Rana Muhammad Imran Arshad https://orcid.org/0000-0002-5687-4634 , Muhammad Hussain Tahir https://orcid.org/0000-0002-2157-3997 , Christophe Chesneau https://orcid.org/0000-0002-1522-9292 , Farrukh Jamal https://orcid.org/0000-0001-6192-9890
ARTICLE

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ABSTRACT

In this paper, we introduce a new family of univariate continuous distributions called the Gamma Kumaraswamy-generated family of distributions. Most of its properties are studied in detail, including skewness, kurtosis, analytical comportments of the main functions, moments, stochastic ordering and order statistics. The next part of the paper focuses on a particular member of the family with four parameters, called the gamma Kumaraswamy exponential distribution. Among its advantages, the following should be mentioned: the corresponding probability density function can have symmetrical, left-skewed, right-skewed and reversed-J shapes, while the corresponding hazard rate function can have (nearly) constant, increasing, decreasing, upside-down bathtub, and bathtub shapes. Subsequently, the inference on the gamma Kumaraswamy exponential model is performed. The method of maximum likelihood is applied to estimate the model parameters. In order to demonstrate the importance of the new model, analyses on two practical data sets were carried out. The results proved more favourable for the studied model than for any of the other eight competitive models.

KEYWORDS

Kumaraswamy distribution, gamma distribution, generalised family, moments, stochastic ordering, maximum likelihood method, data analysis

REFERENCES

ABRAMOWITZ, M., STEGUN, I. A., (1965). Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Dover Publications.

ALDAHLAN, M. A., JAMAL, F., CHESNEAU, C., ELGARHY, M. and ELBATAL, I., (2019). The truncated Cauchy power family of distributions with inference and applications, Entropy, 22, p. 346.

ALEXANDER, C., CORDEIRO, G. M., ORTEGA, E. M. M. and SARABIA, J. M., (2012). Generalized beta-generated distributions. Computational Statistics & Data Analysis, 56, pp. 1880–1897.

ALJARRAH, M. A., LEE, C. and FAMOYE, F., (2014). On generating T-X family of distributions using quantile functions. Journal of Statistical Distributions and Applications, 1, Article No. 2.

ALZAATREH, A., FAMOYE, F. and LEE, C., (2014). T-normal family of distributions: A new approach to generalize the normal distribution. Journal of Statistical Distributions and Applications, 1, Article No. 16.

ALZAATREH, A., LEE, C. and FAMOYE, F., (2013a). A new method for generating families of distributions. Metron, 71, pp. 63–79.

ALZAGHAL, A., LEE, C. and FAMOYE, F., (2013b). Exponentiated T-X family of distributions with some applications. International Journal of Probability and Statistics, 2, pp. 31–49.

AMINI, M., MIRMOSTAFAEE, S. M. T. K. and AHMADI, J., (2014). Log-gammagenerated families of distributions. Statistics, 48, pp. 913–932.

ASGHARZADEH, A., BAKOUCH, H. S. and HABIBI, M., (2016). A generalized binomial exponential 2 distribution: modeling and applications to hydrologic events. Journal of Applied Statistics, 44, pp. 2368–2387.

BOURGUIGNON, M., SILVA, R. B. and CORDEIRO, G. M., (2014). The Weibull-G family of probability distributions. Journal of Data Science, 12, pp. 53–68.

CORDEIRO, G. M., ALIZADEH, M. and ORTEGA, E. M. M., (2014). The exponentiated half-logistic family of distributions: Properties and applications. Journal of Probability and Statistics Article ID 864396, 21 pages.

CORDEIRO, G. M., DE CASTRO, M., (2011). A new family of generalized distributions. Journal of Statistical Computation and Simulation, 81, pp. 883–893.

CORDEIRO, G. M., ORTEGA, E. M. M. and DA CUNHA, D. C. C., (2013). The exponentiated generalized class of distributions. Journal of Data Science, 11, pp. 1–27.

CORDEIRO, G. M., ORTEGA, E. M. M. and NADARAJAH, S., (2010). The Kumaraswamy Weibull distribution with application to failure data. Journal of the Franklin Institute, 347, pp. 1399–1429.

DAVID, H. A., NAGARAJA, H. N., (2003). Order Statistics. John Wiley and Sons, New Jersey.

DE PASCOA, M. A. R., ORTEGA, E. M. M. and CORDEIRO, G. M., (2011). The Kumaraswamy Weibull distribution with application to failure data. Journal of Franklin Institute, 347, pp. 1399–1429.

EUGENE, N., LEE, C. and FAMOYE, F., (2002). Beta-normal distribution and its applications. Communications in Statistics - Theory and Methods, 31, pp. 497–512.

GOMES, A. E., DA SILVA, C. Q., CORDEIRO, G. M. and ORTEGA, E. M. M., (2014). A new lifetime model: The Kumaraswamy generalized Rayleigh distribution. Journal of Statistical Computation and Simulation, 84, pp. 290–309.

GRADSHTEYN, I. S., RYZHIK, I. M., (2000). Table of Integrals, Series and Products. Academic Press, New York.

HOSSEINI, B., AFSHARI, M. and ALIZADEH, M., (2018). The Generalized Odd Gamma- G Family of Distributions: Properties and Applications. Austrian Journal of Statistics, 47, pp. 69–89.

JAMAL, F., CHESNEAU, C. and ELGARHY, M., (2020). Type II general inverse exponential family of distributions, Journal of Statistics and Management Systems 23, 3, pp. 617–641.

JAMAL, F., NASIR, M. A., OZEL, G., ELGARHY, M. and KHAN, N. M., (2019). Generalized inverted Kumaraswamy generated family of distributions: theory and applications. Journal of Applied Statistics, 46, pp. 2927–2944.

JAMAL, F., NASIR, M. A., TAHIR, M. H. and MONTAZERI, N. H., (2017). The odd Burr-III family of distributions. Journal of Statistics Applications and Probability, 6, pp. 105–122.

JONES, M. C., (2004). Families of distributions arising from the distributions of order statistics. Test, 13, pp. 1–43.

JONES, M. C., (2008). Kumaraswamy’s distribution: A beta-type distribution with some tractability advantages. Statistical Methodology, 6, pp. 70–81.

KENNEY, J., KEEPING, E., (1962). Mathematics of Statistics. Vol. 1, 3rd edition, Princeton: NJ, Van Nostrand.

LEE, C., FAMOYE, F. and OLUMOLADE, O., (2007). Beta-Weibull Distribution: Some Properties and Applications to Censored Data. Journal of Modern Applied Statistical Methods, 6, pp. 173–186.

MOORS, J. J. A., (1998). A quantile alternative for kurtosis. Statistician, 37, pp. 25–32.

NASIR, A., BAKOUCH, H. S. and JAMAL, F., (2018). Kumaraswamy Odd Burr G Family of Distributions with Applications to Reliability Data. Studia Scientiarum Mathematicarum Hungarica, 55, pp. 1–21.

NASIR, M. A., TAHIR, M. H., CHESNEAU, C., JAMAL, F. and SHAH, M. A. A., (2020). The odds generalized gamma-G family of distributions: Properties, regressions and applications. Statistica, 80, 1, pp. 3–38.

OGUNTUNDE, P. E., ODETUNMIBI, O. A. and ADEJUMO, A. O., (2015). On the Exponentiated GeneralizedWeibull Distribution: A Generalization of theWeibull Distribution. Indian Journal of Science and Technology, 8, pp. 1–7.

OLUYEDE, B. O., PU, S., MAKUBATE, B. and QIU, Y., (2018). The Gamma-Weibull-G Family of Distributions with Applications. Austrian Journal of Statistics, 47, pp. 45–76.

PARANAIBA, P. F., ORTEGA, E. M. M., CORDEIRO, G. M. and de Pascoa, M. A. D., (2012). The Kumaraswamy Burr XII distribution: Theory and practice. Journal of Statistical Computation and Simulation, 82, pp. 1–27.

RAMOS, M.W. A., (2014). Some new extended distributions: theory and applications, 88 f. Tese (Doutorado em Matem´atica Computacional). Universidade Federal de Pernambuco. Recife.

RISTIC´ , M. and BALAKRISHNAN, N., (2012). The gamma-exponentiated exponential distribution. Journal of Statistical Computation and Simulation, 82, pp. 1191–1206.

RODRIGUES, J. A., SILVA, A. P. C., (2015). The exponentiated Kumaraswamyexponential distribution. British Journal of Applied Science and Technology, 10, pp. 1–12.

SHAKED, M., SHANTHIKUMAR, J. G., (1994). Stochastic orders and their applications. Academic Press, New York.

TORABI, H., MONTAZARI, N. H., (2012). The gamma-uniform distribution and its application. Kybernetika, 48, pp. 16–30.

TORABI, H., MONTAZARI, N. H., (2014). The logistic-uniform distribution and its application, Communications in Statistics - Simulation and Computation, 43, pp. 2551–2569.

ZOGRAFOS, K., BALAKRISHNAN, N., (2009). On families of beta- and generalized gamma-generated distributions and associated inference. Statistical Methodology, 6, pp. 344–362.

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