### The Inverted Exponentiated Gamma Distribution: A Heavy-Tailed Model with Upside Down Bathtub Shaped Hazard Rate

#### Abstract

In this article, we proposed and studied the inverted exponentiated gamma distribution (IEGD). IEGD is obtained by considering the inverse transformation of exponentiated gamma variate. This distribution has been motivated by the extensive use of the exponentiated gamma model in many applied areas and also due to the fact that this new generalization provides more flexibility to analyze real data with upside down bathtub (UBT) hazard rate. The shape of the distribution has been traced mathematically and found that the proposed model is compatible with UBT hazard rate models. The tail area property is also presented based on the idea of Marshall and Olkin (2007) and it is concluded that the new model belongs to the family of heavy-tailed distributions. Some other characteristics such as reliability, hazard, the quantile function, skewness and kurtosis, stochastic ordering, stress-strength reliability and order statistics have been explicitly derived. The classical and Bayesian estimation procedures have been discussed to estimate the unknown parameter of IEGD. The performances of classical and Bayes estimators are studied in terms of average mean square error (MSE) by conducting Monte Carlo simulations. Finally, a real data set with UBT type hazard rate is analyzed for the illustrative purpose of the study.

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D. D. BOOS (1981). Minimum distance estimators for location and goodness of fit. Journal of the American Statistical association, 76, no. 375, pp. 663–670.

A. L. BOWLEY (1920). Elements of Statistics, vol. 2. PS King & Son, London.

R. C. H.CHENG,N.A. K.AMIN (1983). Estimating parameters in continuous univariate distributions with a shifted origin. Journal of the Royal Statistical Society, Series B, 45, no. 3, pp. 394–403.

R. S. CHHIKARA, J. L. FOLKS (1977). The inverse Gaussian distribution as a lifetime model. Technometrics, 19, pp. 461–468.

K. CHOI,W. G. BULGREN (1968). An estimation procedure for mixtures of distributions. Journal of the Royal Statistical Society, Series B, 30, no. 3, pp. 444–460.

F. P. A. COOLEN, M. J. NEWBY (1990). A Note on the Use of the Product of Spacings in Bayesian Inference, vol. 9035 of Memorandum COSOR. Technische Universiteit Eindhoven, Eindhoven.

F. DOMMA, F. CONDINO, B. V. POPOVIC (2017). A new generalized weighted Weibull distribution with decreasing, increasing, upside-down bathtub, n-shape and m-shape hazard rate. Journal of Applied Statistics, 44, no. 16, pp. 2979–2993.

S. FOSS, D. KORSHUNOV, S. ZACHARY (2011). An Introduction to Heavy-Tailed and Subexponential Distributions, vol. 6. Springer, New York.

R. E. GLASER (1980). Bathtub and related failure rate characterizations. Journal of the American Statistical Association, 75, no. 371, pp. 667–672.

S. A. KLUGMAN, H. H. PANJER, G. E. WILLMOT (2012). Loss Models: From Data to Decisions, vol. 715. John Wiley & Sons, New York.

S. KOTZ, Y. LUMELSKII, , M. PENSKY (2003). The Stress-Strength Model and its Generalizations: Theory and Applications. New York: World Scientific.

C. LIN, B. S. DURAN, T. O. LEWIS (1989). Inverted gamma as a life distribution. Microelectron Reliability, 29, no. 4, pp. 619–626.

D. V. LINDLEY (1980). Approximate Bayes method. Trabajos De Estadistica, 31, pp. 223–237.

P. D. M. MACDONALD (1971). Comment on "An estimation procedure for mixtures of distributions" by Choi and Bulgren. Journal of the Royal Statistical Society, Series B, 33, no. 2, pp. 326–329.

A. W. MARSHALL, I. OLKIN (2007). Life Distributions, vol. 13. Springer, New York.

J. J. A. MOORS (1988). A quantile alternative for kurtosis. The Statistician, 37, no. 1, pp. 25–32.

J. NAIR, A. WIERMAN, B. ZWART (2013). The fundamentals of heavy-tails: Properties, emergence, and identification. In ACM SIGMETRICS Performance Evaluation Review. vol. 41, pp. 387–388.

M. M. RISTIC , B. V. POPOVIC , K. ZOGRAFOS, N. BALAKRISHNAN (2018). Discrimination among bivariate beta-generated distributions. Statistics, 52 (2), pp. 303–320.

M. SHAKED, J. SHANTHIKUMAR (1994). Stochastic Orders and Their Applications. Academic Press, New York.

V. K. SHARMA, S. K. SINGH, U. SINGH, V. AGIWAL (2015). The inverse Lindley distribution: A stress-strength reliability model with application to head and neck cancer data. Journal of Industrial and Production Engineering, 32, no. 3, pp. 162–173.

A. I. SHAWKY, R. A. BAKOBAN (2006). Certain characterizations of the exponentiated gamma distribution. Journal of Statistics Science, 3, no. 2, pp. 151–164.

A. I. SHAWKY, R. A. BAKOBAN (2008). Bayesian and non-Bayesian estimations on the exponentiated gamma distribution. Applied Mathematical Sciences, 2, no. 51, pp. 2521–2530.

A. I. SHAWKY, R. A. BAKOBAN (2012). Exponentiated gamma distribution: Different methods of estimation. Journal of Applied Mathematics, pp. 1–24.

S. K. SINGH, U. SINGH, D. KUMAR (2011). Bayesian estimation of the exponentiated gamma parameter and reliability function under asymmetric loss function. REVSTAT - Statistical Journal, 9, no. 3, pp. 247–260.

U. SINGH, S. K. SINGH, R. K. SINGH (2014). A comparative study of traditional estimation methods and maximum product spacings method in generalized inverted exponential distribution. Journal of Statistics Applications & Probability, 3, no. 2, pp. 153–169.

J. J. SWAIN, S. VENKATRAMAN, J. R. WILSON (1988). Least-squares estimation of distribution functions in Johnson's translation system. Journal of Statistical Computation and Simulation, 29, no. 4, pp. 271–297.

A. S. YADAV, S. S. MAITI, M. SAHA (2019). The inverse xgamma distribution: Statistical properties and different methods of estimation. Annals of Data Science, doi.org/10.1007/s40745-019-00211-w.

DOI: 10.6092/issn.1973-2201/8782