### The Extended Exponentiated Weibull Distribution and its Applications

#### Abstract

In this paper, we introduce a univariate four-parameter distribution. Several known distributions like exponentiated Weibull or extended generalized exponential distribution can be obtained as special case of this distribution. The new distribution is quite flexible and can be used quite effectively in analysing survival or reliability data. It can have a decreasing, increasing, decreasing-increasing-decreasing (DID), upside-down bathtub (unimodal) and bathtub-shaped failure rate function depending on its parameters. We provide a comprehensive account of the mathematical properties of the new distribution. In particular, we derive expressions for the moments, mean deviations, Rényi and Shannon entropy. We discuss maximum likelihood estimation of the unknown parameters of the new model for censored and complete sample using the profile and modified likelihood functions. One empirical application of the new model to real data are presented for illustrative purposes.

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E. K. AL-HUSSAINI, M. AHSANULLAH (2015). Exponentiated Distributions. Atlantis Press, Paris.

D. F. ANDREWS, A. M. HERZBERG (1985). Data: A Collection of Problems from Many Fields for the Student and Research Worker. Springer, New York.

B. C. ARNOLD, J. M. SARABIA (2018). Families of Lorenz curves. In B. C. ARNOLD, J. M. SARABIA (eds.), Majorization and the Lorenz Order with Applications in Applied Mathematics and Economics, Springer, Cham, pp. 115-143.

C. E. BONFERRONI (1930). Elementi di Statistica Generale. Seeber, Firenze.

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

K. COORAY, M. M. A. ANANDA (2008). A generalization of the halfnormal distribution wit applications to lifetime data. Communication in Statistics - Theory and Methods, 37, pp. 1323-1337.

M. H. GAIL, J. L. GASTWIRTH (1978). A scale-free goodness-of-fit test for the exponential distribution based on the Lorenz curve. Journal of the American Statistical Association, 73, no. 364, pp. 787-793.

M. E. GHITANY (1998). On a recent generalization of gamma distribution. Communications in Statistics - Theory and Methods, 27, no. 1, pp. 223-233.

G. M. GIORGI, M. CRESCENZI (2001). A look at the Bonferroni inequality measure in a reliability framework. Statistica, 4, pp. 571-583.

J. A. GREENWOOD, J. M. LANDWEHR, N. C. MATALAS, J. R. WALLIS (1979). Probability weighted moments: Definition and relation to parameters of several distributions expressable in inverse form.Water Resources Research, 15, no. 5, pp. 1049-1054.

R. D. GUPTA, D. KUNDU (1999). Generalized exponential distribution. Australian and New Zealand Journal of Statistics, 41, pp. 173-188.

R. D.GUPTA, D. KUNDU (2007). Generalized exponential distribution: Existing methods and recent developments. Journal of Statistical Planning and Inference, 137, pp. 3537-3547.

R. D. GUPTA, D. KUNDU (2011). An extension of the generalized exponential distribution. Statistical Methodology, 8, pp. 485-496.

R. A.HARVEY, J. D.HAYDEN, P. S. KAMBLE, J. R. BOUCHARD, J. C.HUANG (2017). A comparison of entropy balance and probability weighting methods to generalize observational cohorts to a population: A simulation and empirical example. Pharmacoepidemiology and Drug Safety, 26, no. 4, pp. 368-377.

J. R. M.HOSKING (1986). The Theory of Probability Weighted Moments. Research Report RC12210, IBM Thomas J. Watson Research Center, New York.

S. B. KANG, Y. S. CHO, J. T. HAN, J. KIM (2012). An estimation of the entropy for a double exponential distribution based on multiply type-II censored samples. Entropy, 14, no. 2, pp. 161-173.

S. KAYAL, S. KUMAR (2013). Estimation of the Shannon's entropy of several shifted exponential populations. Statistics and Probability Letters, 83, no. 4, pp. 1127-1135.

S. KAYAL, S. KUMAR, P. VELLAISAMY (2015). Estimating the Rényi entropy of several exponential populations. Brazilian Journal of Probability and Statistics, 29, no. 1, pp. 94-111.

E. MAHMOUDI (2011). The beta generalized Pareto distribution with application to life time data. Mathematics and Computers in Simulation, 81, pp. 2414-2430.

J. MI (1995). Bathtub failure rate and upside-down bathtub mean residual life. IEEE Transactions on Reliability, 44, no. 3, pp. 388-391.

G. S. MUDHOLKAR, D. K. SRIVASTAVA (1993). Exponentiated Weibull family for analysing bathtub failure-rate data. IEEE Transactions on Reliability, 42, pp. 299-302.

G. S. MUDHOLKAR, D. K. SRIVASTAVA, G. D. KOLLIA (1995). A generalization of the Weibull family: A reanalysis of the bus motor failure data. Technometrics, 37, pp. 436-445.

G. S. MUDHOLKAR, D. K. SRIVASTAVA, G. D. KOLLIA (1996). A generalization of the Weibull distribution with application to the analysis of survival data. Journal of the American Statistical Association, 91, pp. 1575-1585.

M. M.NASSAR, F. H. EISSA (2003). On the exponentiated Weibull distribution, Communications in Statistics - Theory and Methods, 32, no. 7, pp. 1317-1336.

K. S. PARK (1985). Effect of burn-in on mean residual life. IEEE Transactions on Reliability, 34, no. 5, pp. 522-523.

A. RÉNYI (1961). On measures of entropy and information. In Proceedings of the Fourth Berkeley Symposium on Mathematical Statistics and Probability, Volume 1: Contributions to the Theory of Statistics, University of California Press, Berkeley, pp. 547-561.

J. I. SEO, S. B. KANG (2014). Entropy estimation of generalized half-logistic distribution (GHLD) based on type-II censored samples. Entropy, 16, no. 1, pp. 443-454.

D.N. SHANBHAG (1970). The characterization of exponential and geometric distributions. Journal of the American Statistical Association, 65, pp. 1256-1259.

R. SHANKER, K. K. SHUKLA, R. SHANKER, T. A. LEONIDA (2017). A three-parameter Lindley distribution. American Journal of Mathematics and Statistics, 7, no. 1, pp. 15-26.

C. E. SHANNON (1948). A mathematical theory of communication. Bell System Technical Journal, 27, pp. 379-423.

R. L. SMITH (1985). Maximum likelihood in a class of non-regular cases. Biometrika, 72, pp. 67-90.

J. G. SURLES, W. J. PADGETT (2001). Inference for reliability and stress-strength for a scaled Burr type X distribution. Lifetime and Data Analysis, 7, pp. 187-200.

L. C. TANG, Y. LU, E. P. CHEW (1999). Mean residual life distributions. IEEE Transactions on Reliability, 48, no. 1, pp. 73-68.

A. P. TARKO (2018). Estimating the expected number of crashes with traffic conflicts and the Lomax distribution - A theoretical and numerical exploration. Accident, Analysis and Prevention, 113, pp. 63-73.

G. S.WATSON (1961). Goodness-of-fit tests on a circle. Biometrika, 48, pp. 109-114.

DOI: 10.6092/issn.1973-2201/7503