claudio Agostinelli & Luca Greco, 2019. "Weighted likelihood estimation of multivariate location and scatter," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 28(3), pages 756-784, September.
    • Handle: RePEc:spr:testjl:v:28:y:2019:i:3:d:10.1007_s11749-018-0596-0
    DOI: 10.1007/s11749-018-0596-0
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    References listed on IDEAS

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    1. Agostinelli, Claudio & Markatou, Marianthi, 1998. "A one-step robust estimator for regression based on the weighted likelihood reweighting scheme," Statistics & Probability Letters, Elsevier, vol. 37(4), pages 341-350, March.
    2. M.C. Jones & D.A. Henderson, 2007. "Miscellanea Kernel-Type Density Estimation on the Unit Interval," Biometrika, Biometrika Trust, vol. 94(4), pages 977-984.
    3. Ayanendranath Basu & Bruce Lindsay, 1994. "Minimum disparity estimation for continuous models: Efficiency, distributions and robustness," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 46(4), pages 683-705, December.
    4. Marco Riani & Anthony C. Atkinson & Andrea Cerioli, 2009. "Finding an unknown number of multivariate outliers," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 71(2), pages 447-466, April.
    5. Cerioli, Andrea & Farcomeni, Alessio & Riani, Marco, 2013. "Robust distances for outlier-free goodness-of-fit testing," Computational Statistics & Data Analysis, Elsevier, vol. 65(C), pages 29-45.
    6. Cerioli, Andrea & Farcomeni, Alessio, 2011. "Error rates for multivariate outlier detection," Computational Statistics & Data Analysis, Elsevier, vol. 55(1), pages 544-553, January.
    7. Song Chen, 2000. "Probability Density Function Estimation Using Gamma Kernels," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 52(3), pages 471-480, September.
    8. He, Xuming & Fung, Wing K., 2000. "High Breakdown Estimation for Multiple Populations with Applications to Discriminant Analysis," Journal of Multivariate Analysis, Elsevier, vol. 72(2), pages 151-162, February.
    9. Agostinelli, Claudio, 2006. "Notes on Pearson residuals and weighted likelihood estimating equations," Statistics & Probability Letters, Elsevier, vol. 76(17), pages 1930-1934, November.
    10. Park, Chanseok & Basu, Ayanendranath & G. Lindsay, Bruce, 2002. "The residual adjustment function and weighted likelihood: a graphical interpretation of robustness of minimum disparity estimators," Computational Statistics & Data Analysis, Elsevier, vol. 39(1), pages 21-33, March.
    11. Luca Greco & Alessio Farcomeni, 2016. "A plug-in approach to sparse and robust principal component analysis," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 25(3), pages 449-481, September.
    12. Salibian-Barrera, Matias & Van Aelst, Stefan & Willems, Gert, 2006. "Principal Components Analysis Based on Multivariate MM Estimators With Fast and Robust Bootstrap," Journal of the American Statistical Association, American Statistical Association, vol. 101, pages 1198-1211, September.
    13. Croux, Christophe & Haesbroeck, Gentiane, 1999. "Influence Function and Efficiency of the Minimum Covariance Determinant Scatter Matrix Estimator," Journal of Multivariate Analysis, Elsevier, vol. 71(2), pages 161-190, November.
    14. Cerioli, Andrea, 2010. "Multivariate Outlier Detection With High-Breakdown Estimators," Journal of the American Statistical Association, American Statistical Association, vol. 105(489), pages 147-156.
    15. Hubert, Mia & Van Driessen, Katrien, 2004. "Fast and robust discriminant analysis," Computational Statistics & Data Analysis, Elsevier, vol. 45(2), pages 301-320, March.
    16. Kuchibhotla, Arun Kumar & Basu, Ayanendranath, 2015. "A general set up for minimum disparity estimation," Statistics & Probability Letters, Elsevier, vol. 96(C), pages 68-74.
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    Cited by:

    1. Giovanni Saraceno & Claudio Agostinelli & Luca Greco, 2021. "Robust estimation for multivariate wrapped models," METRON, Springer;Sapienza Università di Roma, vol. 79(2), pages 225-240, August.
    2. Luca Greco, 2022. "Robust fitting of mixtures of GLMs by weighted likelihood," AStA Advances in Statistical Analysis, Springer;German Statistical Society, vol. 106(1), pages 25-48, March.
    3. Jonathan Gillard & Emily O’Riordan & Anatoly Zhigljavsky, 2023. "Polynomial whitening for high-dimensional data," Computational Statistics, Springer, vol. 38(3), pages 1427-1461, September.
    4. Luca Greco & Antonio Lucadamo & Claudio Agostinelli, 2021. "Weighted likelihood latent class linear regression," Statistical Methods & Applications, Springer;Società Italiana di Statistica, vol. 30(2), pages 711-746, June.
    5. Luca Greco & Giovanni Saraceno & Claudio Agostinelli, 2021. "Robust Fitting of a Wrapped Normal Model to Multivariate Circular Data and Outlier Detection," Stats, MDPI, vol. 4(2), pages 1-18, June.

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