Modeling Time-to-Pain-Relief of Analgesic Drug: A Three-Shape-Parameter Modified Exponential Distribution Approach

  • Authors

    https://doi.org/10.14419/pwam8k81

    Received date: November 17, 2025

    Accepted date: January 9, 2026

    Published date: January 14, 2026

  • Probability Distributions; Probability Generators; Kumaraswamy Modified Fréchet Generator; Exponential; Shape Parameter; Model Fitting; ‎Analgesic Data; Simulation

  • Abstract

    In this study, the three-shape parameter modified exponential, also called the Kumaraswamy Modified Fréchet Exponential, is developed to ‎model time to pain relief. Some important properties of this new model are derived: These include the moments, moment generating function, ‎inequality measures, and mean residual life. The parameter estimates are computed under the maximum likelihood estimator. A simulation study ‎is carried out to investigate the behavior of parameters under the maximum likelihood estimation method. The quantile function is used to generate random numbers for the simulation study. We applied the Monte-Carlo simulation technique. The results show that the parameters with ‎their average bias (AB) and root mean square error (RMSE) decreased alongside increasing sample sizes: 50, 150, 250, 500, and 1000. ‎The results show that the three-shape parameter modified exponential behaves well with the maximum likelihood estimator. The three-‎shape parameter modified exponential model is applied to time-to-pain relief data of patients who received analgesic (pain killer medication). ‎Smaller Akaike and Bayesian Information criteria values are achieved better than all the competitive models; thus, the three-shape parameter ‎modified exponential model demonstrates a better fit for the data used‎.

  • References

    1. Yuzhu, T., Maozai, T., and Qianqian, Z. (2014). Transmuted Linear Exponential Distribution: A New Generalization of the Linear Exponential Dis-tribution. Communications in Statistics - Simulation and Computation, 43(10), 2661-2677. https://doi.org/10.1080/03610918.2013.763978.
    2. Khan, M.S., King, R. and Hudson, I. (2015a). Transmuted generalized Exponential distribution: A generalization of the Exponential distribution with applications to survival data. Communications in Statistics: Simulation and Computation.
    3. Das, S.; Kundu, D. On weighted exponential distribution and its length biased version. Journal of the Indian Society of Probability and Statistics. 2016, 17, 57–77 https://doi.org/10.1007/s41096-016-0001-9.
    4. Roy, S.; Adnan, M.A.S. Wrapped weighted exponential distributions. Statistics and Probability Letters. 2012, 82, 77–83 https://doi.org/10.1016/j.spl.2011.08.023.
    5. Hussian, M.A. A weighted inverted exponential distribution. International Journal of Advanced Statistics and Probability. 2013, 1, 142–150. https://doi.org/10.14419/ijasp.v1i3.1361
    6. Oguntunde, P.E. On the exponentiated weighted exponential distribution and its basic statistical properties. Applied Science Reports. 2015, 10, 160–167 https://doi.org/10.15192/PSCP.ASR.2015.10.3.160167.
    7. Perveen, Z.; Ahmed, Z.; Ahmad, M. On size-biased double weighted exponential distribution (SDWED). Open Journal of Statistics. 2016, 6, 917–930. https://doi.org/10.4236/ojs.2016.65076.
    8. Mallick, A.; Ghosh, I.; Dey, S.; Kumar, D. Bounded weighted exponential distribution with applications. American Journal of Mathematical and Management Science. 2022, 40, 68–87. https://doi.org/10.1080/01966324.2020.1834893.
    9. Bakouch, H.S.; Hussain, T.; Chesneau, C.; Khan, M.N. On a weighted exponential distribution with a logarithmic weight: Theory and applications. African Mathematical Journal. 2021, 32, 789–810. https://doi.org/10.1007/s13370-020-00861-7.
    10. Yang, Y.; Tian, W.; Tong, T. Generalized Mixtures of Exponential Distribution and Associated Inference. Mathematics 2021, 9, 1371 https://doi.org/10.3390/math9121371.
    11. M. Alizadeh, A. Z. Afify, M. S. Eliwa, and S. Ali, “The odd log logistic Lindley-G family of distributions: properties, Bayesian and non-Bayesian estimation with applications,” Computational Statistics, vol. 35, no. 1, pp. 281–308, 2020. https://doi.org/10.1007/s00180-019-00932-9.
    12. A. Z. Afify and M. Alizadeh, “The odd Dagum family of distributions: properties and applications,” Journal of Applied Probability and Statistics, vol. 15, no. 1, pp. 45–72, 2020.
    13. Afify A. Z., Altun E. Alizadeh M, Ozel G., and Hamedani G. G., “The odd exponentiated half-logistic-G family: properties, characterizations and applications,” Chilean Journal of Statistics, vol. 8, no. 2, pp. 65–91, 2017. https://doi.org/10.1080/09720510.2018.1495157.
    14. H. Reyad, M. Alizadeh, F. Jamal, and S. Othman, “The Topp Leone odd Lindley-G family of distributions: properties and applications,” Journal of Statistics and Management Systems, vol. 21, no. 7, pp. 1273–1297, 2018.
    15. M. Nassar, D. Kumar, S. Dey, G. M. Cordeiro, and A. Z. Afify, “The Marshall-Olkin alpha power family of distributions with applications,” Jour-nal of Computational and Applied Mathematics, vol. 351, pp. 41–53, 2019. https://doi.org/10.1016/j.cam.2018.10.052.
    16. H. Yousof, M. Mansoor, M. Alizadeh, A. Afify, and I. Ghosh, “The Weibull-G Poisson family for analyzing lifetime data,” Pakistan Journal of Sta-tistics and Operation Research, vol. 16, no. 1, pp. 131–148, 2020. https://doi.org/10.18187/pjsor.v16i1.2840.
    17. Alamgir K., Abdullah A. H. A. ,Muhammad A., Wali K. M. ,Shokrya S. A., and Zabidin S. (2021), A Novel Method for Developing Efficient Probability Distributions with Applications to Engineering and Life Science Data. Journal of Mathematics . Volume 2021, Article ID 4479270, page 1-13. https://doi.org/10.1155/2021/4479270.
    18. Cordeiro, G.M., Nadarajah, S. and Ortega, E.M. (2013). General Results for the Beta Weibull Distribution. Journal of Statistical Computation and Simulation, 83: 1082-1114. core team.
    19. Gleaton, J.U.; Lynch, J.D.(2006). Properties of generalized log-logistic families of lifetime distributions. Jounal of Probabability Statistical Science. 2006, 4, 51–64. https://doi.org/10.1080/00949655.2011.649756.
    20. Gupta, R. D., Gupta, P. L., Gupta, R. C. (1998). Modeling failure time data by Lehman alternatives. Communication statistics-Theory and Methods, 27:(4): 888-910. https://doi.org/10.1080/03610929808832134.
    21. Kumaraswamy, P. (1980). A Generalized Probability Density Function for Double-Bounded Random Processes . Journal of Hydrology, 46:79-88. https://doi.org/10.1016/0022-1694(80)90036-0.
    22. Marshall,A. W. and Olkin, I. (1997). A new methods for adding a parameter to a family of distributions with application to the exponential and Weibull families. Biometrika, 84, pp. 641– 652. https://doi.org/10.1093/biomet/84.3.641.
    23. Nasiru,S. and Abubakari, A.G.(2020). Complementary Generalized Power Weibull Power Series Family of Distributions.Eurasian Bulletin of Math-ematics. 3(1):20-37.
    24. Shaw, W.T. and Buckley, I. R. C. (2009). The Alchemy of Probability Distributions: beyond Gram-Charlier Expansions, and a Skew-Kurtotic-Normal Distribution from a Rank Transmutation Map, pp:1-8.
    25. Tahir, M., Cordeiro, G.M., Alzaatreh, A., Mansoor, M., Zubair, M., (2016). The Logistic-X family of distributions and its applications. Commun. Stat. Theory Methods 45(24), 7326–7349. [26] Eugene, N., Lee, C., Famoye, F., “Beta-normal distribution and its applications”, Communication in Statistics – Theory and Methods 31: 497–512, (2002).
    26. Cordeiro, G.M., Nadarajah, S. and Ortega, E.M.M. (2011). The Kumaraswamy Gumbel.
    27. distribution. Statistical Methods and Applications, to appear.Bonferroni, C.E.(1930). Elementi di statistica general, Libereria Seber, Firenze, 1930. Page 5-9 https://doi.org/10.1080/03610926.2014.980516.
    28. Gleaton, J.U.; Lynch, J.D.(2006). Properties of generalized log-logistic families of lifetime distributions. Journal of Probability Statistical Science. 2006, 4, 51–64.
    29. Navarro, J., Hernandez P.J., (2008). Mean residual life functions of finite mixtures, order statistics and coherent systems. Metrika 67(3):277–298. https://doi.org/10.1007/s00184-007-0133-8.
    30. Gross A. J. and Clark V. A., Survival Distributions: Reliability Applications in the Biomedical Sciences, John Wiley & Sons, New York, NY, USA, 1975.
    31. Alamgir, K.,Abdullah A., H. A, Muhammad, A., Wali K., M., , S.,S., A., and Z.,S. (2021),A Novel Method for Developing Efficient Probability Distributions with Applications to Engineering and Life Science Data, Journal of Mathematics.Volume 2021, Article ID 4479270.13 pages. https://doi.org/10.1155/2021/4479270.

  • Downloads