A comparative study of methods for transforming unbounded distributions to bounded domains: Estimation and applications

Abstract

This study presents four transformation methods: Inverse exponential, arctangent, rational, and reciprocal for converting unbounded probability distributions into bounded forms defined over the interval $(0, 1)$. These methods are applied to the new XLindley distribution, a flexible model whose unbounded nature limits its applicability to enhance its utility in a bounded domain $(0, 1)$. A comprehensive comparison assesses the effectiveness of each transformation through five important dimensions as follows: Monotonicity, mathematical properties including moments, skewness, kurtosis, and entropy, parameter estimation methods with a focus on maximum likelihood estimation and the method of Anderson-Darling, numerical simulation performance assessing convergence and stability, and practical applications as demonstrated by three real-world datasets. The results highlight each method’s weaknesses as well as strengths, with one transformation outperforming the others in terms of flexibility, estimation accuracy, and practical adaptability.

Keywords

• Entropy
• XLindley distribution
• parameter estimation
• skewness
• simulation

References

1. [1] K. Nawel, A.M. Gemeay, H. Zeghdoudi, K. Karakaya, A.M. Alshangiti, M.E. Bakr, O.S. Balogun, A.H. Muse and E. Hussam, Modeling voltage real data set by a new version of Lindley distribution. IEEE Access, 11, 67220–67229, 2023.
2. [2] A. Rényi, On measures of entropy and information. In Proc. fourth Berkeley symp. math. stat. probab., volume 1, contributions to the theory of statistics, volume 4, pages 547–562. University of California Press, 1961.
3. [3] C. Tsallis, Possible generalization of Boltzmann-Gibbs statistics. J. stat. phys., 52, 479–487, 1988.
4. [4] J. Havrda and F. Charvát, Quantification method of classification processes. Concept of structural a-entropy. Kybernetika, 3 (1), 30–35, 1967.
5. [5] C.P. Quesenberry and C. Hales, Concentration bands for uniformity plots. J. Stat. Comput. Simul., 11 (1), 41–53, 1980.
6. [6] M. Sirajo, J.Y. Falgore, M.S. Najmuddeen and A.A. Umar, A new reflected minimax distribution on a bounded domain: Theory and application. Proceeding 5th Int. Conf. PSSN, 5 (1), 2021.
7. [7] N. Alotaibi, A.F. Hashem, I. Elbatal, S.A. Alyami, A.S. Al-Moisheer and M. Elgarhy, Inference for a Kavya–Manoharan inverse length biased exponential distribution under progressive-stress model based on progressive type-II censoring. Entropy, 24 (8), 1033, 2022.
8. [8] F. Tricomi, Sulle funzioni ipergeometriche confluenti. Ann. Mat. Pura Appl., 26, 141–175, 1947.

9. [9] R.A. Fisher, On the mathematical foundations of theoretical statistics. Philos. trans. R. Soc. Lond., contain. pap. math. phys. character, 222 (594-604), 309–368, 1922.
10. [10] T.W. Anderson and D.A. Darling, Asymptotic theory of certain goodness of fit criteria based on stochastic processes. ann. math. stat., pages 193–212, 1952.
11. [11] S. Chowdhury and A.K. Nanda, A new lifetime distribution with applications in inventory and insurance. Int. J. Qual. Reliab. Manag., 35 (2), 527–544, 2018.
12. [12] A. Genç, Estimation of $P(X > Y )$ with Topp-Leone distribution. J. Stat. Comput. Simul., 83 (2), 326–339, 2013.
13. [13] M.C. Korkmaz, E. Altun, C. Chesneau and H.M. Yousof, On the unit-Chen distribution with associated quantile regression and applications. Math. Slovaca, 72 (3), 765–786, 2022.
14. [14] M.C. Korkmaz, C. Chesneau and Z.S. Korkmaz, The unit folded normal distribution: A new unit probability distribution with the estimation procedures, quantile regression modeling and educational attainment applications. J. Reliab. Stat. Stud., 15 (1), 261–298, 2022.
15. [15] A.M.T. Abd El-Bar, H.S. Bakouch and S. Chowdhury, A new trigonometric distribution with bounded support and an application. Rev. Unión Mat. Argent., 62 (2), 459–473, 2021.
16. [16] A.M.T. Abd El-Bar, H. Athar, M. Kayid, R.M. Sayed, O.S. Balogun and A.M. Felifel, The cosine-sine model: Dual generalized order statistics, characterization, and estimation methods with applications to physics and radiation. J. Radiat. Res. Appl. Sci., 18 (2), 101324, 2025.
17. [17] 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 (1), 279–295, 2023.
18. [18] F.A. Bhatti, A. Ali, G. Hamedani, M.C. Korkmaz and M. Ahmad, The unit generalized log Burr XII distribution: Properties and application. AIMS Math., 6 (9), 10222–10252, 2021.
19. [19] A.M.T. Abd El-Bar, M.D.C.S. Lima and M. Ahsanullah, Some inferences based on a mixture of power function and continuous logarithmic distribution. J. Taibah Univ. Sci., 14 (1), 1116–1126, 2020.
20. [20] Ş. Sağlam and K. Karakaya, An alternative bounded distribution: regression model and applications. J. Supercomput., 80 (14), 20861–20890, 2024.
21. [21] J. Mazucheli, A.F.B. Menezes and S. Dey, Unit-Gompertz Distribution with Applications. Statistica, 79 (1), 25–43, 2019.
22. [22] J. Mazucheli, A.F.B. Menezes and S. Dey, The unit-Birnbaum-Saunders distribution with applications. Chil. J. Stat., 9, 47–57, 2018.
23. [23] M.E. Ghitany, J. Mazucheli, A.F.B. Menezes and F. Alqallaf, The unit-inverse Gaussian distribution: A new alternative to two-parameter distributions on the unit interval. Commun. Stat.-Theory Methods, 48 (14), 3423–3438, 2019.
24. [24] F.A. Althubyani, A.M.T. Abd El-Bar, M.A. Fawzy and A.M. Gemeay, A new 3- parameter bounded beta distribution: Properties, estimation, and applications. Axioms, 11 (10), 504, 2022.
25. [25] A.M. Gemeay, L.P. Sapkota, Y.A. Tashkandy, M.E. Bakr, O.S. Balogun and E. Hussam, New bounded probability model: Properties, estimation, and applications. Heliyon, 10 (23), e38965, 2024.
26. [26] A. Saeed, I. Elbatal, A. Saboor, F. Jamal, S. Khan and A.B. Ghorbal, Novel parameters based trigonometric probability distribution having bathtub shape with bounded support. J. Radiat. Res. Appl. Sci., 18 (2), 101443, 2025.
27. [27] S.A. Ogumeyo, F.C. Opone, A.G. Abubakari and J.C. Ehiwario, A bounded lifetime distribution specified by a trigonometric function: Properties, regression model, and applications. Int. J. Math. Math. Sci., 2024 (1), 5583105, 2024.
28. [28] Y.Y. Abdelall, G. Ismail and H. Nagy, A new bounded distribution: Covid-19 application. Egypt. Stat. J., 69 (1), 1–29, 2025.
29. [29] M.R. Irshad, S. Aswathy, R. Maya, A.I. Al-Omari and G. Alomani, A flexible model for bounded data with bathtub shaped hazard rate function and applications. AIMS Math., 9 (9), 24810–24831, 2024.
30. [30] A. Krishna, R. Maya, C. Chesneau and M.R. Irshad, The unit Teissier distribution and its applications. Math. Comput. Appl., 27 (1), 12, 2022.
31. [31] 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.
32. [32] M. Korkmaz and Z.S. Korkmaz, The unit log–log distribution: A new unit distribution with alternative quantile regression modeling and educational measurements applications. J. Appl. Stat., 50 (4), 889–908, 2023.
33. [33] M.Ç. Korkmaz and C. Chesneau, On the unit Burr-XII distribution with the quantile regression modeling and applications. Comput. Appl. Math., 40 (1), 29, 2021.
34. [34] A.F.B. Menezes, J. Mazucheli and S. Dey, The unit-logistic distribution: Different methods of estimation. Pesqui. Oper., 38, 555–578, 2018.
35. [35] 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.
36. [36] A. Shafiq, T.N. Sindhu, Z. Hussain, J. Mazucheli and B. Alves, A flexible probability model for proportion data: Unit Gumbel type-II distribution, development, properties, different method of estimations and applications. Aust. J. Stat., 52 (2), 116–140, 2023.
37. [37] G. Altun, M. Alizadeh, E. Altun and G. Ozel, Odd Burr Lindley distribution with properties and applications. Hacet. J. Math. Stat., 46 (2), 255–276, 2017.
38. [38] A.M.T. Abd El-Bar, A.R. El-Saeed and A.M. Gemeay, A bounded Lindley exponential model with applications to industrial engineering and epidemiological data. AIMS Math., 10 (7), 16233–16263, 2025.
39. [39] K. Karakaya and Ş. Sağlam, Unit Gamma-Lindley distribution: Properties, estimation, regression analysis and practical applications. Gazi Univ. J. Sci., pages 1–1, 2025.
40. [40] H. Karakuş, F.Z. Doğru and F.G. Akgül, Unit power Lindley distribution: Properties and estimation. Gazi Univ. J. Sci., 38 (1), 502–526, 2025.