Abstract
A few historical remarks on the notion of correlation, as well as a brief review of robust estimators of the correlation coefficient are given. A family of M-estimators of the correlation coefficient for bivariate independent component distributions is proposed. Consistency and asymptotic normality of these estimators are established, and the explicit expression for their asymptotic variance is obtained. A minimax variance (in the Huber sense) M-estimator of the correlation coefficient for \(\varepsilon\)-contaminated bivariate normal distributions is designed. Although the structure of this new result generally is similar to the former minimax variance M-estimator of the correlation coefficient proposed by Shevlyakov and Vilchevski (Stat. Probab. Lett. 57, 91–100, 2002b), the efficiency of this new estimator is considerably greater than that of the former one as it generalizes the maximum likelihood estimator of the correlation coefficient of the bivariate normal distribution. Furthermore, highly efficient and robust estimators of correlation are obtained by applying highly efficient and robust estimators of scale. Under the \(\varepsilon\)-contaminated bivariate normal, t- and independent component Cauchy distributions, the proposed robust estimators dominate over the sample correlation coefficient. The comparative analytical and Monte Carlo study of various robust estimators confirm the effectiveness of the proposed M-estimator of the correlation coefficient.
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Acknowledgements
Here it is the right place to thank Prof. Hannu Oja who recommended one of the coauthors of this paper to use the term “independent component distributions” for the corresponding family of bivariate distributions. Also we thank him for the support of our research on robust correlation.
Furthermore, we thank the reviewers whose comments and remarks helped us much to improve our paper.
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Shevlyakov, G., Smirnov, P. (2015). M-Estimators of the Correlation Coefficient for Bivariate Independent Component Distributions. In: Nordhausen, K., Taskinen, S. (eds) Modern Nonparametric, Robust and Multivariate Methods. Springer, Cham. https://doi.org/10.1007/978-3-319-22404-6_9
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DOI: https://doi.org/10.1007/978-3-319-22404-6_9
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