Modeling Extreme Values with Alpha Power Inverse Pareto Distribution

Ihtisham, Shumaila; Manzoor, Sadaf; Khalil, Alamgir; Badshah, Sareer; Ijaz, Muhammad; Atta, Hadia

doi:10.2478/msr-2023-0007

Abstract

The study focuses on the development of a new probability distribution with applications to extreme values. The distribution is proposed by incorporating an additional parameter into the inverse Pareto distribution using the α-Power Transformation. Various properties of the new distribution are derived. The paper also explores the estimation of the parameters by the Maximum Likelihood Estimation (MLE) technique. Simulations are performed to evaluate the performance of the MLEs. In addition, two real data sets with extreme values are used to evaluate the efficacy of the proposed model. It is concluded that the proposed model performs well in the case of extreme values compared to the existing distributions.

References

[1] Hamdi, Y., Haigh, I. D., Parey, S., Wahl, T. (2021). Preface: Advances in extreme value analysis and application to natural hazards. Natural Hazards and Earth System Sciences, 21 (5), 1461-1465. https://doi.org/10.5194/nhess-21-1461-2021
Search in Google Scholar Back to article
[2] Azzalini, A. (1985). A class of distributions which includes the normal ones. Scandinavian Journal of Statistics, 12 (2), 171-178. https://www.jstor.org/stable/4615982
Search in Google Scholar Back to article
[3] Marshall, A. W., Olkin, I. (1997). A new method for adding a parameter to a family of distributions with application to the exponential and Weibull families. Biometrika, 84 (3), 641-652. https://www.jstor.org/stable/2337585
Search in Google Scholar Back to article
[4] Eugene, N, Lee, C, Famoye, F. (2002). Beta-normal distribution and its applications. Communications in Statistics - Theory and Methods, 31 (4), 497-512. https://doi.org/10.1081/STA-120003130
Search in Google Scholar Back to article
[5] Jones, M. C. (2009). Kumaraswamy’s distribution: A beta-type distribution with some tractability advantages. Statistical Methodology, 6 (1), 70-81. https://doi.org/10.1016/j.stamet.2008.04.001
Search in Google Scholar Back to article
[6] Zografos, K., Balakrishnan, N. (2009). On families of beta- and generalized gamma-generated distributions and associated inference. Statistical Methodology, 6 (4), 344-362. https://doi.org/10.1016/j.stamet.2008.12.003
Search in Google Scholar Back to article
[7] Alzaatreh, A., Lee, C., Famoye, F. (2013). A new method for generating families of continuous distributions. Metron, 71 (1), 63-79. https://doi.org/10.1007/s40300-013-0007-y
Search in Google Scholar Back to article
[8] Mahdavi, A., Kundu, D. (2017). A new method for generating distributions with an application to exponential distribution. Communications in Statistics - Theory and Methods, 46 (13), 6543-6557. https://doi.org/10.1080/03610926.2015.1130839
Search in Google Scholar Back to article
[9] Nassar, M., Alzaatreh, A., Mead, M., Abo-Kasem, O. (2017). Alpha power Weibull distribution: Properties and applications. Communications in Statistics - Theory and Methods, 46 (20), 10236-10252. https://doi.org/10.1080/03610926.2016.1231816
Search in Google Scholar Back to article
[10] Dey, S., Ghosh, I., Kumar, D. (2019). Alpha-power transformed Lindley distribution: Properties and associated inference with application to earthquake data. Annals of Data Science, 6, 623-650. https://doi.org/10.1007/s40745-018-0163-2
Search in Google Scholar Back to article
[11] Dey, S., Nassar, M., Kumar, D. (2019). 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. https://doi.org/10.1016/j.cam.2018.03.037
Search in Google Scholar Back to article
[12] Ihtisham, S., Khalil, A., Manzoor, S., Khan, S. A., Ali, A. (2019). Alpha-Power Pareto distribution: Its properties and applications. PloS ONE, 14 (6), e0218027. https://doi.org/10.1371/journal.pone.0218027
Search in Google Scholar Back to article
[13] Ali, M., Khalil, A., Ijaz, M., Saeed, N. (2021). Alpha-Power Exponentiated Inverse Rayleigh distribution and its applications to real and simulated data. PloS ONE, 16 (1), e0245253. https://doi.org/10.1371/journal.pone.0245253
Search in Google Scholar Back to article
[14] Eghwerido, J.T., Nzei, L.C., Zelibe, S.C. (2022). The alpha power extended generalized exponential distribution. Journal of Statistics and Management Systems, 25 (1), 187-210. https://doi.org/10.1080/09720510.2021.1872692
Search in Google Scholar Back to article
[15] Gharib, M., Mohammed, B. I., Aghel, W. E. R. (2017). Marshll-Olkin extended inverse Pareto distribution and its application. International Journal of Statistics and Probability, 6 (6), 71-84. https://doi.org/10.5539/ijsp.v6n6p71
Search in Google Scholar Back to article
[16] Bashir, S., Naqvi, I. B. (2018). The Gompertz inverse Pareto distribution and extreme value theory. American Review of Mathematics and Statistics, 6 (2), 30-37. https://doi.org/10.15640/arms.v6n2a4
Search in Google Scholar Back to article
[17] Shaked, M, Shanthikumar, J. G. (eds.) (2007). Stochastic Orders. Springer. https://doi.org/10.1007/978-0-387-34675-5
Search in Google Scholar Back to article
[18] Srinivasa Rao, G., Kantam, R. R. L., Rosaiah, K., Pratapa Reddy, J. (2013). Estimation of stress–strength reliability from inverse Rayleigh distribution. Journal of Industrial and Production Engineering, 30 (4), 256-263. https://doi.org/10.1080/21681015.2013.828787
Search in Google Scholar Back to article
[19] Guo, L., Gui, W. (2018). Bayesian and classical estimation of the inverse Pareto distribution and its application to strength-stress models. American Journal of Mathematical and Management Sciences, 37 (1), 80-92. https://doi.org/10.1080/01966324.2017.1383217
Search in Google Scholar Back to article
[20] Bourguignon, M., Silva, R. B., Zea, L. M., Cordeiro, G. M. (2013). The kumaraswamy Pareto distribution. Journal of Statistical Theory and Applications, 12 (2), 129-144. https://doi.org/10.2991/jsta.2013.12.2.1
Search in Google Scholar Back to article
[21] Aarset, M. V. (1987). How to identify a bathtub hazard rate. IEEE Transactions on Reliability, 36 (1), 106-108. https://doi.org/10.1109/TR.1987.5222310
Search in Google Scholar Back to article

Modeling Extreme Values with Alpha Power Inverse Pareto Distribution

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