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The unit-Cauchy quantile regression model with variates observed on (0, 1): percentages, proportions, and fractions

Year 2025, Volume: 54 Issue: 2, 633 - 655, 28.04.2025

Talha Arslan , Keming Yu

https://doi.org/10.15672/hujms.1533205

Abstract

In this study, a new parametric quantile regression model is introduced as an alternative to the beta regression and Kumaraswamy quantile regression model. The proposed quantile regression model is obtained by reparametrization of the unit-Cauchy distribution in terms of its quantiles. The model parameters are estimated using the maximum likelihood method. A Monte-Carlo simulation study is conducted to show the efficiency of the maximum likelihood estimation of the model parameters. The implementation of the proposed quantile regression model is shown by using real datasets. Quantile regression models based on unit-Weibull, unit generalized half normal, and unit Burr XII are also considered in the applications. The application results show that the proposed quantile regression model is preferable over its rivals when several comparison criteria are taken into account. In addition, the fitting plots indicate that the proposed quantile regression model fits extreme observations on the right tail better than its strong rivals, which is important in quantile regression modeling.

Keywords

Maximum likelihood Monte-Carlo simulation parametric model quantile regression unit-Cauchy

Thanks

The editor and reviewers are thanked for comments that led to presentational improvements. Talha Arslan would like to thank Brunel University London (UK) for a visiting position in 2023 to collaborate with Prof. Keming Yu and providing a peaceful environment to conduct this study.

References

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  • [2] G. Brys, M. Hubert and A. Struyf, A comparison of some new measures of skewness, in: Developments in Robust Statistics (R. Dutter, P. Filzmoser, U. Gather and P. J. Rousseeuw, eds.), 98–113, Physica, Heidelberg, 2003.
  • [3] G.M. Cordeiro, G.M. Rodrigues, F. Prataviera and E.M.M. Ortega, A new quantile regression model with application to human development index, Comput. Stat. 39, 2925–2948 2024.
  • [4] D.R. Cox and E.J. Snell, A general definition of residuals, J. R. Stat. Soc. B 30 (2), 248–275, 1968.
  • [5] P.K. Dunn and G.K. Smyth, Randomized quantile residuals, J. Comput. Graph. Stat. 5 (3), 236–244, 1996.
  • [6] S. Ferrari and F. Cribari-Neto, Beta regression for modelling rates and proportions, J. Appl. Stat. 31 (7), 799–815, 2004.
  • [7] A. Henningsen and O.Toomet, maxlik: A package for maximum likelihood estimation in R, Comput. Stat 26, 443–458, 2011.
  • [8] D.V. Hinkley, On power transformations to symmetry, Biometrika 62 (1), 101–111, 1975.
  • [9] N.L. Johnson, Systems of frequency curves generated by methods of translation, Biometrika 36 (1), 149–176, 1949.
  • [10] R. Kieschnick and B.D. McCullough, Regression analysis of variates observed on (0, 1): percentages, proportions and fractions, Stat. Model. 3 (3), 193–213, 2003.
  • [11] R. Koenker, Quantile Regression, Cambridge University Press, Cambridge, 2005.
  • [12] R. Koenker and G. Bassett, Regression quantiles, Econometrica 46 (1), 33–50, 1978.
  • [13] M.C. Korkmaz, The unit generalized half-normal distribution: A new bounded distribution with inference and application, U.P.B. Sci. Bull. Ser. A 82 (2), 133–140, 2020.
  • [14] S. Kotz, N.L. Johnson and N. Balakrishnan, Univariate continuous distributions- Volume I, John Wiley & Sons, 1994.
  • [15] K. Krishnamoorthy, Handbook of statistical distributions with applications, Chapman and Hall/CRC, 2016.
  • [16] P. Kumaraswamy, Generalized probability density function for double-bounded random processes, J. Hydrol. 46 (1), 79–88, 1980.
  • [17] H.Y. Lee, H.J. Park and H.M. Kim, A clarification of the Cauchy distribution, Commun. Stat. Appl. Methods 21 (2), 183–191, 2014.
  • [18] Y.S. Maluf, S.L.P. Ferrari and F.F. Queiroz, Robust beta regression through the logit transformation, Metrika 88, 61–81, 2025.
  • [19] J. Mazucheli, A.F.B. Menezes and M.E. Ghitany, The unit-Weibull distribution and associated inference, J. Appl. Probab. Stat. 13 (2), 1–22, 2018.
  • [20] J. Mazucheli, A.F.B. Menezes and S. Chakraborty, On the one parameter unit-Lindley distribution and its associated regression model for proportion data, J. Appl. Stat. 46 (4), 700–714, 2019.
  • [21] J. Mazucheli, A.F.B. Menezes, L.B. Fernandes, R.P. de Oliveira and M.E. Ghitany, The unit-Weibull distribution as an alternative to the Kumaraswamy distribution for the modeling of quantiles conditional on covariates, J. Appl. Stat. 47 (6), 954–974, 2020.
  • [22] J. Mazucheli, B. Alves, A.F.B. Menezes and V. Leiva, An overview on parametric quantile regression models and their computational implementation with applications to biomedical problems including COVID-19 data, Comput. Methods Programs Biomed. 221, 106816, 2022.
  • [23] J. Mazucheli, M.C. Korkmaz, A.F.B. Menezes and V. Leiva, The unit generalized half-normal quantile regression model: formulation, estimation, diagnostics, and numerical applications, Soft Comput. 27, 279–295, 2023.
  • [24] P.A. Mitnik and S. Baek, The Kumaraswamy distribution: median-dispersion reparameterizations for regression modeling and simulation-based estimation, Stat. Pap. 54, 177–192, 2013.
  • [25] J.A.A. Moors, A quantile alternative for kurtosis, J. R. Stat. Soc. D 37 (1), 25–32, 1988.
  • [26] N.J.D. Nagelkerke, A note on a general definition of the coefficient of determination, Biometrika 78 (3), 691–692, 1991.
  • [27] A. Noufaily and M.C. Jones, Parametric quantile regression based on the generalized gamma distribution, J. R. Stat. Soc. C 62 (5), 723–740, 2013.
  • [28] J. Pedro and M.D. Jimenez-Gamero, A quantile regression model for bounded response based on the exponential-geometric distribution, Revstat Stat. J. 18 (4), 415–436, 2020.
  • [29] G.H.A. Pereira, On quantile residuals in beta regression, Commun. Stat. Simul. Comput. 48 (1), 302–316, 2019.
  • [30] A.R. Qader, A unit-Cachy distribution and unit-Cauchy-generated family of distributions, MSc Thesis, Van Yüzüncü Yıl University, 2021.
  • [31] A.R. Qader and T. Arslan, A unit-Cauchy distribution: Definition, properties and application, In: 2nd Int. Appl. Stat. Conf., p. 225, 2021.
  • [32] R Core Team, R: A language and environment for statistical computing, R Found. Stat. Comput., Vienna, 2023.
  • [33] A.E. Raftery, Bayesian model selection in social research, Sociol. Methodol. 25, 111- 163, 1995.
  • [34] T.F. Ribeiro, G.M. Cordeiro, F.A. Pe˜na-Ramìrez and R.R. Guerra, A new quantile regression for the COVID-19 mortality rates in the United States, Comput. Appl. Math. 40, 255, 2021.
  • [35] T.K.A. Ribeiro and S. Ferrari, Robust estimation in beta regression via maximum Lq-likelihood, Stat. Pap. 64, 321–353, 2023.
  • [36] J.T. Schmit and K. Roth, Cost effectiveness of risk management practices, J. Risk Insur. 57 (3), 455–470, 1990.
  • [37] C.W. Topp and F.C. Leone, A family of J-shaped frequency functions, J. Am. Stat. Assoc. 50 (269), 209–219, 1955.
  • [38] J.T. Townsend and H. Colonius, Variability of the max and min statistic: a theory of the quantile spread as a function of sample size, Psychometrika 70 (4), 759–772, 2005.
  • [39] Q.H. Vuong, Likelihood ratio tests for model selection and non-nested hypotheses, Econometrica 57 (2), 307–333, 1989.
  • [40] K. Yu and M.C. Jones, Local linear quantile regression, J. Am. Stat. Assoc. 93 (441), 228–237, 1998.
  • [41] K. Yu and R. Moyeed, Bayesian quantile regression, Stat. Probab. Lett. 54 (4), 437–447, 2001.
  • [42] K. Yu, Z. Lu and J. Stander, Quantile regression: applications and current research areas, J. R. Stat. Soc. D 52 (3), 331–350, 2003.
  • [43] H. Zhou and X. Huang, Bayesian beta regression for bounded responses with unknown supports, Comput. Stat. Data Anal. 167, 107345, 2022.

Year 2025, Volume: 54 Issue: 2, 633 - 655, 28.04.2025

Talha Arslan , Keming Yu

https://doi.org/10.15672/hujms.1533205

Abstract

References

  • [1] T. Arslan, A new family of unit-distribution: Definition, properties and applications, TWMS J. App. Eng. Math. 13 (2), 782–791, 2023.
  • [2] G. Brys, M. Hubert and A. Struyf, A comparison of some new measures of skewness, in: Developments in Robust Statistics (R. Dutter, P. Filzmoser, U. Gather and P. J. Rousseeuw, eds.), 98–113, Physica, Heidelberg, 2003.
  • [3] G.M. Cordeiro, G.M. Rodrigues, F. Prataviera and E.M.M. Ortega, A new quantile regression model with application to human development index, Comput. Stat. 39, 2925–2948 2024.
  • [4] D.R. Cox and E.J. Snell, A general definition of residuals, J. R. Stat. Soc. B 30 (2), 248–275, 1968.
  • [5] P.K. Dunn and G.K. Smyth, Randomized quantile residuals, J. Comput. Graph. Stat. 5 (3), 236–244, 1996.
  • [6] S. Ferrari and F. Cribari-Neto, Beta regression for modelling rates and proportions, J. Appl. Stat. 31 (7), 799–815, 2004.
  • [7] A. Henningsen and O.Toomet, maxlik: A package for maximum likelihood estimation in R, Comput. Stat 26, 443–458, 2011.
  • [8] D.V. Hinkley, On power transformations to symmetry, Biometrika 62 (1), 101–111, 1975.
  • [9] N.L. Johnson, Systems of frequency curves generated by methods of translation, Biometrika 36 (1), 149–176, 1949.
  • [10] R. Kieschnick and B.D. McCullough, Regression analysis of variates observed on (0, 1): percentages, proportions and fractions, Stat. Model. 3 (3), 193–213, 2003.
  • [11] R. Koenker, Quantile Regression, Cambridge University Press, Cambridge, 2005.
  • [12] R. Koenker and G. Bassett, Regression quantiles, Econometrica 46 (1), 33–50, 1978.
  • [13] M.C. Korkmaz, The unit generalized half-normal distribution: A new bounded distribution with inference and application, U.P.B. Sci. Bull. Ser. A 82 (2), 133–140, 2020.
  • [14] S. Kotz, N.L. Johnson and N. Balakrishnan, Univariate continuous distributions- Volume I, John Wiley & Sons, 1994.
  • [15] K. Krishnamoorthy, Handbook of statistical distributions with applications, Chapman and Hall/CRC, 2016.
  • [16] P. Kumaraswamy, Generalized probability density function for double-bounded random processes, J. Hydrol. 46 (1), 79–88, 1980.
  • [17] H.Y. Lee, H.J. Park and H.M. Kim, A clarification of the Cauchy distribution, Commun. Stat. Appl. Methods 21 (2), 183–191, 2014.
  • [18] Y.S. Maluf, S.L.P. Ferrari and F.F. Queiroz, Robust beta regression through the logit transformation, Metrika 88, 61–81, 2025.
  • [19] J. Mazucheli, A.F.B. Menezes and M.E. Ghitany, The unit-Weibull distribution and associated inference, J. Appl. Probab. Stat. 13 (2), 1–22, 2018.
  • [20] J. Mazucheli, A.F.B. Menezes and S. Chakraborty, On the one parameter unit-Lindley distribution and its associated regression model for proportion data, J. Appl. Stat. 46 (4), 700–714, 2019.
  • [21] J. Mazucheli, A.F.B. Menezes, L.B. Fernandes, R.P. de Oliveira and M.E. Ghitany, The unit-Weibull distribution as an alternative to the Kumaraswamy distribution for the modeling of quantiles conditional on covariates, J. Appl. Stat. 47 (6), 954–974, 2020.
  • [22] J. Mazucheli, B. Alves, A.F.B. Menezes and V. Leiva, An overview on parametric quantile regression models and their computational implementation with applications to biomedical problems including COVID-19 data, Comput. Methods Programs Biomed. 221, 106816, 2022.
  • [23] J. Mazucheli, M.C. Korkmaz, A.F.B. Menezes and V. Leiva, The unit generalized half-normal quantile regression model: formulation, estimation, diagnostics, and numerical applications, Soft Comput. 27, 279–295, 2023.
  • [24] P.A. Mitnik and S. Baek, The Kumaraswamy distribution: median-dispersion reparameterizations for regression modeling and simulation-based estimation, Stat. Pap. 54, 177–192, 2013.
  • [25] J.A.A. Moors, A quantile alternative for kurtosis, J. R. Stat. Soc. D 37 (1), 25–32, 1988.
  • [26] N.J.D. Nagelkerke, A note on a general definition of the coefficient of determination, Biometrika 78 (3), 691–692, 1991.
  • [27] A. Noufaily and M.C. Jones, Parametric quantile regression based on the generalized gamma distribution, J. R. Stat. Soc. C 62 (5), 723–740, 2013.
  • [28] J. Pedro and M.D. Jimenez-Gamero, A quantile regression model for bounded response based on the exponential-geometric distribution, Revstat Stat. J. 18 (4), 415–436, 2020.
  • [29] G.H.A. Pereira, On quantile residuals in beta regression, Commun. Stat. Simul. Comput. 48 (1), 302–316, 2019.
  • [30] A.R. Qader, A unit-Cachy distribution and unit-Cauchy-generated family of distributions, MSc Thesis, Van Yüzüncü Yıl University, 2021.
  • [31] A.R. Qader and T. Arslan, A unit-Cauchy distribution: Definition, properties and application, In: 2nd Int. Appl. Stat. Conf., p. 225, 2021.
  • [32] R Core Team, R: A language and environment for statistical computing, R Found. Stat. Comput., Vienna, 2023.
  • [33] A.E. Raftery, Bayesian model selection in social research, Sociol. Methodol. 25, 111- 163, 1995.
  • [34] T.F. Ribeiro, G.M. Cordeiro, F.A. Pe˜na-Ramìrez and R.R. Guerra, A new quantile regression for the COVID-19 mortality rates in the United States, Comput. Appl. Math. 40, 255, 2021.
  • [35] T.K.A. Ribeiro and S. Ferrari, Robust estimation in beta regression via maximum Lq-likelihood, Stat. Pap. 64, 321–353, 2023.
  • [36] J.T. Schmit and K. Roth, Cost effectiveness of risk management practices, J. Risk Insur. 57 (3), 455–470, 1990.
  • [37] C.W. Topp and F.C. Leone, A family of J-shaped frequency functions, J. Am. Stat. Assoc. 50 (269), 209–219, 1955.
  • [38] J.T. Townsend and H. Colonius, Variability of the max and min statistic: a theory of the quantile spread as a function of sample size, Psychometrika 70 (4), 759–772, 2005.
  • [39] Q.H. Vuong, Likelihood ratio tests for model selection and non-nested hypotheses, Econometrica 57 (2), 307–333, 1989.
  • [40] K. Yu and M.C. Jones, Local linear quantile regression, J. Am. Stat. Assoc. 93 (441), 228–237, 1998.
  • [41] K. Yu and R. Moyeed, Bayesian quantile regression, Stat. Probab. Lett. 54 (4), 437–447, 2001.
  • [42] K. Yu, Z. Lu and J. Stander, Quantile regression: applications and current research areas, J. R. Stat. Soc. D 52 (3), 331–350, 2003.
  • [43] H. Zhou and X. Huang, Bayesian beta regression for bounded responses with unknown supports, Comput. Stat. Data Anal. 167, 107345, 2022.
There are 43 citations in total.

Details

Primary Language English
Subjects Computational Statistics, Statistical Analysis, Statistical Theory
Journal Section Statistics
Authors

Talha Arslan 0000-0002-4630-4857

Keming Yu 0000-0001-6341-8402

Early Pub Date February 19, 2025
Publication Date April 28, 2025
Submission Date August 14, 2024
Acceptance Date February 15, 2025
Published in Issue Year 2025 Volume: 54 Issue: 2

Cite

APA Arslan, T., & Yu, K. (2025). The unit-Cauchy quantile regression model with variates observed on (0, 1): percentages, proportions, and fractions. Hacettepe Journal of Mathematics and Statistics, 54(2), 633-655. https://doi.org/10.15672/hujms.1533205
AMA Arslan T, Yu K. The unit-Cauchy quantile regression model with variates observed on (0, 1): percentages, proportions, and fractions. Hacettepe Journal of Mathematics and Statistics. April 2025;54(2):633-655. doi:10.15672/hujms.1533205
Chicago Arslan, Talha, and Keming Yu. “The Unit-Cauchy Quantile Regression Model With Variates Observed on (0, 1): Percentages, Proportions, and Fractions”. Hacettepe Journal of Mathematics and Statistics 54, no. 2 (April 2025): 633-55. https://doi.org/10.15672/hujms.1533205.
EndNote Arslan T, Yu K (April 1, 2025) The unit-Cauchy quantile regression model with variates observed on (0, 1): percentages, proportions, and fractions. Hacettepe Journal of Mathematics and Statistics 54 2 633–655.
IEEE T. Arslan and K. Yu, “The unit-Cauchy quantile regression model with variates observed on (0, 1): percentages, proportions, and fractions”, Hacettepe Journal of Mathematics and Statistics, vol. 54, no. 2, pp. 633–655, 2025, doi: 10.15672/hujms.1533205.
ISNAD Arslan, Talha - Yu, Keming. “The Unit-Cauchy Quantile Regression Model With Variates Observed on (0, 1): Percentages, Proportions, and Fractions”. Hacettepe Journal of Mathematics and Statistics 54/2 (April 2025), 633-655. https://doi.org/10.15672/hujms.1533205.
JAMA Arslan T, Yu K. The unit-Cauchy quantile regression model with variates observed on (0, 1): percentages, proportions, and fractions. Hacettepe Journal of Mathematics and Statistics. 2025;54:633–655.
MLA Arslan, Talha and Keming Yu. “The Unit-Cauchy Quantile Regression Model With Variates Observed on (0, 1): Percentages, Proportions, and Fractions”. Hacettepe Journal of Mathematics and Statistics, vol. 54, no. 2, 2025, pp. 633-55, doi:10.15672/hujms.1533205.
Vancouver Arslan T, Yu K. The unit-Cauchy quantile regression model with variates observed on (0, 1): percentages, proportions, and fractions. Hacettepe Journal of Mathematics and Statistics. 2025;54(2):633-55.

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