Main Article Content
Abstract
In this paper, the authors introduce a new three-parameter lifetime probability distribution known as the Marshall-Olkin-generated log-logistic (LL) distribution. They thoroughly examine and describe this distribution, providing insights into its characteristics and its suitability for various applications. This newly constructed distribution's density function exhibits characteristics of both symmetry and right-skewness, providing modelling flexibility across a range of datasets. Because of its skewness coefficient, which can take both positive and negative values, a wide range of data asymmetries can be represented. The Marshall-Olkin generated LL distribution's corresponding hazard rate displays a variety of characteristics, including monotonic increase, increasing-constant, constant, upside-down, and monotonic drop. Because of its adaptability, the distribution can successfully capture various risk or failure rate patterns across time. Using a number of techniques, the researchers expand this distribution to the bivariate domain. Its utility in modelling multivariate lifetime data and inter-variable relationships is improved by these extensions. The researchers use the maximum likelihood method to estimate the parameters of the distribution, which ensures reliable and effective parameter estimation from observed data. They carry out an extensive simulation research to analyse biases and mean squared errors in a range of scenarios and sample sizes in order to evaluate the finite behaviour of the maximum likelihood estimators. In real-life and reliability applications, this meticulous methodology aids in evaluating the estimators' precision and dependability. Because it may offer a comprehensive and nuanced knowledge of high financial risks, the Peaks Over a Random Threshold Value-at-Risk (PORT-VaR) study is crucial for evaluating Norwegian fire insurance claims. This financial analysis is given extra consideration.
Keywords
Article Details
This work is licensed under a Creative Commons Attribution 4.0 International License.
Authors who publish with this journal agree to the following License
CC BY: This license allows reusers to distribute, remix, adapt, and build upon the material in any medium or format, so long as attribution is given to the creator. The license allows for commercial use.
References
- Aarset, M. V. (1987). How to identify a bathtub hazard rate. IEEE Transactions on Reliability, 36(1), 106-108.
- Alqasem, O. A., Alsheikh, S. M. A., Alsuhabi, H., Aldallal, R., Alsolmi, M. M., Yousof, H. M., ... & Hashem, A. F. (2024). A novel compound G family with properties, copulas, modelling the asymmetric and positive-skewed engineering and medical data. Alexandria Engineering Journal, 98, 80-96.
- Alkhayyat, S. L., Mohamed, H. S., Butt, N. S., Yousof, H. M., & Ali, E. I. (2023). Modeling the Asymmetric Reinsurance Revenues Data using the Partially Autoregressive Time Series Model: Statistical Forecasting and Residuals Analysis. Pakistan Journal of Statistics and Operation Research, 425-446.
- Atkinson, A. B. and Harrison, A. J. (1978). Distribution of Personal Wealth in Britain (Cambridge University Press, Cambridge).
- Alizadeh, M., Afshari, M., Contreras-Reyes, J. E., Mazarei, D. and H. M. Yousof, "The Extended Gompertz Model: Applications, Mean of Order P Assessment and Statistical Threshold Risk Analysis Based on Extreme Stresses Data," in IEEE Transactions on Reliability, doi: 10.1109/TR.2024.3425278.
- Alizadeh, M., Afshari, M., Ranjbar, V., Merovci, F., & Yousof, H. M. (2023). A novel XGamma extension: applications and actuarial risk analysis under the reinsurance data. São Paulo Journal of Mathematical Sciences, 1-31.
- Altun, E., Yousof, H. M. and Hamedani, G. G. (2018a). A new log-location regression model with influence diagnostics and residual analysis. Facta Universitatis, Series: Mathematics and Informatics, 33(3), 417-449.
- Altun, E., Yousof, H. M., Chakraborty, S. and Handique, L. (2018b). Zografos-Balakrishnan Burr XII distribution: regression modeling and applications. International Journal of Mathematics and Statistics, 19(3), 46-70.
- Aryal, G. R., Ortega, E. M., Hamedani, G. G. and Yousof, H. M. (2017). The Topp Leone Generated Weibull distribution: regression model, characterizations and applications, International Journal of Statistics and Probability, 6, 126-141.
- Atkinson, A.B. and Harrison, A.J. (1978). Distribution of Personal Wealth in Britain (Cambridge University Press, Cambridge).
- Balkema, A.A. and de Hann, L. Residual life at great age, Annals of Probability 2, 972-804, 1974.
- Beirlant, J., Goegebeur, Y., Segers, J., & Teugels, J. (2004). Statistics of Extremes: Theory and Applications. Wiley.
- Bhatti, F. A., Hamedani, G. G., Korkmaz, M. Ç., Yousof, H. M., & Ahmad, M. (2023). On The New Modified Burr XII Distribution: Development, Properties, Characterizations and Applications. Pakistan Journal of Statistics and Operation Research, 19(2), 327-348.
- Bryson, M. C. (1974). Heavy-tailed distribution: properties and tests, Technimetrics, 16, 161-68.
- Chahkandi, M. and Ganjali, M. (2009). On some lifetime distributions with decreasing failure rates, Computational Statistics and Data Analysis 53, 4433-4440.
- Chesneau, C. and Yousof, H. M. (2021). On a special generalized mixture class of probabilistic models. Journal of Nonlinear Modeling and Analysis, 3(1), 71-92.
- Cordeiro, G. M., Yousof, H. M., Ramires, T. G. and Ortega, E. M. M. (2018). The Burr XII system of densities: properties, regression model and applications. Journal of Statistical Computation and Simulation, 88(3), 432-456.
- Durbey, S. D. (1970). Compound gamma, beta and F distributions, Metrika 16, 27-31.
- Elbatal, I., Diab, L. S., Ghorbal, A. B., Yousof, H. M., Elgarhy, M., & Ali, E. I. (2024). A new losses (revenues) probability model with entropy analysis, applications and case studies for value-at-risk modeling and mean of order-P analysis. AIMS Mathematics, 9(3), 7169-7211.
- Elgohari, H. and Yousof, H. M. (2020a). A Generalization of Lomax Distribution with Properties, Copula and Real Data Applications. Pakistan Journal of Statistics and Operation Research, 16(4), 697-711. https://doi.org/10.18187/pjsor.v16i4.3260
- Elgohari, H. and Yousof, H. M. (2021b). A New Extreme Value Model with Different Copula, Statistical Properties and Applications. Pakistan Journal of Statistics and Operation Research, 17(4), 1015-1035. https://doi.org/10.18187/pjsor.v17i4.3471
- Elgohari, H. and Yousof, H. M. (2020c). New Extension of Weibull Distribution: Copula, Mathematical Properties and Data Modeling. Statistics, Optimization & Information Computing, 8(4), 972-993. https://doi.org/10.19139/soic-2310-5070-1036
- Elgohari, H., Ibrahim, M. and Yousof, H. M. (2021). A New Probability Distribution for Modeling Failure and Service Times: Properties, Copulas and Various Estimation Methods. Statistics, Optimization & Information Computing, 8(3), 555-586.
- Embrechts, P., Klüppelberg, C., & Mikosch, T. (2013). Modelling Extremal Events: For Insurance and Finance (2nd ed.). Springer.
- Elsayed, H. A., & Yousof, H. M. (2019a). A new Lomax distribution for modeling survival times and taxes revenue data sets. Journal of Statistics and Applications, 2(1), 35-58.
- Elsayed, H. A. H. and Yousof, H. M. (2019b). The Burr X Nadarajah Haghighi distribution: statistical properties and application to the exceedances of flood peaks data. Journal of Mathematics and Statistics, 15, 146-157.
- Elsayed, H. A. H., & Yousof, H. M. (2020). The generalized odd generalized exponential Fréchet model: univariate, bivariate and multivariate extensions with properties and applications to the univariate version. Pakistan Journal of Statistics and Operation Research, 529-544.
- Elsayed, H. A. H. and Yousof, H. M. (2021). Extended Poisson Generalized Burr XII Distribution. Journal of Applied Probability and Statistics, 16(1), 01-30.
- Gupta, R. C., Gupta, P. L. and Gupta, R. D. (1998). Modeling failure time data by Lehman alternatives. Communications in Statistics-Theory and methods, 27(4), 887-904.
- Corbellini, A., Crosato, L., Ganugi, P and Mazzoli, M. (2007). Fitting Pareto II distributions on firm size: Statistical methodology and economic puzzles. Paper presented at the International Conference on Applied Stochastic Models and Data Analysis, Chania, Crete.
- Cordeiro, G. M., Ortega, E. M. and Popovic, B. V. (2015). The gamma-Lomax distribution. Journal of Statistical computation and Simulation, 85(2), 305-319.
- Elbiely, M. M., & Yousof, H. M. (2018). A new extension of the Lomax distribution and its applications. J. Stat. Appl, 2, 18-34.
- Elbiely, M. M., & Yousof, H. M. (2019a). A new flexible Weibull Burr XII distribution. Journal of Statistics and Applications, 2(1), 59-77.
- Elbiely, M. M. and Yousof, H. M. (2019b). A New Inverse Weibull Distribution: Properties and Applications. Journal of Mathematics and Statistics, 15(1), 30-43.
- Emam, W.; Tashkandy, Y.; Hamedani, G.G.; Shehab, M.A.; Ibrahim, M.; Yousof, H.M. (2023a). A Novel Discrete Generator with Modeling Engineering, Agricultural and Medical Count and Zero‐Inflated Real Data with Bayesian, and Non‐Bayesian Inference. Mathematics 2023, 11, 1125. https://doi.org/10.3390/math110511.
- Emam, W.; Tashkandy, Y.; Goual, H.; Hamida, T.; Hiba, A.; Ali, M.M.; Yousof, H.M.; Ibrahim, M. (2023b). A New One-Parameter Distribution for Right Censored Bayesian and Non-Bayesian Distributional Validation under Various Estimation Methods. Mathematics 2023, 11, 897. https://doi.org/10.3390/math1104089
- Harris, C.M. (1968). The Pareto distribution as a queue service descipline, Operations Research, 16, 307-313.
- Hashem, A. F., Abdelkawy, M. A., Muse, A. H., & Yousof, H. M. (2024). A novel generalized Weibull Poisson G class of continuous probabilistic distributions with some copulas, properties and applications to real-life datasets. Scientific Reports, 14(1), 1741.
- Hashempour, M., Alizadeh, M., & Yousof, H. M. (2023). A New Lindley Extension: Estimation, Risk Assessment and Analysis Under Bimodal Right Skewed Precipitation Data. Annals of Data Science, 1-40.
- Hosking, J. R. M., & Wallis, J. R. (1987). Parameter and quantile estimation for the generalized Pareto distribution. Technometrics, 29(3), 339-349.
- Ibrahim, M. and Yousof, H. M. (2020). A new generalized Lomax model: statistical properties and applications, Journal of Data Science, 18(1), 190 - 217.
- Ibrahim, M. (2019). A new extended Fréchet distribution: properties and estimation. Pak. J. Stat. Oper. Res.,15 (3), 773-796.
- Ibrahim, M. (2020a). The compound Poisson Rayleigh Burr XII distribution: properties and applications. Journal of Applied Probability and Statistics, 15(1), 73-97.
- Ibrahim, M. (2020b). The generalized odd Log-logistic Nadarajah Haghighi distribution: statistical properties and different methods of estimation. J. Appl. Probab. Stat, 15, 61-84.
- Ibrahim, M., Yadav, A. S., Yousof, H. M., Goual, H. and Hamedani, G. G. (2019). A new extension of Lindley distribution: modified validation test, characterizations and different methods of estimation. Communications for Statistical Applications and Methods, 26(5), 473-495.
- Jansen, D., & de Vries, C. G. (1991). On the frequency of large stock returns: Putting booms and busts into perspective. The Review of Economics and Statistics, 73(1), 18-24.
- Lemonte, A. J. and Cordeiro, G. M. (2013). An extended Lomax distribution. Statistics, 47(4), 800-816.
- Lomax, K.S. (1954). Business failures: Another example of the analysis of failure data, Journal of the American Statistical Association, 49, 847-852.
- Gad, A. M., Hamedani, G. G., Salehabadi, S. M. and Yousof, H. M. (2019). The Burr XII-Burr XII distribution: mathematical properties and characterizations. Pakistan Journal of Statistics, 35(3), 229-248.
- Ghosh, I. and Ray. S. (2016). Some alternative bivariate Kumaraswamy type distributions via copula with application in risk management. Journal of Statistical Theory and Practice 10, 693-706.
- Gleaton, J. U. and Lynch, J.D. (2006). Properties of generalized loglogistic families of lifetime distributions. Journal of Probability and Statistical Science, 4, 51-64.
- Goual, H. and Yousof, H. M. (2020). Validation of Burr XII inverse Rayleigh model via a modified chi-squared goodness-of-fit test. Journal of Applied Statistics, 47(3), 393-423.
- Goual, H., Yousof, H. M. and Ali, M. M. (2019). Validation of the odd Lindley exponentiated exponential by a modified goodness of fit test with applications to censored and complete data. Pakistan Journal of Statistics an
- Operation Research, 15(3), 745-771.
- Goual, H., Yousof, H. M. and Ali, M. M. (2020). Lomax inverse Weibull model: properties, applications and a modified Chi-squared goodness-of-fit test for validation, Journal of Nonlinear Science and Applications. 13(6), 330-353.
- Gumbel, E. J. (1961). Bivariate logistic distributions. Journal of the American Statistical Association, 56(294), 335-349.
- Gumbel, E. J. (1960) Bivariate exponential distributions. Journ. Amer. Statist. Assoc., 55, 698-707.
- Hashem, A. F., Alotaibi, N., Alyami, S. A., Abdelkawy, M. A., Elgawad, M. A. A., Yousof, H. M., & Abdel-Hamid, A. H. (2024). Utilizing Bayesian inference in accelerated testing models under constant stress via ordered ranked set sampling and hybrid censoring with practical validation. Scientific Reports, 14(1), 14406.
- Hamedani, G. G., Goual, H., Emam, W., Tashkandy, Y., Ahmad Bhatti, F., Ibrahim, M. and Yousof, H. M. (2023). A new right-skewed one-parameter distribution with mathematical characterizations, distributional validation, and actuarial risk analysis, with applications. Symmetry, 15(7), 1297.
- Klugman, S. A., Panjer, H. H., & Willmot, G. E. (2012). Loss Models: From Data to Decisions (4th ed.). Wiley.
- Korkmaz, M. Ç., Cordeiro, G. M., Yousof, H. M., Pescim, R. R., Afify, A. Z., & Nadarajah, S. (2019). The Weibull Marshall–Olkin family: Regression model and application to censored data. Communications in Statistics-Theory and Methods, 48(16), 4171-4194.
- Kotz, S. and Johnson, N. L. (1992). Breakthroughs in Statistics: Foundations and basic theory. Springer,Volume 1.
- Loubna, H., Goual, H., Alghamdi, F. M., Mustafa, M. S., Tekle Mekiso, G., Ali, M. M., ... & Yousof, H. M. (2024). The quasi-xgamma frailty model with survival analysis under heterogeneity problem, validation testing, and risk analysis for emergency care data. Scientific Reports, 14(1), 8973.
- Mansour, M. M., Ibrahim, M., Aidi, K., Shafique Butt, N., Ali, M. M., Yousof, H. M. and Hamed, M. S. (2020a). A New Log-Logistic Lifetime Model with Mathematical Properties, Copula, Modified Goodness-of-Fit Test for Validation and Real Data Modeling. Mathematics, 8(9), 1508.
- Mansour, M., Rasekhi, M., Ibrahim, M., Aidi, K., Yousof, H. M. and Elrazik, E. A. (2020b). A New Parametric Life Distribution with Modified Bagdonavičius-Nikulin Goodness-of-Fit Test for Censored Validation, Properties, Applications, and Different Estimation Methods. Entropy, 22(5), 592.
- Mansour, M., Yousof, H. M., Shehata, W. A. M. and Ibrahim, M. (2020c). A new two parameter Burr XII distribution: properties,copula, different estimation methods and modeling acute bone cancer data, Journal of Nonlinear Science and Applications, 13, 223-238.
- McNeil, A. J., Frey, R., & Embrechts, P. (2015). Quantitative Risk Management: Concepts, Techniques, and Tools. Princeton University Press.
- Mohamed, H. S., Cordeiro, G. M., Minkah, R., Yousof, H. M., & Ibrahim, M. (2024). A size-of-loss model for the negatively skewed insurance claims data: applications, risk analysis using different methods and statistical forecasting. Journal of Applied Statistics, 51(2), 348-369.
- Morgenstern, D. (1956). Einfache beispiele zweidimensionaler verteilungen. Mitteilingsblatt fur Mathematische Statistik, 8, 234-235.
- Murthy, D.N.P. Xie, M. and Jiang, R. (2004). Weibull Models, Wiley.
- Nasir, M. A., Korkmaz, M. C., Jamal, F. and Yousof, H. M. (2018). On a new Weibull Burr XII distribution for lifetime data. Sohag Journal of Mathematics, 5(2), 47-56.
- Pougaza, D. B. and Djafari, M. A. (2011). Maximum entropies copulas. Proceedings of the 30th international workshop on Bayesian inference and maximum Entropy methods in Science and Engineering, 329-336.
- Poon, S. H., & Rockinger, M. (2003). The conditional distribution of extreme stock market returns: An empirical analysis. Journal of Empirical Finance, 10(4), 373-400.
- Rasekhi, M., Altun, E., Alizadeh, M. and Yousof, H. M. (2022). The Odd Log-Logistic Weibull-G Family of Distributions with Regression and Financial Risk Models. Journal of the Operations Research Society of China, 10(1), 133-158.
- Rasekhi, M., Saber, M. M., & Yousof, H. M. (2020). Bayesian and classical inference of reliability in multicomponent stress-strength under the generalized logistic model. Communications in Statistics-Theory and Methods, 50(21), 5114-5125.
- Rodriguez-Lallena, J. A. and Ubeda-Flores, M. (2004). A new class of bivariate copulas. Statistics and Probability Letters, 66, 315-25.
- Salem, M., Emam, W., Tashkandy, Y., Ibrahim, M., Ali, M. M., Goual, H., & Yousof, H. M. (2023). A new lomax extension: Properties, risk analysis, censored and complete goodness-of-fit validation testing under left-skewed insurance, reliability and medical data. Symmetry, 15(7), 1356.
- Shehata, W. A., Goual, H., Hamida, T., Hiba, A., Hamedani, G. G., Al-Nefaie, A. H., ... & Yousof, H. M. (2024). Censored and Uncensored Nikulin-Rao-Robson Distributional Validation: Characterizations, Classical and Bayesian estimation with Censored and Uncensored Applications. Pakistan Journal of Statistics and Operation Research, 11-35.
- Shrahili, M.; Elbatal, I. and Yousof, H. M. Asymmetric Density for Risk Claim-Size Data: Prediction and Bimodal Data Applications. Symmetry 2021, 13, 2357. https://doi.org/10.3390/sym13122357
- Teghri, S., Goual, H., Loubna, H., Butt, N. S., Khedr, A. M., Yousof, H. M., ... & Salem, M. (2024). A New Two-Parameters Lindley-Frailty Model: Censored and Uncensored Schemes under Different Baseline Models: Applications, Assessments, Censored and Uncensored Validation Testing. Pakistan Journal of Statistics and Operation Research, 109-138.
- Reynkens, T., Verbelen, R., Beirlant, J., & Antonio, K. (2017). Modelling censored losses using splicing: A global fit strategy with mixed Erlang and extreme value distributions. Insurance: Mathematics and Economics, 77, 65-77.
- Tahir, M. H., Cordeiro, G. M., Mansoor, M., & Zubair, M. (2015). The Weibull-Lomax distribution: properties and applications. Hacettepe Journal of Mathematics and Statistics, 44(2), 461-480.
- Yadav, A.S., Goual, H., Alotaibi, R.M. Rezk, H., Ali, M.M. and Yousof, H.M. (2020). Validation of the Topp-Leone-Lomax model via a modified Rao-Robson-Nikulin goodness-of-fit test with different methods of estimation. Symmetry, 12, 1-26. doi: 10.3390/sym12010057
- Yousof, H. M., Afify, A. Z., Abd El Hadi, N. E., Hamedani, G. G., & Butt, N. S. (2016). On six-parameter Fréchet distribution: properties and applications. Pakistan Journal of Statistics and Operation Research, 281-299.
- Yousof, H. M., Ahsanullah, M. and Khalil, M. G. (2019). A New Zero-Truncated Version of the Poisson Burr XII Distribution: Characterizations and Properties. Journal of Statistical Theory and Applications, 18(1), 1-11.
- Yousof, H. M., Alizadeh, M., Jahanshahiand, S. M. A., Ramires, T. G., Ghosh, I. and Hamedani, G. G. (2017). The Topp-Leone Transmuted G family of distributions: theory, characterizations and applications. Journal of Data Science, 15(4), 723-740.
- Yousof, H. M., Korkmaz, M. Ç., K., Hamedani, G. G and Ibrahim, M. (2022). A novel Chen extension: theory, characterizations and different estimation methods. Eur. J. Stat, 2(2022), 1-20.
- Yousof, H. M., Altun, E., Ramires, T. G., Alizadeh, M. and Rasekhi, M. (2018). A new family of distributions with properties, regression models and applications, Journal of Statistics and Management Systems, 21, 1, 163-188.
- Yousof, H. M., Ansari, S. I., Tashkandy, Y., Emam, W., Ali, M. M., Ibrahim, M., Alkhayyat, S. L. (2023a). Risk Analysis and Estimation of a Bimodal Heavy-Tailed Burr XII Model in Insurance Data: Exploring Multiple Methods and Applications. Mathematics. 2023; 11(9):2179. https://doi.org/10.3390/math11092179
- Yousof, H. M., Goual, H., Khaoula, M. K., Hamedani, G. G., Al-Aefaie, A. H., Ibrahim, M., ... & Salem, M. (2023b). A Novel Accelerated Failure Time Model: Characterizations, Validation Testing, Different Estimation Methods and Applications in Engineering and Medicine. Pakistan Journal of Statistics and Operation Research, 691-717.
- Yousof, H. M., Majumder, M., Jahanshahi, S. M. A., Ali, M. M. and Hamedani G. G. (2018). A new Weibull class of distributions: theory, characterizations and applications, Journal of Statistical Research of Iran, 15, 45-83.
- Yousof, H. M., Saber, M. M., Al-Nefaie, A. H., Butt, N. S., Ibrahim, M., & Alkhayyat, S. L. (2024). A discrete claims-model for the inflated and over-dispersed automobile claims frequencies data: Applications and actuarial risk analysis. Pakistan Journal of Statistics and Operation Research, 261-284.
- Yousof, H.M.; Tashkandy, Y.; Emam, W.; Ali, M.M.; Ibrahim, M. (2023c). A New Reciprocal Weibull Extension for Modeling Extreme Values with Risk Analysis under Insurance Data. Mathematics 2023, 11, 966. https://doi.org/10.3390/ math11040966