Skip to main content

Sampling Bias Correction in the Model of Mixtures with Varying Concentrations


Model of mixture with varying concentrations is a generalization of the classical finite mixture model in which the mixing probabilities (concentrations) vary from observation to observation. We consider the case when the concentrations of the mixture components are known, but no assumptions on the distributions of the observed variable are made. The problem is to estimate the moments of the components’ distributions. We propose a modification of the Horvitz-Thompson weighting for moments estimation by observations from mixture with varying concentrations in presence of sampling bias. Consistency of obtained estimators is demonstrated. Results of simulations are presented.

This is a preview of subscription content, access via your institution.


  1. Autin F, Pouet Ch (2011) Test on the components of mixture densities. Stat & Risk Modeling 28(4):389–410

    Article  MATH  MathSciNet  Google Scholar 

  2. Lohr SL (2009) Sampling: design and analysis, 2nd edn. Cengage Learning, Boston

    Google Scholar 

  3. Maiboroda R, Kubaichuk O (2005) Improved estimators for moments constructed from observations of a mixture. Theory Probab Math Stat 70:83–92

    Article  MathSciNet  Google Scholar 

  4. Maiboroda R, Sugakova O (2012) Statistics of mixtures with varying concentrations with application to DNA microarray data analysis. J Nonparametr Stat 24(1):201–215

    Article  MATH  MathSciNet  Google Scholar 

  5. McLachlan GJ, Peel D (2000) Finite mixture models. Wiley, New York

    Book  MATH  Google Scholar 

  6. Nocedal J, Wright SJ (2006) Numerical optimization, 2nd edn. Springer, NY

    MATH  Google Scholar 

  7. Pokhyl’ko D (2005) Wavelet estimators of a density constructed from observations of a mixture. Theory Probab Math Stat 70:135–145

    Article  Google Scholar 

  8. Shao J (2003) Mathematical statistics. Springer, NY

    Book  MATH  Google Scholar 

  9. Van den Vaart AW (2000) Asymptotic statistics. Cambridge University Press

  10. Vapnik V, Čhervonenkis A (1974) Theory of pattern recognition. Nauka, Moskow [in Russian]

    MATH  Google Scholar 

Download references

Author information



Corresponding author

Correspondence to Rostyslav Maiboroda.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Sugakova, O., Maiboroda, R. Sampling Bias Correction in the Model of Mixtures with Varying Concentrations. Methodol Comput Appl Probab 17, 223–234 (2015).

Download citation


  • Biased sampling
  • Horvitz-Thompson weights
  • Finite mixture model
  • Mixture with varying concentrations
  • Consistency
  • Nonparametric estimation

AMS 2000 Subject Classification

  • MSC 62G05
  • MSC 62D05
  • MSC 65C60