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Robust median estimator for generalized linear models with binary responses
Kybernetika, Tome 48 (2012) no. 4, pp. 768-794 Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

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The paper investigates generalized linear models (GLM's) with binary responses such as the logistic, probit, log-log, complementary log-log, scobit and power logit models. It introduces a median estimator of the underlying structural parameters of these models based on statistically smoothed binary responses. Consistency and asymptotic normality of this estimator are proved. Examples of derivation of the asymptotic covariance matrix under the above mentioned models are presented. Finally some comments concerning a method called enhancement and robustness of median estimator are given and results of simulation experiment comparing behavior of median estimator with other robust estimators for GLM's known from the literature are reported.
The paper investigates generalized linear models (GLM's) with binary responses such as the logistic, probit, log-log, complementary log-log, scobit and power logit models. It introduces a median estimator of the underlying structural parameters of these models based on statistically smoothed binary responses. Consistency and asymptotic normality of this estimator are proved. Examples of derivation of the asymptotic covariance matrix under the above mentioned models are presented. Finally some comments concerning a method called enhancement and robustness of median estimator are given and results of simulation experiment comparing behavior of median estimator with other robust estimators for GLM's known from the literature are reported.
Classification : 62F10, 62F12, 62F35
Keywords: generalized linear models; binary responses; statistical smoothing; statistical enhancing; maximum likelihood estimator; median estimator; consistency; asymptotic normality; efficiency; robustness 
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Hobza, Tomáš; Pardo, Leandro; Vajda, Igor. Robust median estimator for generalized linear models with binary responses. Kybernetika, Tome 48 (2012) no. 4, pp. 768-794. http://geodesic.mathdoc.fr/item/KYB_2012_48_4_a7/

[1] G. Adimari, L. Ventura: Robust inference for generalized linear models with application to logistic regression. Statist. Probab. Lett. 55 (2001), 4, 413-419. | DOI | MR | Zbl

[2] A. M. Bianco, V. J. Yohai: Robust estimation in the logistic regression model. In: Robust Statistics, Data Analysis, and Computer Intensive Methods (Schloss Thurnau, 1994), pp. 17-34. Lecture Notes in Statist. 109 Springer, New York 1996. | MR | Zbl

[3] C. Croux, G. Haesbroeck: Implementing the Bianco and Yohai estimator for logistic regression. Computat. Statist. Data Anal. 44 (2003), 273-295. | DOI | MR

[4] J. E. Dennis, Jr., R. B. Schnabel: Numerical Methods for Unconstrained Optimization and Nonlinear Equations. Prentice-Hall, Englewood Cliffs, New Jersey 1983. | MR | Zbl

[5] D. Gervini: Robust adaptive estimators for binary regression models. J. Statist. Plann. Inference 131 (2005), 297-311. | DOI | MR | Zbl

[6] F. R. Hampel, P. J. Rousseeuw, E. M. Ronchetti, W. A. Stahel: Robust Statistics: The Approach Based on Influence Functions. Wiley, New York 1986. | MR | Zbl

[7] T. Hobza, L. Pardo, I. Vajda: Median Estimators in Generalized Logistic Regression. Research Report DAR-UTIA 2005/40. Institute of Information Theory, Prague 2005 (available at http://dar.site.cas.cz/?publication=1007)

[8] T. Hobza, L. Pardo, I. Vajda: Robust Median Estimators in Logistic Regression. Research Report DAR-UTIA 2006/31. Institute of Information Theory, Prague 2006 (available at http://dar.site.cas.cz/?publication=1089)

[9] T. Hobza, L. Pardo, I. Vajda: Robust median estimator in logistic regression. J. Statist. Plann. Inference 138 (2008), 3822-3840. | DOI | MR | Zbl

[10] J. Jurečková, P. K. Sen: Robust Statistical Procedures. Wiley, New York 1996. | MR | Zbl

[11] N. Kordzakhia, G. D. Mishra, L. Reiersølmoen: Robust estimation in the logistic regression model. J. Statist. Plann. Inference 98 (2001), 211-223. | DOI | MR

[12] F. Liese, I. Vajda: $M$-estimators of structural parameters in pseudolinear models. Appl. Math. 44 (1999), 245-270. | DOI | MR | Zbl

[13] F. Liese, I. Vajda: A general asymptotic theory of $M$-estimators I. Math. Methods Statist. 12 (2003) 454-477. | MR

[14] F. Liese, I. Vajda: A general asymptotic theory of $M$-estimators II. Math. Methods Statist. 13 (2004) 82-95. | MR | Zbl

[15] P. McCullagh, J. A. Nelder: Generalized Linear Models. Chapman and Hall, London 1989. | MR | Zbl

[16] M. L. Menéndez, L. Pardo, M. C. Pardo: Preliminary test estimators and phi-divergence measures in generalized linear models with binary data. J. Multivariate Anal. 99 (2009), 10, 2265-2284. | DOI | MR | Zbl

[17] J. Moré, G. Burton, H. Kenneth: User Guide for MINPACK-1. Argonne National Laboratory Report ANL-80-74, Argonne 1980.

[18] S. Morgenthaler: Least-absolute-deviations fits for generalized linear models. Biometrika 79 (1992), 747-754. | DOI | Zbl

[19] J. Nagler: Scobit: An alternative estimator to logit and Probit. Amer. J. Political Sci. 38 (1994), 1, 230-255. | DOI

[20] D. Pregibon: Resistant lits for some commonly used logistic models with medical applications. Biometrics 38 (1982), 485-498. | DOI

[21] R. L. Prentice: A generalization of the probit and logit methods for dose-response curves. Biometrika 32 (1976), 761-768.

[22] P. J. Rousseeuw, A. Christmann: Robustness against separation and outliers in logistic regression. Comput. Statist. Data Anal. 43 (2003), 315-332. | DOI | MR