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Unit Gamma-Lindley Distribution: Properties, Estimation, Regression Analysis, and Practical Applications

Year 2025, Volume: 38 Issue: 2, 1021 - 1040, 01.06.2025
https://doi.org/10.35378/gujs.1549073

Abstract

This study proposes the unit Gamma-Lindley distribution, a novel bounded statistical model that extends the flexibility of existing distributions for modeling data on the (0,1) interval. The proposed distribution is characterized, by closed-form expressions derived for its cumulative distribution, probability density, and hazard rate functions. Some statistical properties, including moments, order statistics, Bonferroni, Lorenz curves, entropy, etc. are examined. To estimate the unknown model parameters, several estimation methods are introduced and their performance is assessed through a Monte Carlo simulation experiment based on bias and mean square error criteria. A real data application focusing on firm management cost-effectiveness highlights the practical utility of the model, demonstrating its superior fit compared to current distributions, such as beta and Kumaraswamy. Furthermore, a novel regression model is developed based on the proposed distribution, with parameter estimation performed using the maximum likelihood method. The new regression model provides an alternative for analyzing bounded response variables. The findings contribute to the statistical literature by offering a flexible and comprehensive modeling framework for bounded data, with theoretical advancements and practical applicability.

References

  • [1] Mazucheli, J., Menezes, A.F.B., Fernandes, L. B., De Oliveira, R.P., and Ghitany, M.E., “The unit-Weibull distribution as an alternative to the Kumaraswamy distribution for the modeling of quantiles conditional on covariates. Journal of Applied Statistics”, 47(6): 954-974, (2020). DOI: 10.1080/02664763.2019.1657813
  • [2] Bhatti, F. A., Ali, A., Hamedani, G., Korkmaz, M. Ç., and Ahmad, M. “The unit generalized log Burr XII distribution: Properties and applications”, AIMS Mathematics. (2021). DOI: 10.3934/math.2021592
  • [3] Ghitany, M. E., Mazucheli, J., Menezes, A. F. B., and Alqallaf, F., “The unit-inverse Gaussian distribution: A new alternative to two-parameter distributions on the unit interval”, Communications in Statistics-Theory and Methods, 48(14): 3423-3438, (2019). DOI: 10.1080/03610926.2018.1476717
  • [4] Guerra, R. R., Pena-Ramirez, F. A., and Bourguignon, M., “The unit extended Weibull families of distributions and its applications”, Journal of Applied Statistics, 48(16): 3174-3192, (2021). DOI: 10.1080/02664763.2020.1796936
  • [5] Korkmaz, M.Ç., Leiva, V., and Martin-Barreiro, C., “The continuous Bernoulli distribution: Mathematical characterization, fractile regression, computational simulations, and applications” Fractal and Fractional, 7(5): 386, (2023). DOI: https://doi.org/10.3390/fractalfract7050386
  • [6] Korkmaz, M. Ç., Altun, E., Alizadeh, M., and El-Morshedy, M., “The log exponential-power distribution: Properties, estimations and quantile regression model”, Mathematics, 9(21): 2634, (2021). DOI: https://doi.org/10.3390/math9212634
  • [7] Korkmaz, M. Ç., Chesneau, C., and Korkmaz, Z. S. “The unit folded normal distribution: A new unit probability distribution with the estimation procedures, quantile regression modeling and educational attainment applications”, Journal of Reliability and Statistical Studies, 261-298, (2022). DOI: 10.13052/jrss0974-8024.15111
  • [8] Maya, R., Jodra, P., Irshad, M. R., and Krishna, A.,” The unit Muth distribution: Statistical properties and applications”, Ricerche di Matematica, 1-24, (2022). DOI: https://doi.org/10.1007/s11587-022-00703-7
  • [9] Mazucheli J, Menezes A.F., and Dey S.,” The unit-Birnbaum-Saunders distribution with applications”, Chilean Journal of Statistics, 9(1): 47-57, (2018).
  • [10] Mazucheli, J., Alves, B., Korkmaz, M. Ç., and Leiva, V., “Vasicek quantile and mean regression models for bounded data: New formulation, mathematical derivations, and numerical applications”, Mathematics, 10(9): 1389, (2022). DOI: https://doi.org/10.3390/ math10091389
  • [11] Mazucheli, J., Korkmaz, M. Ç., Menezes, A. F., and Leiva, V., “The unit generalized half-normal quantile regression model: formulation, estimation, diagnostics, and numerical applications”, Soft Computing, 27(1): 279-295, (2023). DOI: https://doi.org/10.1007/s00500-022-07278-3
  • [12] Mazucheli, J., Alves, B., and Korkmaz, M. Ç., “The Unit-Gompertz Quantile Regression Model for the Bounded Responses”, Mathematica Slovaca, 73(4): 1039-1054, (2023). DOI: https://doi.org/10.1515/ms-2023-0077
  • [13] Altun, E., and Cordeiro, G. M., “The unit-improved second-degree Lindley distribution: inference and regression modeling”, Computational Statistics, 35: 259-279, (2020). DOI: https://doi.org/10.1007/s00180-019-00921-y
  • [14] Korkmaz, M. C¸., Chesneau, C., and Korkmaz, Z. S., “Transmuted unit Rayleigh quantile regression model: Alternative to beta and Kumaraswamy quantile regression models”, University Politehnica of Bucharest Scientific Bulletin-Series A-Applied Mathematics and Physics, 83: 149-158, (2021).
  • [15 Ribeiro, T.F., Cordeiro, G.M., Pena-Ramirez, F.A., and Guerra, R.R., “A new quantile regression for the COVID-19 mortality rates in the United States”, Computational and Applied Mathematics, 40: 1-16, (2021). DOI: https://doi.org/10.1007/s40314-021-01553-z
  • [16] Abdi, M., Asgharzadeh, A., Bakouch, H. S., and Alipour, Z., “A new compound gamma and Lindley distribution with application to failure data”, Austrian Journal of Statistics, 48(3): 54-75, (2019). DOI: https://doi.org/10.17713/ajs.v48i3.843
  • [17] Canuto, C., Hussaini, M. Y., Quarteroni, A., and Zang, T. A., “Spectral methods: evolution to complex geometries and applications to fluid dynamics”, Springer Science and Business Media, (2007).
  • [18] Bonferroni, C., Elmenti di statistica generale [elements of general statistics]. Firenze: Libreria Seber, (1930).
  • [19] Casella, G., Robert, C. P., and Wells, M. T., “Generalized accept-reject sampling schemes”, Lecture notes-monograph series, 342-347, (2004).
  • [20] Amigó, J. M., Balogh, S. G., and Hernández, S., “A brief review of generalized entropies”, Entropy, 20(11): 813, (2018). DOI: https://doi.org/10.3390/e20110813
  • [21] Kumaraswamy, P., “A generalized probability density function for double bounded random processes”, Journal of Hydrology, 46(1-2): 79-88, (1980). DOI: https://doi.org/10.1016/0022-1694(80)90036-0
  • [22] Korkmaz, M. C¸., and Chesneau, C., “On the unit Burr-XII distribution with the quantile regression modeling and applications”, Computational and Applied Mathematics, 40(1): 29, (2021). DOI: https://doi.org/10.1007/s40314-021-01418-5
  • [23] Abd El-Bar, A., Bakouch, H. S., and Chowdhury, S., “A new trigonometric distribution with bounded support and an application”, Revista de la Union Matematica Argentina, 62(2): 459-473, (2021). DOI: https://doi.org/10.33044/revuma.1872
  • [24] Gomez-Deniz, E., Sordo, M. A., and Calderin-Ojeda, E., “The Log–Lindley distribution as an alternative to the beta regression model with applications in insurance”, Insurance: Mathematics and Economics, 54: 49-57, (2014). DOI: https://doi.org/10.1016/j.insmatheco.2013.10.017
  • [25] Jodra, P., and Jimenez-Gamero, M. D., “A quantile regression model for bounded responses based on the exponential-geometric distribution”, REVSTAT-Statistical Journal, 18(4): 415-436, (2020).
Year 2025, Volume: 38 Issue: 2, 1021 - 1040, 01.06.2025
https://doi.org/10.35378/gujs.1549073

Abstract

References

  • [1] Mazucheli, J., Menezes, A.F.B., Fernandes, L. B., De Oliveira, R.P., and Ghitany, M.E., “The unit-Weibull distribution as an alternative to the Kumaraswamy distribution for the modeling of quantiles conditional on covariates. Journal of Applied Statistics”, 47(6): 954-974, (2020). DOI: 10.1080/02664763.2019.1657813
  • [2] Bhatti, F. A., Ali, A., Hamedani, G., Korkmaz, M. Ç., and Ahmad, M. “The unit generalized log Burr XII distribution: Properties and applications”, AIMS Mathematics. (2021). DOI: 10.3934/math.2021592
  • [3] Ghitany, M. E., Mazucheli, J., Menezes, A. F. B., and Alqallaf, F., “The unit-inverse Gaussian distribution: A new alternative to two-parameter distributions on the unit interval”, Communications in Statistics-Theory and Methods, 48(14): 3423-3438, (2019). DOI: 10.1080/03610926.2018.1476717
  • [4] Guerra, R. R., Pena-Ramirez, F. A., and Bourguignon, M., “The unit extended Weibull families of distributions and its applications”, Journal of Applied Statistics, 48(16): 3174-3192, (2021). DOI: 10.1080/02664763.2020.1796936
  • [5] Korkmaz, M.Ç., Leiva, V., and Martin-Barreiro, C., “The continuous Bernoulli distribution: Mathematical characterization, fractile regression, computational simulations, and applications” Fractal and Fractional, 7(5): 386, (2023). DOI: https://doi.org/10.3390/fractalfract7050386
  • [6] Korkmaz, M. Ç., Altun, E., Alizadeh, M., and El-Morshedy, M., “The log exponential-power distribution: Properties, estimations and quantile regression model”, Mathematics, 9(21): 2634, (2021). DOI: https://doi.org/10.3390/math9212634
  • [7] Korkmaz, M. Ç., Chesneau, C., and Korkmaz, Z. S. “The unit folded normal distribution: A new unit probability distribution with the estimation procedures, quantile regression modeling and educational attainment applications”, Journal of Reliability and Statistical Studies, 261-298, (2022). DOI: 10.13052/jrss0974-8024.15111
  • [8] Maya, R., Jodra, P., Irshad, M. R., and Krishna, A.,” The unit Muth distribution: Statistical properties and applications”, Ricerche di Matematica, 1-24, (2022). DOI: https://doi.org/10.1007/s11587-022-00703-7
  • [9] Mazucheli J, Menezes A.F., and Dey S.,” The unit-Birnbaum-Saunders distribution with applications”, Chilean Journal of Statistics, 9(1): 47-57, (2018).
  • [10] Mazucheli, J., Alves, B., Korkmaz, M. Ç., and Leiva, V., “Vasicek quantile and mean regression models for bounded data: New formulation, mathematical derivations, and numerical applications”, Mathematics, 10(9): 1389, (2022). DOI: https://doi.org/10.3390/ math10091389
  • [11] Mazucheli, J., Korkmaz, M. Ç., Menezes, A. F., and Leiva, V., “The unit generalized half-normal quantile regression model: formulation, estimation, diagnostics, and numerical applications”, Soft Computing, 27(1): 279-295, (2023). DOI: https://doi.org/10.1007/s00500-022-07278-3
  • [12] Mazucheli, J., Alves, B., and Korkmaz, M. Ç., “The Unit-Gompertz Quantile Regression Model for the Bounded Responses”, Mathematica Slovaca, 73(4): 1039-1054, (2023). DOI: https://doi.org/10.1515/ms-2023-0077
  • [13] Altun, E., and Cordeiro, G. M., “The unit-improved second-degree Lindley distribution: inference and regression modeling”, Computational Statistics, 35: 259-279, (2020). DOI: https://doi.org/10.1007/s00180-019-00921-y
  • [14] Korkmaz, M. C¸., Chesneau, C., and Korkmaz, Z. S., “Transmuted unit Rayleigh quantile regression model: Alternative to beta and Kumaraswamy quantile regression models”, University Politehnica of Bucharest Scientific Bulletin-Series A-Applied Mathematics and Physics, 83: 149-158, (2021).
  • [15 Ribeiro, T.F., Cordeiro, G.M., Pena-Ramirez, F.A., and Guerra, R.R., “A new quantile regression for the COVID-19 mortality rates in the United States”, Computational and Applied Mathematics, 40: 1-16, (2021). DOI: https://doi.org/10.1007/s40314-021-01553-z
  • [16] Abdi, M., Asgharzadeh, A., Bakouch, H. S., and Alipour, Z., “A new compound gamma and Lindley distribution with application to failure data”, Austrian Journal of Statistics, 48(3): 54-75, (2019). DOI: https://doi.org/10.17713/ajs.v48i3.843
  • [17] Canuto, C., Hussaini, M. Y., Quarteroni, A., and Zang, T. A., “Spectral methods: evolution to complex geometries and applications to fluid dynamics”, Springer Science and Business Media, (2007).
  • [18] Bonferroni, C., Elmenti di statistica generale [elements of general statistics]. Firenze: Libreria Seber, (1930).
  • [19] Casella, G., Robert, C. P., and Wells, M. T., “Generalized accept-reject sampling schemes”, Lecture notes-monograph series, 342-347, (2004).
  • [20] Amigó, J. M., Balogh, S. G., and Hernández, S., “A brief review of generalized entropies”, Entropy, 20(11): 813, (2018). DOI: https://doi.org/10.3390/e20110813
  • [21] Kumaraswamy, P., “A generalized probability density function for double bounded random processes”, Journal of Hydrology, 46(1-2): 79-88, (1980). DOI: https://doi.org/10.1016/0022-1694(80)90036-0
  • [22] Korkmaz, M. C¸., and Chesneau, C., “On the unit Burr-XII distribution with the quantile regression modeling and applications”, Computational and Applied Mathematics, 40(1): 29, (2021). DOI: https://doi.org/10.1007/s40314-021-01418-5
  • [23] Abd El-Bar, A., Bakouch, H. S., and Chowdhury, S., “A new trigonometric distribution with bounded support and an application”, Revista de la Union Matematica Argentina, 62(2): 459-473, (2021). DOI: https://doi.org/10.33044/revuma.1872
  • [24] Gomez-Deniz, E., Sordo, M. A., and Calderin-Ojeda, E., “The Log–Lindley distribution as an alternative to the beta regression model with applications in insurance”, Insurance: Mathematics and Economics, 54: 49-57, (2014). DOI: https://doi.org/10.1016/j.insmatheco.2013.10.017
  • [25] Jodra, P., and Jimenez-Gamero, M. D., “A quantile regression model for bounded responses based on the exponential-geometric distribution”, REVSTAT-Statistical Journal, 18(4): 415-436, (2020).
There are 25 citations in total.

Details

Primary Language English
Subjects Statistical Theory, Applied Statistics
Journal Section Statistics
Authors

Kadir Karakaya 0000-0002-0781-3587

Şule Sağlam 0000-0002-1851-8217

Early Pub Date April 26, 2025
Publication Date June 1, 2025
Submission Date September 12, 2024
Acceptance Date February 13, 2025
Published in Issue Year 2025 Volume: 38 Issue: 2

Cite

APA Karakaya, K., & Sağlam, Ş. (2025). Unit Gamma-Lindley Distribution: Properties, Estimation, Regression Analysis, and Practical Applications. Gazi University Journal of Science, 38(2), 1021-1040. https://doi.org/10.35378/gujs.1549073
AMA Karakaya K, Sağlam Ş. Unit Gamma-Lindley Distribution: Properties, Estimation, Regression Analysis, and Practical Applications. Gazi University Journal of Science. June 2025;38(2):1021-1040. doi:10.35378/gujs.1549073
Chicago Karakaya, Kadir, and Şule Sağlam. “Unit Gamma-Lindley Distribution: Properties, Estimation, Regression Analysis, and Practical Applications”. Gazi University Journal of Science 38, no. 2 (June 2025): 1021-40. https://doi.org/10.35378/gujs.1549073.
EndNote Karakaya K, Sağlam Ş (June 1, 2025) Unit Gamma-Lindley Distribution: Properties, Estimation, Regression Analysis, and Practical Applications. Gazi University Journal of Science 38 2 1021–1040.
IEEE K. Karakaya and Ş. Sağlam, “Unit Gamma-Lindley Distribution: Properties, Estimation, Regression Analysis, and Practical Applications”, Gazi University Journal of Science, vol. 38, no. 2, pp. 1021–1040, 2025, doi: 10.35378/gujs.1549073.
ISNAD Karakaya, Kadir - Sağlam, Şule. “Unit Gamma-Lindley Distribution: Properties, Estimation, Regression Analysis, and Practical Applications”. Gazi University Journal of Science 38/2 (June 2025), 1021-1040. https://doi.org/10.35378/gujs.1549073.
JAMA Karakaya K, Sağlam Ş. Unit Gamma-Lindley Distribution: Properties, Estimation, Regression Analysis, and Practical Applications. Gazi University Journal of Science. 2025;38:1021–1040.
MLA Karakaya, Kadir and Şule Sağlam. “Unit Gamma-Lindley Distribution: Properties, Estimation, Regression Analysis, and Practical Applications”. Gazi University Journal of Science, vol. 38, no. 2, 2025, pp. 1021-40, doi:10.35378/gujs.1549073.
Vancouver Karakaya K, Sağlam Ş. Unit Gamma-Lindley Distribution: Properties, Estimation, Regression Analysis, and Practical Applications. Gazi University Journal of Science. 2025;38(2):1021-40.