Electronic Journal of Statistics

A Wald-type test statistic for testing linear hypothesis in logistic regression models based on minimum density power divergence estimator

Ayanendranath Basu, Abhik Ghosh, Abhijit Mandal, Nirian Martín, and Leandro Pardo

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In this paper a robust version of the classical Wald test statistics for linear hypothesis in the logistic regression model is introduced and its properties are explored. We study the problem under the assumption of random covariates although some ideas with non random covariates are also considered. A family of robust Wald type tests are considered here, where the minimum density power divergence estimator is used instead of the maximum likelihood estimator. We obtain the asymptotic distribution and also study the robustness properties of these Wald type test statistics. The robustness of the tests is investigated theoretically through the influence function analysis as well as suitable practical examples. It is theoretically established that the level as well as the power of the Wald-type tests are stable against contamination, while the classical Wald type test breaks down in this scenario. Some classical examples are presented which numerically substantiate the theory developed. Finally a simulation study is included to provide further confirmation of the validity of the theoretical results established in the paper.

Article information

Electron. J. Statist. Volume 11, Number 2 (2017), 2741-2772.

Received: July 2016
First available in Project Euclid: 4 July 2017

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Zentralblatt MATH identifier

Primary: 62F35: Robustness and adaptive procedures 662F05

Influence function logistic regression minimum density power divergence estimators random explanatory variables robustness Wald-type test statistics

Creative Commons Attribution 4.0 International License.


Basu, Ayanendranath; Ghosh, Abhik; Mandal, Abhijit; Martín, Nirian; Pardo, Leandro. A Wald-type test statistic for testing linear hypothesis in logistic regression models based on minimum density power divergence estimator. Electron. J. Statist. 11 (2017), no. 2, 2741--2772. doi:10.1214/17-EJS1295. https://projecteuclid.org/euclid.ejs/1499133753

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  • Basu, A., Harris, I. R., Hjort, N. L. and Jones, M. C. (1998). Robust and efficient estimation by minimizing a density power, divergence.Biometrika,85, 549–559.
  • Basu, A., Shioya, H. and Park, C., (2011).The minimum distance approach. Monographs on Statistics and Applied Probability. CRC Press, Boca Raton.
  • Basu, A., Mandal, A., Martín, N. and Pardo, L. (2013). Testing statistical hypotheses based on the density power, divergence.Annals of the Institute of Statistical Mathematics,65, 319–348.
  • Basu, A., Mandal, A., Martín, N. and Pardo, L. (2015). Robust tests for the equality of two normal means based on the density power, divergence.Metrika,78, 611–634.
  • Basu, A., Mandal, A., Martín, N., Pardo, L. (2016). Generalized Wald-type tests based on minimum density power divergence, estimators.Statistics,50, 1–26.
  • Bianco, A. M. and Martinez, E. (2009). Robust testing in the logistic regression, model.Computational Statistics and Data Analysis,53, 4095–4105.
  • Bianco, A. M. and Yohai, V. J. (1996). Robust Estimation in the Logistic Regression Model,in Robust Statistics, Data Analysis and Computer Intensive Methods, 17–34; Lecture Notes in Statistics 109, Springer Verlag, Ed. H. Rieder. New, York
  • Bondell, H. D. (2005). Minimum distance estimation for the logistic regression, model.Biometrika,92, 724–731.
  • Bondell, H. D. (2008). A characteristic function approach to the biased sampling model, with application to robust logistic, regression.Journal of Statistical Planning and Inference,138, 742–755.
  • Brown, B. W. (1980). Prediction analysis for binary data, inBiostatistics Casebook, R. G. Miller, B. Efron, B. W. Brown and L. E. Moses, eds., John Wiley and Sons, New York, pp. 3–18.
  • Carroll, R. J. and Pederson, S. (1993). On Robustness in the logistic regression, model.Journal of the Royal Statistical Society: Series B,55, 669–706.
  • Copas, J. B. (1988). Binary regression models for contaminated, data.Journal of the Royal Statistical Society: Series B,50, 225–265.
  • Croux, C. and Haesbroeck, G. (2003). Implementing the Bianco and Yohai estimator for logistic, regression.Computational Statistics and Data Analysis,44, 273–295.
  • Christmann, A. (1994). Least Median of Weighted Squares in Logistic Regression with Large, Strata.Biometrika,81, 413–417.
  • Christmann, A. and Rousseeuw, P. J. (2001). Measuring overlap in binary, regression,Comp. Statistics & Data Analysis,37, 65–75.
  • Cook, R. D. and Weisberg, S., (1982).Residuals and Influence in Regression, Chapman & Hall, London.
  • Feigl, P. and Zelen, M. (1965). Estimation of exponential probabilities with concomitant, information.Biometrics,21, 826–838.
  • Finney, D. J. (1947). The estimation from individual records of the relationship between dose and quantal, response.Biometrika,34, 320–334.
  • Ghosh, A. and Basu, A. (2013). Robust Estimation for Independent but Non-Homogeneous Observations using Density Power Divergence with application to Linear, Regression.Electronic Journal of Statistics,7, 2420–2456.
  • Ghosh, A. and Basu, A. (2015). Robust estimation for non-homogeneous data and the selection of the optimal tuning parameter: the density power divergence, approach.Journal of Applied Statistics,42(9), 2056–2072.
  • Ghosh, A., Basu, A. and Pardo, L. (2015). On the robustness of a divergence based test of simple statistical, hypotheses,Journal of Statistical Planning and Inference,161, 91–108.
  • Ghosh, A., Harris, I. R., Maji, A., Basu, A. and Pardo, L. (2016a). A Generalized Divergence for Statistical, Inference.Bernoulli,23(4A), 2746–2783.
  • Ghosh, A., Mandal, A., Martín, N. and Pardo, L. (2016b). Influence Analysis of Robust Wald-type, Tests.Journal of Multivariate Analysis,147, 102–126.
  • Ghosh, A., Martin, N., Basu, A. and Pardo, L. (2016c). A New Class of Robust Two-Sample Wald-Type, Tests.eprint arXiv:1702.04552.
  • A. Ghosh and A. Basu (2016d). Robust Estimation in Generalized Linear Models: The Density Power Divergence, Approach.TEST,25(2), 269–290.
  • Greene, W. H., (2003).Econometric Analysis.Upper SaddleRiver: Prentice Hall Inc.
  • Hampel, F. R., Ronchetti, E. M., Rousseeuw, P. J. and Stahel, W. A., (1986).Robust statistics: The approach based on influence functions.Wiley Series in Probability and Mathematical Statistics: Probability and Mathematical Statistics. John Wiley & Sons, Inc., New York.
  • Hobza, T., Pardo, L. and I. Vajda (2008). Robust median estimator in logistic, regression.Journal of Statistical Planning and Inference,138, 3822–3840.
  • Hobza, T., Martín, N. and Pardo, L. (2017). A Wald-type test statistic based on robust modified median estimator in logistic regression, models.Journal of Statistical Computation and Simulation,87, 2309–2333.
  • Hong, C. and Y. Kim (2001). Automatic selection of the tuning parameter in the minimum density power divergence, estimation.Journal of the Korean Statistical Association,30, 453–465.
  • Johnson, W. (1985). Influence measures for logistic regression: Another point of, view.Biometrics,72, 59–65.
  • Maronna, R. A., Martin, R. D. and Yohai, V. J., (2006).Robust Statistics. Theory and Methods. Wiley Series in Probability and Statistics.
  • Martín, N. and Pardo, L. (2009). On the asymptotic distribution of Cook’s distance in logistic regression, models.Journal of Applied Statistics,36, 1119–1146.
  • Muñoz-Garcia, J., Muñoz-Pichardo, J. M. and Pardo, L. (2006). Cressie and Read power-divergences as influence measures for logistic regression, models.Comput. Statist. Data Anal.,50, 3199–3221.
  • Morgenthaler, S. (1992), Least-absolute-deviations fits for generalized linear, models.Biometrika,79, 747–754.
  • Pregibon, D. (1981). Logistic regression, diagnostics.Annals of Statistics,9, 705–724.
  • Pregibon, D. (1982), Resistant lits for some commonly used logistic models with medical, applications,Biometrics,38,485–498.
  • Rousseeuw, P. J. and Christmann, A. (2003), Robustness against separation and outliers in logistic, regression.Computational Statistics and Data Analysis,43, 315–332.
  • Rousseeuw, P. J. and Ronchetti, E. (1979) The influence curve for, tests.Research Report21, Fachgruppe fur Statistik, ETH Zurich.
  • Toma, A. and Broniatowski, M. (2011). Dual divergence estimators and tests: Robustness, results.Journal of Multivariate Analysis,102, 20–36.
  • Victoria-Feser, M. (2000). Robust Logistic Regression for Binomial Responses. Available at, SSRN:https://ssrn.com/abstract=1763301orhttp://dx.doi.org/10.2139/ssrn.1763301
  • Yohai, V. J. (1987). High breakdown-point and high efficiency robust estimates for, regression.Annals Statistics,15,692–656.
  • Warwick, J. and Jones, M. C. (2005). Choosing a robustness tuning, parameter.Journal of Statistical Computation and Simulation,75, 581–588.
  • Zelterman, D., (2005).Models for Discrete Data. Oxford University Press, New York.