An Extended Generalized Frechet (EGFr) Distribution: Properties and It Applications to Reliability Data

Abstract

This paper introduces a novel Extended Generalized Frechet (EGFr) distribution, a flexible extension of the Frechet distribution. The EGFr incorporates additional parameters that provide enhanced flexibility for modeling diverse data sets, especially those with complex patterns or extreme values. The probability density function of the EGFr is derived from the T-X family of distributions and can be expressed as a linear combination of Frechet densities. We investigate the statistical properties of the EGFr, including moments, quantiles, hazard functions and order statistics. Maximum likelihood estimation is used to estimate the model parameters. Extensive simulations demonstrate the consistency and efficiency of the EGFr in parameter estimation. Real-life applications to reliability datasets demonstrate the superior performance of the EGFr over existing Frechet-based distributions. The EGFr’s ability to accurately capture complex data patterns and provide reliable estimates makes it a valuable tool for researchers and practitioners in the fields of reliability engineering and sciences.

Keywords

akaike information criteria / bayesian information criteria / log-likelihood / simulation / reliability function / flexibility

Supplementary Materials

The following supporting information can be downloaded at: https://jmasm.com/index.php/jmasm/article/view/1269/s1

Author Contributions

Conceptualization: J.O.B., I.S., O.A.B. and F.M.C.; Data curation: J.O.B., I.S., O.A.B. and F.M.C.; Formal analysis: J.O.B., I.S., O.A.B. and F.M.C.; Validation: J.O.B., I.S., O.A.B. and F.M.C.; Methodology: J.O.B., I.S.,O.A.B. and F.M.C.; Software: J.O.B., I.S., O.A.B. and F.M.C.; Visualisation: J.O.B., I.S., O.A.B. and F.M.C.;Supervision: F.M.C.; Writing—original draft: J.O.B., I.S., O.A.B. and F.M.C.; Writing—revision and editing: J.O.B., I.S., O.A.B. and F.M.C. All authors have read and agreed to the published version of the manuscript.

Funding

This research received no specific grant from any funding agency in the public, commercial, or not-forprofit sectors.

Data Availability Statement

No new data were generated or analyzed during this study. This research involved a secondary analysis of existing datasets which are publicly available at https://onlinelibrary.wiley.com/doi/10.1155/2023/4458562.

Acknowledgments

The University of the Free State, South Africa and Tertiary Education Trust Fund (TETFund), Nigeria is acknowledged by the support of the first author.

Conflicts of Interest

The authors declare no conflicts of interest.

Use of AI and AI-Assisted Technologies

The authors declare that they did not use AI tools for data analysis, image generation, or text drafting in this study.

References

[1] Kotz S.; Nadarajah S. Extreme Value Distributions: Theory and Applications; World Scientific: Singapore, 2001.

[2] Coles S. An Introduction to Statistical Modeling of Extreme Values. In Springer Series in Statistics; Springer: Berlin/Heidelberg, Germany, 2001.

[3] Kislov A.; Matveeva T. Extreme values of wind speed over the kara sea based on the ERA 5 dataset. Atmos. Clim. Sci. 2021, 11, 98-113. https://doi.org/10.4236/acs.2021.111007.

[4] Harlow D.G. Applications of the Frechet distribution function. Int. J. Mater. Prod. Technol. 2002, 17, 482. https://doi.org/10.1504/IJMPT.2002.005472.

[5] Resnick S.I. Preliminaries Series: Springer Series in Operations Research and Financial Engineering; Springer: Berlin/Heidelberg, Germany, 1987; pp. 1-37. https://doi.org/10.978-0-387-75953-11.

[6] Srivastava R.; Sengupta D. Consistent estimation of non-band limited spectral density from uniformly spaced samples. IEEE Trans. Inf. Theory 2010, 56, 3642-3659. https://doi.org/10.1109/TIT.2010.2050807.

[7] Campbell A.; O’Rourke S. Spectrum of heavy-tailed elliptic random matrices. Electron. J. Probab. 2022, 27, 1-56. https://doi.org/10.1214/22-EJP849.

[8] Kratz M. Normex, a new method for evaluating the distribution of aggregated heavy tailed risks: Application to risk measures. Extremes 2014, 17, 661-691. https://doi.org/10.1007/s10687-014-0197-6.

[9] Silva R.V.D.; Andrade T.A.N.D.; Maciel D.B.M.; et al. A new lifetime model: The gamma extended frechet distribution. J. Stat. Theory Appl. 2013, 12, 39-54. https://doi.org/10.2991/jsta.2013.12.1.4.

[10] Nadarajah S.; Kotz S. The Exponentiated Fréchet Distribution. InterStat. Electron. J. 2003, 14, 1-7.

[11] Arshad M.Z.; Iqbal M.Z.; Ahmad M. Transmuted Exponentiated Moment Pareto distribution. Open J. Stat. 2018, 8, 939-961. https://doi.org/10.4236/ojs.2018.86063.

[12] Ogunde A.A.; Ajao I.O.; Olalude G.A. On the application of Nadarajah Haghighi Gompertz distribution as a life time distribution. Open J. Stat. 2020, 10, 850-862. https://doi.org/10.4236/ojs.2020.105049.

[13] Enogwe S.U.; Ibeh G.C. Beta-exponentiated ishita distribution and its applications. Open J. Stat. 2021, 11, 690-712. https://doi.org/10.4236/ojs.2021.115041.

[14] Mead M.E.; Afify A.Z.; Hamedani G.G.; et al. The beta exponential Fréchet distribution with applications. Austrian J. Stat. 2017, 46, 41-63. https://doi.org/10.17713/ajs.v46i1.144.

[15] Mahmoud M.R.; Mandouh R.M. On the transmuted Frechet distribution. J. Appl. Sci. Res. 2013, 9, 5553-5561.

[16] Deka D.; Das B.; Baruah B.K.; et al. Some properties on Fréchet-Weibull distribution with application to real life data. Math. Stat. 2021, 9, 8-15. https://doi.org/10.13189/ms.2021.090102.

[17] Baharith L.A.; Alamoudi H.H. The Exponentiated Fréchet generator of distributions with applications. Symmetry 2021, 13, 572. https://doi.org/10.3390/sym13040572.

[18] Niyoyunguruza A.; Odongo L.O.; Nyarige E.; et al. Marshall-Olkin Exponentiated Frechet distribution. J. Data Anal. Inf. Process. 2023, 11, 262-292. https://doi.org/10.4236/jdaip.2023.113014.

[19] Abubakari A.G. Actuarial measures, regression, and applications of Exponentiated Fréchet loss distribution. Int. J. Math. Math. Sci. 2022, 2022, 3155188. https://doi.org/10.1155/2022/3155188.

[20] Alzaghal A.; Famoye F.; Lee C. Exponentiated T-X family of distributions with some applications. Int. J. Probab. Stat. 2013, 2, 31-49.

[21] Reyad H.M.; Alizadeh M.; Jamal F.; et al. The Exponentiated Generalized Topp Leone G Family of Distributions: Properties and Applications. Pak. J. Stat. Oper. Res. 2019, 15, 1-24. https://doi.org/10.18187/pjsor.v15i1.2166.

[22] Awopeju K.A.; Abiodun A.A. Exponentially Generated Modified Chen Distribution with Applications to Lifetime Dataset. J. Probab. Stat. 2024, 2023, 4458562. https://doi.org/10.21203/rs.3.rs-4125728/v1.