A Novel Two-Parameter Sine Model: Bayesian and Non-Bayesian Analysis with Application

Abstract

Statistical methods are widely used in various real-life applications. Researchers are particularly interested in creating new probability models through different functions. However, few studies have explored the use of trigonometric functions to create new probability models. This paper introduces the sine power inverted Topp-Leone (SPITL) model, a novel two-parameter model constructed through a trigonometric-based transformation. The SPITL model features a hazard rate function that can take various forms, including inverted bathtub-shaped, increasing, decreasing, reversed J-shaped, or upside-down, along with a probability density function that is unimodal and positively skewed. The probability density function, cumulative distribution function, quantile function, moment-generating function, incomplete moments, Lorenz and Bonferroni curves, residual life and reversed residual life functions, stochastic ordering, stress-strength reliability, and entropy measures are among the important characteristics of the model that are examined in this work. Parameter estimation is conducted using both maximum likelihood and Bayesian methods with simulations to verify the model's effectiveness. Finally, two real-world datasets from agriculture and water resources are analyzed to demonstrate the model's flexibility. The SPITL model is compared with several competing distributions and is shown to outperform them according to standard statistical criteria. Furthermore, the likelihood ratio test is employed to assess the compatibility of the sine-inverted Topp-Leone model as a special case. The results confirm that the SPITL framework effectively captures the data's characteristics while maintaining statistical consistency with its sub-models, offering fresh avenues for further study in these fields.

Keywords

• Sine family
• Power inverted Topp-Leone model
• Stochastic ordering
• Lorenz and Bonferroni curves
• Stress-strength model

References

1. Eugene, N., Lee, C., and Famoye, F., “Beta-normal distribution and its applications”, Communications in Statistics-Theory and Methods, 31(4): 497–512, (2002). DOI: https://doi.org/10.1081/STA-120003130.
2. Marshall, A.W., and Olkin, I., “A new method for adding a parameter to a family of distributions with application to the exponential and Weibull families”, Biometrika, 92(2): 505–505, (2005). DOI: https://doi.org/10.1093/biomet/84.3.641.
3. Alzaatreh, A., Lee, C., and Famoye, F., “A new method for generating families of continuous distributions”, Metron, 71(1): 63–79, (2013). DOI: https://doi.org/10.1007/s40300-013-0007-y.
4. Hassan, A.S., and Nassr, S.G., “Power Lindley-G family of distributions”, Annals of Data Science, 6: 189–210, (2019). DOI: https://doi.org/10.1007/s40745-018-0159-y.
5. Tahir, M.H., Hussain, M.A., Cordeiro, G.M., El-Morshedy, M., and Eliwa, M.S., “A new Kumaraswamy generalized family of distributions with properties, applications, and bivariate extension”, Mathematics, 8(11): 1989, (2020). DOI: https://doi.org/10.3390/math8111989.
6. Hassan, A.S., Sabry, M., and Elsehetry, A., “A new probability distribution family arising from truncated power Lomax distribution with application to Weibull model”, Pakistan Journal of Statistics and Operation Research, 16(4): 661–674, (2020). DOI: http://dx.doi.org/10.18187/pjsor.v16i4.3442.
7. Eghwerido, J.T., Nzei, L.C., Omotoye, A.E., and Agu, F.I., “The Teissier-G family of distributions: Properties and applications”, Mathematica Slovaca, 72(5): 1301–1318, (2022). DOI: https://doi.org/10.1515/ms-2022-0089.
8. Kumar, D., Singh, U., and Singh, S.K., “A new distribution using sine function-its application to bladder cancer patients data”, Journal of Statistics Applications and Probability, 4(3): 417, (2015). DOI: https://dx.doi.org/10.12785/jsap/040309.

9. Hassan, A.S., Saudi, O.A., and Nagy, H.F., “A new three-parameter inverted exponentiated Weibull distribution: statistical inference and application”, The Egyptian Statistical Journal, 68(2): 34–64, (2024). DOI: http://doi.org/10.21608/esju.2024.310699.1038.
10. Jamal, F., Chesneau, C., and Aidi, K., “The sine extended odd Fréchet-G family of distribution with applications to complete and censored data”, Mathematica Slovaca, 71(4): 961–982, (2021). DOI: https://doi.org/10.1515/ms-2021-0033.
11. Rahman, M.M., “Arcsine-G family of distributions”, Journal of Statistics Applications and Probability Letters, 8(3): 169–179, (2021). DOI: http://dx.doi.org/10.18576/jsap/080303.
12. Isa, A.M., Doguwa, S.I., Alhaji, B.B., and Dikko, H.G., “Sine Type II Topp-Leone G family of probability distribution: Mathematical properties and application”, Arid Zone Journal of Basic and Applied Research, 2(4): 124–138, (2023). DOI: http://doi.org/10.55639/607.757473.
13. Hassan, A.S., Seyam, E.A., Saudi, O.A., and Gira, A.A., “Survival analysis of the novel sine distribution: Theoretical framework, graphical insights, and data applications”, Scientific African: e03009, (2025). DOI: https://doi.org/10.1016/j.sciaf.2025.e03009.
14. Barco, K.V.P., Mazucheli, J., and Janeiro, V., “The inverse power Lindley distribution”, Communications in Statistics-Simulation and Computation, 46(8): 6308–6323, (2017). DOI: https://doi.org/10.1080/03610918.2016.1202274.
15. Lee, S., Noh, Y., and Chung, Y., “Inverted exponentiated Weibull distribution with applications to lifetime data”, Communications for Statistical Applications and Methods, 24(3): 227–240, (2017). DOI: https://doi.org/10.5351/CSAM.2017.24.3.227.
16. Tahir, M.H., Cordeiro, G.M., Ali, S., Dey, S., and Manzoor, A., “The inverted Nadarajah–Haghighi distribution: estimation methods and applications”, Journal of Statistical Computation and Simulation, 88(14): 2775–2798, (2018). DOI: https://doi.org/10.1080/00949655.2018.1487441.
17. Hassan, A.S., Abdalla, G.S.S., Elgarhy, M., Mendy, J.T., and Almetwally, E.M., “Efficient inference of P(W< Z) using ranked set sampling: The inverse power half-logistic model”, Journal of Statistics Applications and Probability 14(5): 761–785, (2025). DOI: http://dx.doi.org/10.18576/jsap/140507.
18. Hassan, A.S., Morgan, Y.S., and Abd-Allah, M., “Parameter estimation and data analysis using a novel extension of Weibull distribution”, The Egyptian Statistical Journal, 69(2): 131–154, (2025). DOI: http://doi.org/10.21608/esju.2025.389962.1098.
19. Eliwa, M., El-Morshedy, M., and Ibrahim, M., “Inverse Gompertz distribution: properties and different estimation methods with application to complete and censored data”, Annals of Data Science, 6: 321–339, (2019). DOI: https://doi.org/10.1007/s40745-018-0173-0.
20. Hassan, A.S., Elgarhy, M., and Ragab, R., “Statistical properties and estimation of inverted Topp-Leone distribution”, Journal of Statistics Applications and Probability, 9(2): 319–331, (2020). DOI: http://dx.doi.org/10.18576/jsap/090212.
21. Yadav, A.S., Maiti, S.S., and Saha, M., “The inverse Xgamma distribution: statistical properties and different methods of estimation”, Annals of Data Science, 8(2): 275–293, (2021). DOI: https://doi.org/10.1007/s40745-019-00211-w.
22. Hassan, A.S., Alsadat, N., Chesneau, C., Elgarhy, M., Kayid, M., Nasiru, S., and Gemeay, A.M., “Inverse power XLindley distribution with statistical inference and applications to engineering data”, Scientific Reports, 15(1): 4385, (2025). DOI: https://doi.org/10.1038/s41598-025-87023-6.
23. Elbatal, I., Hassan, A.S., Gemeay, A.M., Diab, L., Ghorbal, A.B., and Elgarhy, M., “Statistical analysis of the inverse power Zeghdoudi model: Estimation, simulation and modeling to engineering and environmental data”, Physica Scripta, 99(6): 065231, (2024). DOI: https://doi.org/10.1088/1402-4896/ad46d0.
24. Abushal, T.A., Hassan, A.S., El-Saeed, A.R., and Nassr, S.G., “Power Inverted Topp–Leone distribution in acceptance sampling plans”, CMC-Computers, Materials and Continua, 67(1): 991–1011, (2021). DOI: http://doi.org/10.32604/cmc.2021.014620.
25. Ibrahim, G.M., Hassan, A.S., Almetwally, E.M., and Almongy, H.M., “Parameter estimation of alpha power inverted Topp-Leone distribution with applications”, Intelligent Automation and Soft Computing, 29: 353–371, (2021). DOI: https://doi.org/10.32604/iasc.2021.017586.
26. Nassr, S.G., Hassan, A.S., Alsultan, R., and El-Saeed, A.R., “Acceptance sampling plans for the three-parameter inverted Topp–Leone model”, Mathematical Biosciences and Engineering, 19: 13628–13659, (2022). DOI: http://doi.org/10.3934/mbe.2022636.
27. El-Saeed, A.R., Hassan, A.S., Elharoun, N.M., Al Mutairi, A., Khashab, R.H., and Nassr, S.G., “A class of power inverted Topp-Leone distribution: Properties, different estimation methods and applications”, Journal of Radiation Research and Applied Sciences, 16(4): 100643, (2023). DOI: http://doi.org/10.1016/j.jrras.2023.100643.
28. Mohamed, R.A., Elgarhy, M., Alabdulhadi, M.H., Almetwally, E.M., and Radwan, T., “Statistical inference of truncated Cauchy power-inverted Topp–Leone distribution under hybrid censored scheme with applications”, Axioms, 12(2): 148, (2023). DOI: https://doi.org/10.3390/axioms12020148.
29. Hassan, A.S., Seyam, E.A., Alzahrani, A.A., and Saudi, O.A., “An innovative discrete distribution for modeling genotoxicity data: Applications in pharmacology and environmental health”, AIMS Mathematics, 11(1): 1145–1174, (2026). DOI: http://doi.org/10.3934/math.20251245.
30. Shaked, M., and Shanthikumar, J.G., “Stochastic Orders”. New York: Springer, (2007).
31. Tsallis, C., “Possible generalization of Boltzmann-Gibbs statistics”, Journal of Statistical Physics, 52: 479–487, (1988). DOI: https://doi.org/10.1007/BF01016429.
32. Dey, S., Singh, S., Tripathi, Y.M., and Asgharzadeh, A., “Estimation and prediction for a progressively censored generalized inverted exponential distribution”, Statistical Methodology, 32: 185–202, (2016). DOI: https://doi.org/10.1016/j.stamet.2016.05.007.
33. Tummala, V.R., and Sathe, P.T., “Minimum expected loss estimators of reliability and parameters of certain lifetime distributions”, IEEE Transactions on Reliability, 27(4): 283–285, (1978). DOI: http://doi.org/10.1109/TR.1978.5220373.
34. Metropolis, N., Rosenbluth, A.W., Rosenbluth, M.N., Teller, A.H., and Teller, E., “Equation of state calculations by fast computing machines”, The Journal of Chemical Physics, 21(6): 1087–1092, (1953). DOI: https://doi.org/10.1063/1.1699114.
35. Kleiber, C., and Kotz, S., “Statistical Size Distributions in Economics and Actuarial Sciences”. Wiley Series in Probability and Statistics. Hoboken, New Jersey: John Wiley and Sons, Inc., (2003).
36. Aldahlan, M.A., “Sine Frèchet Model: Modeling of COVID‐19 Death Cases in Kingdom of Saudi Arabia”, Mathematical Problems in Engineering, 2022(1): 2039076, (2022). DOI: https://doi.org/10.1155/2022/2039076.
37. Keller, A.Z., Kamath, A.R.R., and Perera, U.D., “Reliability analysis of CNC machine tools”, Reliability Engineering, 3(6): 449–473, (1982). DOI: https://doi.org/10.1016/0143-8174(82)90036-1.
38. Alyami, S.A., Elbatal, I., Alotaibi, N., Almetwally, E.M., and Elgarhy, M., “Modeling to factor productivity of the United Kingdom food chain: Using a new lifetime-generated family of distributions”, Sustainability, 14(14): 8942, (2022). DOI: https://doi.org/10.3390/su14148942.