### Statistical Modeling using A New Hybrid Form of The Inverted Exponential Distribution with Different Estimation Methods

#### Abstract

This paper introduces a new four-parameter distribution called the exponentiated Gompertz generated inverted exponential (EGG_{IE}) distribution. Explicit expressions of the structural properties such as the ordinary and incomplete moments, probability weighted moments, quantile function, Lorenz and Bonferroni curves, entropies, and order statistics are derived. The empirical findings indicate that the maximum likelihood procedure dominates the other estimators in the simulation study while the Cramer-Von Mises procedure dominates in the two real datasets applications. We demonstrate the superiority of the EGG_{IE} distribution over the Gompertz Lomax, odd Fréchet Inverse exponential, generalized inverse exponential, generalized inverse exponential, exponential inverse exponential, and Gompertz Weibull distribution using the maximum likelihood procedure utilizing two real datasets applications. The findings show that the EGG_{IE} distribution yields the best goodness of fit to the two datasets.

**Keywords: **exponentiated Gompertz generated family; inverse exponential distribution; Kolmogorov-Smirnov statistic; Anderson-Darling; maximum product spacing.

**Abstrak**

*Paper ini memperkenalkan distribusi 4-parameter baru yang disebut dengan distribusi* exponentiated Gompertz generated inverted exponential (EGGIE). *Ekspresi eksplisit sifat struktural dari distribusi ini diturunkan, seperti momen biasa dan momen tak lengkap, momen probabilitas terboboti, fungsi kuartil, kurva* Lorenz *dan* Bonferroni, *entropi, dan statistik urutan. Temuan empiris menunjukan bahwa prosedur maksimum likelihood mendominasi estimator lainnya pada studi simulasi, sementara prosedur* Cramer-Von Mises *mendominasi pada aplikasi dua dataset nyata. Peneliti menunjukkan keunggulan dari distribusi* EGGIE *dibandingkan distribusi* Gompertz Lomax, odd Frechet Inverse exponential, generalized inverse exponential, exponential inverse exponential, *dan* Gompertz Weibull *menggunakan metode maksimum likelihood yang diaplikasikan pada dua dataset nyata. Hasil menunjukan bahwa distribusi *EGGIE* menghasilkan kecocokan model yang baik pada kedua dataset.*

**Kata Kunci:** *keluarga bangkitan *exponentiated Gompertz; *distribusi* inverse exponential; Kolmogorov-Smirnov statistic; Anderson-Darling; maximum product spacing.

**2020MSC:** 62E10

#### Keywords

#### References

A. Z. Keller and A. R. Kamath, "Reliability analysis of CNC machine tools," Reliability Engineering, vol. 3, pp. 449-473, 1982.

C. T. Lin, B. S. Duran and T. O. Lewis, "Inverted Gamma as a life distribution," Microelectronics Reliab, vol. 29, p. 619–626, 1989.

A. Sharifah, "The odd Fréchet inverse exponential distribution with application," Journal of Nonlinear Sciences and Applications, vol. 12, pp. 535-542, 2019.

B. Singh and R. Goel, "The beta inverted exponential distribution: Properties and applications," Int. J. Applied Sci. Math., vol. 2, p. 132–141, 2015.

J. T. Eghwerido, S. C. Zelibe and E. Efe-Eyefia, "Gompertz Alpha-power inverse exponential distribution: properties and applications," Thailand Statistician, vol. 18, no. 3, pp. 319-332, 2020.

T. G. Leren and J. Abdullahi, "Properties and applications of a two-parameter inverse exponential distribution with a decreasing failure rate," Pak. J. Statist., vol. 36, no. 3, pp. 183-206, 2020.

S. S. Abdulkadir, J. Jerry and T. G. Leren, "Statistical properties of Lomax inverse exponential distribution and application to real life data," FUDMA Journal of Sciences (FJS), vol. 4, no. 2, pp. 680-694, 2020.

B. O. Sule, "A new extended generalized inverted exponential distribution: properties and applications," Asian Journal of Probability and Statistics, vol. 11, no. 2, pp. 30-46, 2021.

J. T. Eghwerido, "A new Weibull inverted exponential distribution: properties and applications," FUPRE Journal of Scientific and Industrial Research, vol. 6, no. 1, pp. 58-72, 2022.

A. Alzaatreh, C. Lee and F. Famoye, "A new method for generating families of continuous distributions," Metron, vol. 71, pp. 63-79, 2013.

G. M. Cordeiro, M. Alizadeh, A. D. C. Nascimento and M. Rasekhi, "The exponentiated Gompertz generated family of distributions: Properties and Applications," Chilean Journal of Statistics, vol. 7, no. 2, pp. 29-50, 2016.

J. F. Kenney and E. Keeping, Mathematics of Statistics, D. Van Nostr and Company, 1962.

J. J. Moors, "A quantile alternative for Kurtosis," The Statistician, vol. 37, pp. 25-32, 1988.

H. A. David, Order statistics, New York: John Wiley & Sons, 1981.

J. A. Greenwood, J. M. Landwehr and N. C. Matalas, "Probability weighted moments: Definitions and relations of parameters of several distributions expressible in inverse form," Water Resources Research, vol. 15, pp. 1049-1054, 1979.

C. E. Shannon, "A mathematical theory of communication," Bell System Technical Journal, vol. 27, pp. 379-423, 1948.

C. Tsallis, "Possible generalization of Boltzmann-Gibbs statistics," Journal of Statistical Physics, vol. 52, no. 1-2, pp. 479-487, 1988.

A. Rényi, Proceeding of the fourth Berkeley symposium on mathematical statistics and probabilities, First Edition, University of California Press Berkeley, 1961.

R. Cheng and N. Amin, "Maximum product of spacing estimation with application to Lognormal distribution. Mathematical Report 79-1," Cardiff, UK, University of Wales, 1979.

R. Cheng and N. Amin, "Estimating parameters in continuous univariate distributions with a shifted origin," J. R. Stat. Soc. Ser. B Methodol, vol. 45, pp. 394-403, 1983.

B. Ranneby, "The maximum spacing method: An estimation method related to the maximum likelihood method," Scand. J. Stat., vol. 11, pp. 93-112, 1984.

A. Luceño, "Fitting the generalized Pareto distribution to data using maximum goodness-of-fit estimators," Comput Stat. Data Anal., vol. 51, pp. 904-917, 2006.

P. MacDonald, "Comment on an estimation procedure for mixtures of distribution by Choi and Bulgren," J. R. Stat. Soc. Ser. B Methodol, vol. 29, pp. 271-329, 1971.

T. Abouelmagd, S. Al-mualim, A. Afify, M. Ahmad and H. Al-Mofleh, "The Odd Lindley Burr XII Distribution with Applications," Pakistan Journal of Statistics, vol. 34, no. 1, pp. 15-32, 2018.

M. E. Mead, G. M. Cordeiro, A. Afify and H. Al-Mofleh, "The Alpha Power Transformation Family: Properties and Applications," Pakistan Journal of Statistics and Operation Research, vol. 15, no. 3, pp. 525-545, 2019.

S. C. Zelibe, J. T. Eghwerido and E. Efe-Eyefia, "Kumaraswamy Alpha Power Inverted Exponential Distribution: Properties and Applications," Journal of the Turkish Statistical Association, vol. 12, no. 1-2, p. 35–48, 2019.

A. A. Al-Bastian, I. Elbasan, H. Al-Mofleh, A. M. Gemeay, A. Z. Afify and A. M. Sarg, "The Flexible Burr X-G Family: Properties, Inference, and Applications in Engineering Science," Symmetry, vol. 13, no. 3, p. 474, 2021.

P. E. Oguntunde, A. O. Adejumo and E. A. Owoloko, "Exponential Inverse Exponential (EIE) Distribution with Applications to Lifetime Data," Asian Journal of Scientific Research, vol. 10, pp. 169-177, 2017a.

P. E. Oguntunde, A. O. Adejumo and O. S. Balogun, "Statistical properties of the exponentiated generalized inverse exponential distribution," Applied Mathematics, vol. 4, no. 2, pp. 47-55, 2014.

A. M. Abouammoh and A. M. Alshingiti, "Reliability estimation of generalized inverted exponential distribution," Journal of Statistical Communication and Simulation, vol. 79, no. 11, pp. 1301-1315, 2009.

P. E. Oguntunde, M. A. Khalee, M. T. Ahmed, A. O. Adejumo and O. A. Odetunmibi, " (2017b). A New Generalization of the Lomax Distribution with Increasing, Decreasing, and Constant Failure Rate," Modelling and Simulation in Engineering, vol. 6043169, 2017.

------------, The Gompertz Weibull distribution. Properties and application, Unpublished.

DOI: 10.15408/inprime.v4i2.26830

### Refbacks

- There are currently no refbacks.