Skip to main content

Unit Log-Logistic Distribution and Unit Log-Logistic Regression Model

Abstract

In this paper, the unit log-logistic distribution was proposed. This distribution is obtained through by transformation of a random variable with log-logistic distribution. The unit log-logistic has closed forms for the cumulative distribution function and quantile function. Subsequently, the unit log-logistic regression model, with parametrization in the median was defined. So, in the presence of outliers, this model is more robust than the models with parametrization in the mean. The maximum likelihood method was used to estimate the parameters. The validity of the estimators of this model is shown through Monte Carlo simulations. Application to real data showed that the new model has a better fit than the popular beta regression model.

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

Fig. 1
Fig. 2
Fig. 3
Fig. 4

Notes

  1. The betareg package can be installed with install.packages(”betareg”) command.

References

  • Cox DR, Snell EJ (1968) A general definition of residuals. J Roy Stat Soc Ser B (Methodol) 30(2):248–265

    MathSciNet  MATH  Google Scholar 

  • Cribari-Neto F, Zeileis A (2010) Beta regression in R. J Stat Softw 34(2):1–24

    Article  Google Scholar 

  • Doornik JA (2018) Ox: an object-oriented matrix programming language. Timberlake Consultants and Oxford, London. https://www.doornik.com/

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

    Google Scholar 

  • Espinheira PL, Ferrari SLP, Cribari-Neto F (2008) On beta regression residuals. J Appl Stat 35(4):407–419

    MathSciNet  Article  Google Scholar 

  • Espinheira PL, Ferrari SLP, Cribari-Neto F (2014) Bootstrap prediction intervals in beta regressions. Comput Stat 29(5):1263–1277

    MathSciNet  Article  Google Scholar 

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

    MathSciNet  Article  Google Scholar 

  • Lima FP, Cribari-Neto F (2020) Bootstrap-based testing inference in beta regressions. Braz J Probab Stat 34(1):18–34

    MathSciNet  Article  Google Scholar 

  • Ospina R, Cribari-Neto F, Vasconcellos KLP (2006) Improved point and interval estimation for a beta regression model. Comput Stat Data Anal 51(2):960–981

    MathSciNet  Article  Google Scholar 

  • R Core Team (2020) R: A language and environment for statistical computing. R Foundation for Statistical Computing, Vienna, Austria. https://www.R-project.org/

  • Simas AB, Barreto-Souza W, Rocha AV (2010) Improved estimators for a general class of beta regression models. Comput Stat Data Anal 54(2):348–366

    MathSciNet  Article  Google Scholar 

Download references

Funding

Not applicable.

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Lucas David Ribeiro-Reis.

Ethics declarations

Conflict of interest

The authors declare that they have no conflict of interest.

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

Verify currency and authenticity via CrossMark

Cite this article

Ribeiro-Reis, L.D. Unit Log-Logistic Distribution and Unit Log-Logistic Regression Model. J Indian Soc Probab Stat 22, 375–388 (2021). https://doi.org/10.1007/s41096-021-00109-y

Download citation

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s41096-021-00109-y

Keywords

  • Unit log-logistic distribution
  • Unit log-logistic regression model
  • Median
  • Quantile function
  • Outliers

Mathematics Subject Classification

  • 60E05
  • 62F12
  • 65C05
  • 65C10