On the Consequences of Model Misspecification for Biased Samples from the Weibull Distribution

  • George Tzavelas
  • Polychronis Economou
Conference paper
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 231)


Model misspecification is common in practice specially when the sampling mechanism is not known. A sized-biased sample arises in case where the probability of a unit of the population to be chosen in a sample is proportional to some nonnegative weight function w(x) of its size x. In this chapter, we study the model misspecification results when a sized-biased sample from the Weibull distribution is treated as a random one as well as when a random sample is treated as biased. Special attention is paid on the misspecification effects on the parameter estimation and on some of the most important characteristics of the distribution, such as the mean, the median, and the variance. It is proven that when we treat a biased sample as a random one, the parameters are overestimated and in the opposite case are underestimated. Simulation results verify the theoretical findings for small as well as for large samples.


Weighted distributions Misspecification r-size biased sampling Parameter estimation 


  1. 1.
    Blumenthal, S.: Proportional sampling in life length studies. Technometrics 9(2), 205–218 (1966)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Gove, J.H.: Moment and maximum likelihood estimators for weibull distributions under length- and area biased sampling. Environ. Ecol. Stat. 10, 455–467 (2003)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Keith, K.: Mathematical Statistics. Chapman & Hall, Boca Raton (2000)zbMATHGoogle Scholar
  4. 4.
    Kiefer, N.M.: Economic duration data and hazard functions. J. Econ. Lit. 26(2), 646–79 (1988)Google Scholar
  5. 5.
    Lv, J., Liu, J.S.: Model selection principles in misspecified models. J. R. Stat. Soc. Ser. B 76(1), 141–167 (2014)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Morimoto, T., Nakagawa, S., Shinji, S.: Bias in the weibull strength estimation of a sic fiber for the small gauge length case. JSME Int. J. Ser. A 48(4), 194–198 (2005)CrossRefGoogle Scholar
  7. 7.
    Patil, G.: Weighted distributions. Encyclopedia of Environmetrics, pp. 2369–2377. Wiley, Chichester (2002)Google Scholar
  8. 8.
    Tzavelas, G., Douli, M., Economou, P.: Model misspecification effects for biased samples. Metrika 80(2), 171–185 (2017)MathSciNetCrossRefGoogle Scholar
  9. 9.
    White, H.: Maximum likelihood estimation in misspecified models. Econometrica 50(1), 1–25 (1982)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Yi, Y.G., Reid, N.: A note on misspecified estimating functions. Statistica Sinica 20, 1749–1769 (2010)MathSciNetzbMATHGoogle Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of Statistics and Insurance SciencesUniversity of PiraeusPiraeusGreece
  2. 2.Department of Civil EngineeringUniversity of PatrasRio AchaiaGreece

Personalised recommendations