# A New Generalization of the Fréchet Distribution: Properties and Application

## DOI:

https://doi.org/10.6092/issn.1973-2201/8462## Keywords:

Fréchet distribution, Hazard rate function, Maximum likelihood estimation, Moments, T-X family of distributions## Abstract

A new generalization of the Fréchet distribution is introduced and studied. Its structural properties including the quantile function, random number generation, moments, moment generating function and order statistics are investigated. The unknown parameters of the model are estimated using maximum likelihood estimation method and a simulation study is carried out to check the performance of the method. The new model is applied to a real data set to prove empirically its flexibility.

## References

K. ABBAS, Y. TANG (2015). Analysis of Fréchet distribution using reference priors. Communications in Statistics-Theory and Methods, 44, pp. 2945–2956.

A. Z. AFIFY, G. G. HAMEDANI, I. GHOSH, M. E. MEAD (2015). The transmuted Marshall-Olkin Fréchet distribution: Properties and applications. International Journal of Statistics and Probability, 4, pp. 132–148.

A. Z. AFIFY, H. M. YOUSOF, G. M. CORDEIRO, E. M. M. ORTEGA, Z. M. NOFAL (2016). The Weibull Fréchet distribution and its applications. Journal of Applied Statistics, 43, pp. 2608–2626.

A. AHMAD, S. P. AHMAD, A. AHMED (2014). Transmuted inverse Rayleigh distribution: A generalization of the inverse Rayleigh distribution. Mathematical Theory and Modeling, 4, pp. 90–98.

W. BARRETO-SOUZA, G. M. CORDEIRO, A. B. SIMAS (2011). Some results for beta Fréchet distribution. Communications in Statistics-Theory and Methods, 40, pp. 798–811.

G. CHEN, N. BALAKRISHNAN (1995). A general purpose approximate goodness-of-fit test. Journal of Quality Technology, 27, pp. 154–161.

R. V. DA SILVA, T. A. N. DE ANDRADE, D. B. M. MACIEL, R. P. S. CAMPOS, G. M. CORDEIRO (2013). A new lifetime model: The gamma extended Fréchet distribution. Journal of Statistical Theory and Applications, 12, pp. 39–54.

I. ELBATAL, G. ASHA, A. V. RAJA (2014). Transmuted exponentiated Fréchet distribution: Properties and applications. Journal of Statistics Applications and Probability, 3, pp. 379–394.

E. J. GUMBEL (1965). A quick estimation of the parameters in Fréchet's distribution. Review of the International Statistical Institute, 33, pp. 349–363.

K. JAYAKUMAR (2003). Mittag-Leffler process. Mathematical and Computer Modelling, 37, pp. 1427–1434.

K. JAYAKUMAR, M. G. BABU (2017). T-transmuted X family of distributions. Statistica, 77, pp. 251–276.

K. JAYAKUMAR, M. G. BABU (2018). Discrete Weibull geometric distribution and its properties. Communications in Statistics- heory and Methods, 47, pp. 1767–1783.

K. JAYAKUMAR, R.N. PILLAI (1993). The first-order autoregressive Mittag-Leffler process. Journal of Applied Probability, 30, pp. 462–466.

A. Z. KELLER, A. R. KAMATH (1982). Reliability analysis of CNC machine tools. Reliability Engineering, 3, pp. 449–473.

S. KOTZ, S. NADARAJAH (2000). Extreme Value Distributions Theory and Applications. World Scientific, London.

E. KRISHNA, K. K. JOSE, T. ALICE, M. M. RISTIC (2013). The Marshall-Olkin Fréchet distribution. Communications in Statistics-Theory and Methods, 42, pp. 4091–4107.

E. T. LEE, J. WANG (2003). Statistical Methods for Survival Data Analysis. John Wiley and Sons, New York.

M. R. MAHMOUD, R. M. MANDOUH (2013). On the transmuted Fréchet distribution. Journal of Applied Sciences Research, 9, pp. 5553–5561.

M. E. MEAD, A. R. ABD-ELTAWAB (2014). Anote on Kumaraswamy Fréchet distribution. Australian Journal of Basic and Applied Sciences, 8, pp. 294–300.

M. E. MEAD, A. Z. AFIFY, G. G. HAMEDANI, I. GHOSH (2017). The beta exponential Fréchet distribution with applications. Austrian Journal of Statistics, 46, pp. 41–63.

S. NADARAJAH, K. JAYAKUMAR, M. M. RISTIC (2013). A newfamily of lifetime models. Journal of Statistical Computation and Simulation, 83, pp. 1389–1404.

S. NADARAJAH, S. KOTZ (2003). The exponentiated Fréchet distribution. InterStat Electronic Journal, 31, pp. 1–7.

S. NADARAJAH, S. KOTZ (2008). Sociological models based on Fréchet random variables. Quality & Quantity, 42, pp. 89–95.

Z. M. NOFAL, M. AHSANULLAH (2019). A new extension of the Fréchet distribution: Properties and its characterization. Communications in Statistics-Theory and Methods, 48, pp. 2267–2285.

P. E. OGUNTUNDE, A. O. ADEJUMO (2015). The transmuted inverse exponential distribution. International Journal of Advanced Statistics and Probability, 3, pp. 1–7.

R. N. PILLAI, K. JAYAKUMAR (1995). Discrete Mittag-Leffler distributions. Statistics & Probability Letters, 23, pp. 271–274.

V. G. VODA (1972). On the inverse Rayleigh distributed random variable. Reports of Statistical Application Research, 19, pp. 13–21.

H. M.YOUSOF,A. Z. AFIFY, A. N. EBRAHEIM, G. G.HAMEDANI,N. S. BUTT (2016). On six-parameter Fréchet distribution: Properties and applications. Pakistan Journal of Statistics and Operation Research, 12, pp. 281–299.

M. ZAYED, N. S. BUTT (2017). The extended Fréchet distribution: Properties and applications. Pakistan Journal of Statistics and Operation Research, 13, pp. 529–543.

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*Statistica*,

*79*(3), 267–289. https://doi.org/10.6092/issn.1973-2201/8462

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