Heavy-Tailed Log-Logistic Distribution: Properties, Risk Measures and Applications
Abstract
Heavy tailed distributions have a big role in studying risk data sets. Statisticians in many cases search and try to find new or relatively new statistical models to fit data sets in different fields. This article introduced a relatively new heavy-tailed statistical model by using alpha power transformation and exponentiated log-logistic distribution which called alpha power exponentiated log-logistic distribution. Its statistical properties were derived mathematically such as moments, moment generating function, quantile function, entropy, inequality curves and order statistics. Five estimation methods were introduced mathematically and the behaviour of the proposed model parameters was checked by randomly generated data sets and these estimation methods. Also, some actuarial measures were deduced mathematically such as value at risk, tail value at risk, tail variance and tail variance premium. Numerical values for these measures were performed and proved that the proposed distribution has a heavier tail than others compared models. Finally, three real data sets from different fields were used to show how these proposed models fitting these data sets than other many wells known and related models.References
M. V. Aarset. How to identify a bathtub hazard rate. IEEE Transactions on Reliability, 36(1):106–108, 1987.
F. S. Adeyinka and A. K. Olapade. On transmuted four parameters generalized log-logistic distribution. International Journal of Statistical Distributions and Applications, 5(2):32–37, 2019.
A. Z. Afify, G. M. Cordeiro, H. M. Yousof, A. Alzaatreh, and Z. M. Nofal. The kumaraswamy transmuted-g family of distributions: properties and applications. Journal of Data Science, 14(2):245–270, 2016.
A. Z. Afify, A. M. Gemeay, and N. A. Ibrahim. The heavy-tailed exponential distribution: Risk measures, estimation, and application to actuarial data. Mathematics, 8(8):1276, 2020.
A. Z. Afify, M. Nassar, G. M. Cordeiro, and D. Kumar. The weibull marshall–olkin lindley distribution: Properties and estimation. Journal of Taibah University for Science, 14(1):192–204, 2020.
Z. Ahmad, E. Mahmoudi, and S. Dey. A new family of heavy tailed distributions with an application to the heavy tailed insurance loss data. Communications in Statistics-Simulation and Computation, pages 1–24, 2020.
A. A. Al-Babtain, I. Elbatal, H. Al-Mofleh, A. M. Gemeay, A. Z. Afify, and A. M. Sarg. The flexible burr xg family: Properties, inference, and applications in engineering science. Symmetry, 13(3):474, 2021.
A. A. Al-Babtain, A. M. Gemeay, and A. Z. Afify. Estimation methods for the discrete Poisson-Lindley and discrete Lindley distributions with actuarial measures and applications in medicine. 2020.
H. Al-Mofleh, A. Z. Afify, and N. A. Ibrahim. A new extended two-parameter distribution: Properties, estimation methods, and applications in medicine and geology. Mathematics, 8(9):1578, 2020.
M. A. Aldahlan. Alpha power transformed log-logistic distribution with application to breaking stress data. Advances in Mathematical Physics, 2020, 2020.
M. A. D. Aldahlan and A. Z. Afify. The odd exponentiated half-logistic exponential distribution: estimation methods and application to engineering data. Mathematics, 8(10):1684, 2020.
N. M. Alfaer, A. M. Gemeay, H. M. Aljohani, and A. Z. Afify. The extended log-logistic distribution: Inference and actuarial applications. Mathematics, 9(12):1386, 2021.
J. A. Almamy. Extended poisson-log-logistic distribution. International Journal of Statistics and Probability, 8(2), 2019.
A. Arcagni and F. Porro. The graphical representation of inequality. Revista Colombiana de estadistica, 37(2):419-437, 2014.
B. C. Arnold. Pareto distributions fairland. Maryland: International Cooperative Publishing House, 1983.
P. Artzner. Application of coherent risk measures to capital requirements in insurance. North American Actuarial Journal, 3(2):11–25, 1999.
M. Bernardi, A. Maruotti, and L. Petrella. Skew mixture models for loss distributions: A bayesian approach. Insur. : Math. Econ., 51(3):617–623, 2012.
U¨ . Ceren, S. Cakmakyapan, and O¨ . Gamze. Alpha power inverted exponential distribution: Properties and application. Gazi University Journal of Science, 31(3):954–965, 2018.
D. Collett. Modelling survival data in medical research (ed 2) 2003 london united kingdom.
K. Cooray and M. M. A. Ananda. Modeling actuarial data with a composite lognormal-Pareto model. Scand. Actuar. J., 2005(5):321–334, 2005.
G. M. Cordeiro, A. Z. Afify, E. M. M. Ortega, A. K. Suzuki, and M. E. Mead. The odd lomax generator of distributions: Properties, estimation and applications. Journal of Computational and Applied Mathematics, 347:222–237, 2019.
G. M. Cordeiro, E. M. M. Ortega, B. V. Popovi´c, and R. R. Pescim. The lomax generator of distributions: Properties, minification process and regression model. Applied Mathematics and Computation, 247:465–486, 2014.
C. Dagum. A model of income distribution and the conditions of existence of moments of finite order. Bulletin of the International Statistical Institute, 46:199–205, 1975.
T. V. F. De Santana, E. M. M. Ortega, G. M. Cordeiro, and G. O. Silva. The kumaraswamy-log-logistic distribution. Journal of Statistical Theory and Applications, 11(3):265–291, 2012.
S. Dey, M. Nassar, and D. Kumar. Alpha power transformed inverse lindley distribution: A distribution with an upside-down bathtub-shaped hazard function. Journal of Computational and Applied Mathematics, 348:130–145, 2019.
P. R. Fisk. The graduation of income distributions. Econometrica: journal of the Econometric Society, pages 171–185, 1961.
J. Galambos. The asymptotic theory of extreme order statistics. R.E. Krieger Pub. Co., 1987.
D. C. T. Granzotto and F. Louzada. The transmuted log-logistic distribution: modeling, inference, and an application to a polled tabapua race time up to first calving data. Communications in Statistics-Theory and Methods, 44(16):3387-3402, 2015.
W. Gui. Marshall-olkin extended log-logistic distribution and its application in minification processes. Appl Math Sci, 7(80):3947–3961, 2013.
G. Hamedani. The zografos-balakrishnan log-logistic distribution: Properties and applications. Journal of Statistical Theory and Applications, 2013.
X. Hao and Q. Tang. Asymptotic ruin probabilities for a bivariate l´evy-driven risk model with heavy-tailed claims and risky investments. Journal of Applied Probability, 49(4):939–953, 2012.
R. V. Hogg and S. A. Klugman. Loss Distributions, vol. 249. John Wiley & Sons, 2009.
R. Ibragimov and A. Prokhorov. Heavy tails and copulas: Topics in dependence modelling in economics and finance. World Scientific, 2017.
S. A. Khan and S. K. Khosa. Generalized log-logistic proportional hazard model with applications in survival analysis. Journal of Statistical Distributions and Applications, 3(1):1–18, 2016.
C. Kleiber and S. Kotz. Statistical size distributions in economics and actuarial sciences, volume 470. John Wiley & Sons, 2003.
S. A. Klugman, H. H. Panjer, and G. E. Willmot. Loss models: From data to decisions, volume 715. John Wiley & Sons, 2012.
M. N. Lane. Pricing risk transfer transactions 1. ASTIN Bull. : J. IAA, 30(2):259–293, 2000.
A. J. Lemonte. The beta log-logistic distribution. Brazilian journal of probability and statistics, 28(3):313–332, 2014.
A. Mahdavi and D. Kundu. A new method for generating distributions with an application to exponential distribution. Communications in Statistics-Theory and Methods, 46(13):6543–6557, 2017.
M. M. Mansour, E. M. A. Elrazik, E. Altun, A. Z. Afify, and Z. Iqbal. Pak. j. statist. 2018 vol. 34 (6), 441-458 a new three- parameter fr´echet distribution: Properties and applications. Pak. J. Statist, 34(6):441–458, 2018.
M. E. Mead, G. M. Cordeiro, A. Z. Afify, and H. Al Mofleh. The alpha power transformation family: Properties and applications. Pakistan Journal of Statistics and Operation Research, 15(3):525–545, 2019.
M. Nassar, A. Z. Afify, S. Dey, and D. Kumar. A new extension of weibull distribution: properties and different methods of estimation. Journal of Computational and Applied Mathematics, 336:439–457, 2018.
M. Nassar, A. Z. Afify, and M. Shakhatreh. Estimation methods of alpha power exponential distribution with applications to engineering and medical data. Pakistan Journal of Statistics and Operation Research, pages 149–166, 2020.
M. Nassar, D. Kumar, S. Dey, G. M. Cordeiro, and A. Z. Afify. The marshall–olkin alpha power family of distributions with applications. Journal of Computational and Applied Mathematics, 351:41–53, 2019.
Y. Qi. On the tail index of a heavy tailed distribution. Annals of the Institute of Statistical Mathematics, 62(2):277- 298, 2010.
K. Rosaiah, R. R. L. Kantam, and S. Kumar. Reliability test plans for exponentiated log-logistic distribution. Economic Quality Control, 21(2):279–289, 2006.
M. M. Shoukri, I. U. H. Mian, and D. S. Tracy. Sampling properties of estimators of the log-logistic distribution with application to canadian precipitation data. Canadian Journal of Statistics, 16(3):223–236,1988.
M. H. Tahir, M. Mansoor, M. Zubair, and G. Hamedani. Mcdonald log-logistic distribution with an application to breast cancer data. Journal of Statistical Theory and Applications, 2014.
Y. Yang, K.Wang, J. Liu, and Z. Zhang. Asymptotics for a bidimensional risk model with two geometric l´evy price processes. Journal of Industrial & Management Optimization, 15(2):481, 2019.
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