Skip to main content
SpringerLink
Log in

Complementary Beta Regression Model for Fitting Bounded Data

  • Original Article
  • Published:
Journal of Statistical Theory and Practice Aims and scope Submit manuscript

Abstract

The beta regression model is the commonly used approach for modeling data in the unit interval. However, there are in the literature some useful and interesting alternatives which often under-used. This paper proposes a novel regression model for bounded data, where the response variable is complementary beta distributed with mean and dispersion parameters. The proposed regression model is a natural strong competitor of the beta regression model. The maximum likelihood method is used for estimating the model parameters. A Monte Carlo experiment is conducted to evaluate the performances of these estimators in finite samples. The usefulness of the new regression model is illustrated by two real applications.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8

Similar content being viewed by others

References

  1. Akaike H (1974) A new look at the statistical model identification. IEEE Trans Autom Control 19(6):716–723

    Article  MathSciNet  Google Scholar 

  2. Atkinson A (1985) Plots. transformation and regression. Clarendon Press, Oxford

    Google Scholar 

  3. Atkinson AC (1981) Two graphical displays for outlying and influential observations in regression. Biometrika 68(1):13–20

    Article  MathSciNet  Google Scholar 

  4. Cepeda-Cuervo E (2001) Variability modeling in generalized linear models. Ph.D. thesis, Mathematics Institute, Universidade Federal do Rio de Janeiro

  5. Cox DR, Hinkley DV (1979) Theoretical statistics. CRC Press, Boca Raton

    Book  Google Scholar 

  6. Dixon AC (1902) Summation of a certain series. Proc London Math Soc 35(1):284–291

    Article  MathSciNet  Google Scholar 

  7. Dunn PK, Smyth GK (1996) Randomized quantile residuals. J Comput Gr Stat 5(3):236–244

    Google Scholar 

  8. Ferrari S, Cribari-Neto F (2004) Beta regression for modelling rates and proportions. J Appl Stat 31(7):799–815

    Article  MathSciNet  Google Scholar 

  9. Gradshteyn I, Ryzhik I (1994) Table of integrals, series, and products. Academic Press, Cambridge

    MATH  Google Scholar 

  10. Gupta AK, Nadarajah S (2004) Handbook of beta distribution and its applications. CRC Press, Boca Raton

    Book  Google Scholar 

  11. Hannan EJ, Quinn BG (1979) The determination of the order of an Autoregression. J R Stat Soc Ser B (Methodol) 41(2):190–195

    MathSciNet  MATH  Google Scholar 

  12. Held L, Sabanés Bové D (2014) Applied statistical inference - likelihood and bayes. Springer, Ney York

    Book  Google Scholar 

  13. Hosking JRM (1990) L-Moments: analysis and estimation of distributions using linear combinations of order statistics. J R Stat Soc Ser B (Methodol) 52(1):105–124

    MathSciNet  MATH  Google Scholar 

  14. Hosking JRM (1992) Moments or L Moments? An example comparing two measures of distributional shape. Am Stat 46(3):186–189

    Google Scholar 

  15. Iacobellis V (2008) Probabilistic model for the estimation of T year flow duration curves. Water Resour Res 44(2):02413

    Article  Google Scholar 

  16. Johnson NL, Kotz S, Balakrishnan N (1995) Continuous univariate distributions, vol 2, 2nd edn. Wiley, New York

    MATH  Google Scholar 

  17. Jones MC (2002) The complementary Beta distribution. J Stat Plan Inference 104(2):329–337

    Article  MathSciNet  Google Scholar 

  18. Jones MC (2018) Families of complementary distributions. Stat Probabil Lett 141:74–81

    Article  MathSciNet  Google Scholar 

  19. Jorgensen B (1997) The theory of dispersion models. Chapman & Hall, USA

    MATH  Google Scholar 

  20. Kamps U (1991) A general recurrence relation for moments of order statistics in a class of probability distributions and characterizations. Metrika 38:215–225

    Article  MathSciNet  Google Scholar 

  21. Kieschnick R, McCullough BD (2003) Regression analysis of variates observed on (0, 1): Percentages, proportions and fractions. Stat Modell 3(3):193–213

    Article  MathSciNet  Google Scholar 

  22. Lemonte AJ, Bazán JL (2015) New class of Johnson distributions and its associated regression model for rates and proportions. Biometrical J 58(4):727–746

    Article  MathSciNet  Google Scholar 

  23. Mazucheli J, Menezes AFB, Chakraborty S (2019) On the one parameter unit-Lindley distribution and its associated regression model for proportion data. J Appl Stat 46(4):700–714

    Article  MathSciNet  Google Scholar 

  24. Mazucheli J, Menezes AFB, Fernandes LB, de Oliveira RP, Ghitany ME (2019) The unit-Weibull distribution as an alternative to the Kumaraswamy distribution for the modeling of quantiles conditional on covariates. J Appl Stat 47:954

    Article  MathSciNet  Google Scholar 

  25. McCullagh P, Nelder J (1989) Generalized linear models, 2nd edn. Chapman and Hall, USA

    Book  Google Scholar 

  26. Menezes AFB, Furriel WO (2019) Beta and Simplex regression models in the analysis of the municipal human development index 2010. Rev Bras Biom 37(3):394–408

    Article  Google Scholar 

  27. Mitnik PA, Baek S (2013) The Kumaraswamy distribution: median-dispersion re-parameterizations for regression modeling and simulation-based estimation. Stat Papers 54(1):177–192

    Article  MathSciNet  Google Scholar 

  28. Nadarajah S, Kotz S (2007) Multitude of Beta distributions with applications. Stat: J Theor Appl Stat 41(2):153–179

    Article  MathSciNet  Google Scholar 

  29. Neyman J, Pearson ES (1928) On the use and interpretation of certain test criteria for purposes of statistical inference. Biometrika 20A(1/2):175–240

    Article  Google Scholar 

  30. SAS, 2010. The NLMIXED Procedure, SAS/STAT® User’s Guide, Version 9.4. Cary, NC: SAS Institute Inc

  31. Schwarz G et al (1978) Estimating the dimension of a model. Annals Stat 6(2):461–464

    Article  MathSciNet  Google Scholar 

  32. Smithson M, Verkuilen J (2006) A better lemon squeezer? maximum-likelihood regression with beta-distributed dependent variables. Psychol Methods 11:54–71

    Article  Google Scholar 

  33. Zhang P, Qiu Z, Shi C (2016) Simplexreg: an R package for regression analysis of proportional data using the Simplex distribution. J Stat Softw 71(11):1–21

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Departamento de Estatística, Universidade Estadual de Campinas, Campinas, SP, Brasil

    André F. B. Menezes

  2. Departamento de Estatística, Universidade Federal do Rio Grande do Norte, Natal, RN, Brasil

    Marcelo Bourguignon

  3. Departamento de Estatística, Universidade Estadual de Maringá, Maringa, PR, Brasil

    Josmar Mazucheli

Authors
  1. André F. B. Menezes
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Marcelo Bourguignon
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Josmar Mazucheli
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Marcelo Bourguignon.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Menezes, A.F.B., Bourguignon, M. & Mazucheli, J. Complementary Beta Regression Model for Fitting Bounded Data. J Stat Theory Pract 16, 25 (2022). https://doi.org/10.1007/s42519-022-00256-w

Download citation

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s42519-022-00256-w

Keywords

Search

Navigation