Statistical Papers

, Volume 60, Issue 3, pp 945–962 | Cite as

Logistic Liu Estimator under stochastic linear restrictions

  • Nagarajah VarathanEmail author
  • Pushpakanthie Wijekoon
Regular Article


In order to overcome the problem of multicollinearity in logistic regression, several researchers proposed alternative estimators when exact linear restrictions are available in addition to sample model. However, in practical situations the linear restrictions are not always exact and mostly their nature is stochastic. In this paper, we propose a new estimator called stochastic restricted Liu maximum likelihood estimator (SRLMLE) by incorporating Liu estimator to the logistic regression model when the linear restrictions are stochastic. Moreover, the conditions for superiority of SRLMLE over the maximum likelihood estimator (MLE), stochastic restricted maximum likelihood estimator (SRMLE) and restricted Liu logistic estimator (RLLE) are derived with respect to mean square error criterion. Finally, the performance of the new estimator over MLE, LLE, SRMLE and RLLE is investigated in the sense of scalar mean squared error by conducting a Monte Carlo simulation and using a numerical example.


Logistic regression Multicollinearity Liu estimator Stochastic restricted Liu maximum likelihood estimator Stochastic linear restrictions 


  1. Aguilera AM, Escabias M, Valderrama MJ (2006) Using principal components for estimating logistic regression with high-dimensional multicollinear data. Comput Stat Data Anal 50:1905–1924MathSciNetCrossRefzbMATHGoogle Scholar
  2. Asar Y, Genc A (2015) New shrinkage parameters for the Liu-type logistic estimators. Commun Stat Simul Comput.
  3. Duffy DE, Santner TJ (1989) On the small sample prosperities of norm-restricted maximum likelihood estimators for logistic regression models. Commun Stat Theor Meth 18:959–980CrossRefzbMATHGoogle Scholar
  4. Inan D, Erdogan BE (2013) Liu-type logistic estimator. Commun Stat Simul Comput 42:1578–1586MathSciNetCrossRefzbMATHGoogle Scholar
  5. Kibria BMG (2003) Performance of some new ridge regression estimators. Commun Stat Theor Meth 32:419–435MathSciNetzbMATHGoogle Scholar
  6. Liu K (1993) A new class of biased estimate in linear regression. Commun Stat Theor Meth 22:393–402MathSciNetCrossRefzbMATHGoogle Scholar
  7. Mansson G, Kibria BMG, Shukur G (2012) On Liu estimators for the logit regression model. The Royal Institute of Techonology, Centre of Excellence for Science and Innovation Studies (CESIS), Sweden, Paper No. 259Google Scholar
  8. McDonald GC, Galarneau DI (1975) A Monte Carlo evaluation of some ridge type estimators. J Am Stat Assoc 70:407–416CrossRefzbMATHGoogle Scholar
  9. Nja ME, Ogoke UP, Nduka EC (2013) The logistic regression model with a modified weight function. J Stat Econom Method 2(4):161–171Google Scholar
  10. Rao CR, Toutenburg H, Shalabh, Heumann C (2008) Linear models and generalizations. Springer, BerlinGoogle Scholar
  11. Rao CR, Toutenburg H (1995) Linear models: least squares and alternatives, 2nd edn. Springer, New YorkCrossRefzbMATHGoogle Scholar
  12. Schaefer RL, Roi LD, Wolfe RA (1984) A ridge logistic estimator. Commun Stat Theor Meth 13:99–113CrossRefGoogle Scholar
  13. Şiray GU, Toker S, Kaçiranlar S (2015) On the restricted Liu estimator in logistic regression model. Commun Stat Simul Comput 44:217–232MathSciNetCrossRefzbMATHGoogle Scholar
  14. Theil H, Goldberger AS (1961) On pure and mixed estimation in economics. Int Econ Rev 2(1):65–77.  10.2307/2525589 CrossRefGoogle Scholar
  15. Trenkler G, Toutenburg H (1990) Mean square error matrix comparisons between biased estimators: an overview of recent results. Stat Pap 31:165–179.  10.1007/BF02924687 CrossRefzbMATHGoogle Scholar
  16. Urgan NN, Tez M (2008) Liu estimator in logistic regression when the data are collinear. In: International conference “continuous optimization and knowledge-based technologies”, pp 323–327Google Scholar
  17. Nagarajah V, Wijekoon P (2015) Stochastic restricted maximum likelihood estimator in logistic regression model. Open J Stat 5:837–851.  10.4236/ojs.2015.57082 CrossRefGoogle Scholar
  18. Varathan N, Wijekoon P (2016) Ridge estimator in logistic regression under stochastic linear restriction. Br J Math Comput Sci 15(3):1.  10.9734/BJMCS/2016/24585 CrossRefzbMATHGoogle Scholar
  19. Wu J (2015) Modified restricted Liu estimator in logistic regression model. Comput Stat. doi: 10.1007/s00180-015-0609-3
  20. Wu J, Asar Y (2016) On almost unbiased ridge logistic estimator for the logistic regression model. Hacet J Math Stat 45(3):989–998.  10.15672/HJMS.20156911030 MathSciNetzbMATHGoogle Scholar
  21. Wu J, Asar Y (2015) More on the restricted Liu estimator in the logistic regression model. Commun Stat Simul Comput. doi: 10.1080/03610918.2015.1100735

Copyright information

© Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  1. 1.Postgraduate Institute of ScienceUniversity of PeradeniyaPeradeniyaSri Lanka
  2. 2.Department of Mathematics and StatisticsUniversity of JaffnaJaffnaSri Lanka
  3. 3.Department of Statistics and Computer ScienceUniversity of PeradeniyaPeradeniyaSri Lanka

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