Kybernetika 48 no. 4, 768-794, 2012

Robust median estimator for generalized linear models with binary responses

Tomáš Hobza, Leandro Pardo and Igor Vajda


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.


asymptotic normality, consistency, robustness, generalized linear models, binary responses, statistical smoothing, statistical enhancing, maximum likelihood estimator, median estimator, efficiency


62F10, 62F12, 62F35


  1. G. Adimari and L. Ventura: Robust inference for generalized linear models with application to logistic regression. Statist. Probab. Lett. 55 (2001), 4, 413-419.   CrossRef
  2. A. M. Bianco and 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.   CrossRef
  3. C. Croux and G. Haesbroeck: Implementing the Bianco and Yohai estimator for logistic regression. Computat. Statist. Data Anal. 44 (2003), 273-295.   CrossRef
  4. J. E. Dennis, Jr. and R. B. Schnabel: Numerical Methods for Unconstrained Optimization and Nonlinear Equations. Prentice-Hall, Englewood Cliffs, New Jersey 1983.   CrossRef
  5. D. Gervini: Robust adaptive estimators for binary regression models. J. Statist. Plann. Inference 131 (2005), 297-311.   CrossRef
  6. F. R. Hampel, P. J. Rousseeuw, E. M. Ronchetti and W. A. Stahel: Robust Statistics: The Approach Based on Influence Functions. Wiley, New York 1986.   CrossRef
  7. T. Hobza, L. Pardo and I. Vajda: Median Estimators in Generalized Logistic Regression. Research Report DAR-UTIA 2005/40. Institute of Information Theory, Prague 2005 (available at   CrossRef
  8. T. Hobza, L. Pardo and I. Vajda: Robust Median Estimators in Logistic Regression. Research Report DAR-UTIA 2006/31. Institute of Information Theory, Prague 2006 (available at   CrossRef
  9. T. Hobza, L. Pardo and I. Vajda: Robust median estimator in logistic regression. J. Statist. Plann. Inference 138 (2008), 3822-3840.   CrossRef
  10. J. Jurečková and P. K. Sen: Robust Statistical Procedures. Wiley, New York 1996.   CrossRef
  11. N. Kordzakhia, G. D. Mishra and L. Reiersølmoen: Robust estimation in the logistic regression model. J. Statist. Plann. Inference 98 (2001), 211-223.   CrossRef
  12. F. Liese and I. Vajda: $M$-estimators of structural parameters in pseudolinear models. Appl. Math. 44 (1999), 245-270.   CrossRef
  13. F. Liese and I. Vajda: A general asymptotic theory of $M$-estimators I. Math. Methods Statist. 12 (2003) 454-477.   CrossRef
  14. F. Liese and I. Vajda: A general asymptotic theory of $M$-estimators II. Math. Methods Statist. 13 (2004) 82-95.   CrossRef
  15. P. McCullagh and J. A. Nelder: Generalized Linear Models. Chapman and Hall, London 1989.   CrossRef
  16. M. L. Menéndez, L. Pardo and 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.   CrossRef
  17. J. Moré, G. Burton and H. Kenneth: User Guide for MINPACK-1. Argonne National Laboratory Report ANL-80-74, Argonne 1980.   CrossRef
  18. S. Morgenthaler: Least-absolute-deviations fits for generalized linear models. Biometrika 79 (1992), 747-754.   CrossRef
  19. J. Nagler: Scobit: An alternative estimator to logit and Probit. Amer. J. Political Sci. 38 (1994), 1, 230-255.   CrossRef
  20. D. Pregibon: Resistant lits for some commonly used logistic models with medical applications. Biometrics 38 (1982), 485-498.   CrossRef
  21. R. L. Prentice: A generalization of the probit and logit methods for dose-response curves. Biometrika 32 (1976), 761-768.   CrossRef
  22. P. J. Rousseeuw and A. Christmann: Robustness against separation and outliers in logistic regression. Comput. Statist. Data Anal. 43 (2003), 315-332.   CrossRef