# GDUS-Modified Topp-Leone Distribution: A New Distribution with Increasing, Decreasing, and Bathtub Hazard Functions

### Abstract

In this paper, we propose an extension to the Topp-Leone distribution, as introduced by [20] using the Generalized-DUS transformation given by [8]. The Topp-Leone distribution is defined on interval (0,1) and has a characteristic J-shaped frequency curve. The newly extended version of Topp-Leone distribution accommodates a variety of shapes of hazard rate functions making it a versatile distribution. We have also derived explicit expressions for some properties like ordinary moments, conditional moments, distribution of order statistics, quantiles, mean deviation, and entropy. Further, we have also discussed results on identifiability, stress-strength reliability, and stochastic ordering that are concerned with two independent random variables. For inference regarding the unknown parameters of the distribution, we derive the equations which give their maximum likelihood estimators. We also present the asymptotic confidence intervals of the unknown parameters of the distribution, based on large sample property, using the Fisher information matrix. To facilitate further studies, a step-by-step algorithm is presented to produce a random sample from the distribution. Further, extensive simulation experiments are done to study the long-term behavior of the maximum likelihood estimators of the parameters through their mean squared error and mean absolute bias on the basis of large number of samples. The consistency of the MLEs is empirically proved. Lastly, the application of the proposed distribution is shown by fitting a real-life dataset over some existing distributions in the same range.

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### References

M. Alizadeh, F. Lak, M. Rasekhi, T. G. Ramires, H. M. Yousof, and E. Altun. The odd log-logistic topp-leone g family of distributions: heteroscedastic regression models and applications. Comput. Stat., 33, 3:1217–1244, 2018.

A. P. Basu and J. K. Ghosh. Identifiability of distributions under competing risks and complementary risks model. Communications in Statistics – Theory and Methods, 9:1515–1525, 1980.

P.C. Consul and G.C. Jain. On the log-gamma distribution and its properties. Stat Pap, 12:100–106, 1971.

E. Gómez-Déniz, M.A. Sordo, and E. Calderín-Ojeda. The log-lindley distribution as an alternative to the beta regression model with applications in insurance. Insur Math Econ, 54:49–57, 2014.

N.L. Johnson. Systems of frequency curves generated by methods of translation. Biometrika, 36(1/2):149–176, 1949.

P. Kumaraswamy. A generalized probability density function for double-bounded random processes. J Hydrol, 46(1-2):79-88, 1980.

M.Ç. Korkmaz and C. Chesneau. On the unit burr-xii distribution with the quantile regression modeling and applications. Comp. Appl. Math., 40(1), 2021.

S.K. Maurya, A. Kaushik, S.K. Singh, and U. Singh. A new class of distribution having decreasing, increasing, and bathtub-shaped failure rate. Commun. Stat. Theory Methods, 46, 20:10359–10372, 2017.

J. Mazucheli and S. Dey A.F. Menezes. The unit-birnbaum-saunders distribution with applications. Chile J Stat, 9(1):47–57, 2018.

J. Mazucheli, A.F.B. Menezes, and M.E. Ghitany. The unit-weibull distribution and associated inference. J Appl Probab Stat, 13:1–22, 2018.

J. Mazucheli, A.F. Menezes, and S. Dey. Unit-gompertz distribution with applications. Statistica, 79(1):25–43, 2019.

S. Nadarajah and S. Kotz. Moments of some j-shaped distributions. J. Appl. Stat., 35, 10:1115–1129, 2003.

A. Renyi. On measures of entropy and information. In Proceedings of the 4th Berkeley symposium on mathematical statistics and probability. Berkeley :University of California Press., 1:547–561, 1961.

R Core Team. R: A Language and Environment for Statistical Computing. R Foundation for Statistical Computing, Vienna, Austria, 2013.

Y. Sangsanit and W. Bodhisuwan. The topp-leone generator of distributions: properties and inferences. Songklanakarin. J. Sci. Technol., 38:537–548, 2016.

M. Shaked and J. Shanthikumar. Stochastic orders and their applications. Academic Press, Boston, 1994.

V.K. Sharma. Topp-leone normal distribution with application to increasing failure rate data. Journal of Statistical Computation and Simulation, 83, 2:326–339, 2018.

V.K. Sharma. R for lifetime data modeling via probability distributions. Handbook of Probabilistic Models, Elsevier Inc., 2020.

K. Shekhawat and V.K. Sharma. An extension of J-shaped distribution with application to tissue damage proportions in blood. Sankhya B, 83:543–574, 2021.

C. W. Topp and F. C. Leone. A family of J-shaped frequency functions. J. Am. Stat. Assoc., 50:209–219, 1955.

M. N. Shahzad, Z. Asghar, Transmuted Power Function Distribution: A More Flexible Distribution, Journal of Statistics and Management Systems 19(4) (2016) 519–539.

Tanış, C. (2021) On Transmuted Power Function Distribution: Characterization, Risk Measures, and Estimation. Journal of New Theory, (34), 72–81.

Karakaya, K., Kınacı, İ., Kuş, C., Akdoğan, Y. (2021). On the DUS-Kumaraswamy Distribution. Istatistik Journal of The Turkish Statistical Association, 13(1), 29–38.

Kumar, D., Singh, U. and Singh, S.K. (2015). A Method of Proposing New Distribution and its Application to Bladder Cancer Patients Data. J. Stat. Appl. Pro. Lett., 2(2), 235–245.