The Weibull-Power Cauchy distribution: model, properties and applications

Year 2017, Volume: 46 Issue: 4, 767 - 789, 01.08.2017

M. H. Tahir M. Zubair Gauss M. Cordeiro Ayman Alzaatreh M. Mansoor

https://izlik.org/JA37SY62XE

Abstract

We propose a new three-parameter distribution with increasing, decreasing, reversed-J and upside-down bathtub shaped hazard rate, called the Weibull-power Cauchy distribution. We obtain explicit expressions for the mode, ordinary, negative and incomplete moments, mean deviations, mean residual life, quantile and generating functions, order statistics, Shannon entropy and reliability. We derive a power series
for the quantile function using exponential partial Bell polynomials. A useful characterization of the new distribution is also presented. The
method of maximum likelihood is used to estimate the model parameters. The importance of the new distribution is proved empirically by
means of three real-life data sets.

Keywords

Cauchy distribution , half-Cauchy distribution , moments , power Cauchy distribution , T-X family , Weibull-X family

References

• Alshawarbeh, E., Famoye, F. and Lee, C. Beta-Cauchy distribution: Some properties and applications, Journal of Statistical Theory and Applications 12, 378391, 2013.
• Alshawarbeh, E., Lee, C. and Famoye, F. The beta-Cauchy distribution, Journal of Probabilty and Statistical Science 10, 4157, 2012.
• Alzaatreh, A., Famoye, F. and Lee, C. Gamma-Pareto distribution and its applica- tions, Journal of Modern Applied Statistical Methods 11, 7894, 2012.
• Alzaatreh, A., Famoye, F. and Lee, C. Weibull-Pareto distribution and its applica- tions, Communications in StatisticsTheory and Methods 42, 16731691, 2013.
• Alzaatreh, A., Famoye, A. and Lee, C. The gamma-normal family of distribution: Properties and applications, Computational Statistics and Data Analysis 69, 6780, 2014.
• Alzaatreh, A. and Ghosh, I. A study of the Gamma-Pareto (IV) distribution and its applications, Communications in StatisticsTheory and Methods in press, 2015.
• Alzaatreh, A., Ghosh, I. and Said, H. On the gamma-logistic distribution, Journal of Modern Applied Statistical Methods 13, 5570, 2014.
• Alzaatreh, A., Lee, C. and Famoye, F. A new method for generating families of continuous distributions, Metron 71, 6379, 2013.
• Alzaatreh, A., Famoye, F. and Lee, C. T-normal family of distributions: A new approach to generalize the normal distribution, Journal of Statistical Distributions and Applications 1, Article 16, 2014.
• Alzaatreh, A. and Knight, K. On the gamma-half Normal distribution and its appli- cations, Journal of Modern Applied Statistical Methods 12, 103119, 2013.
• Chen, G. and Balakrishnan, N. A general purpose approximate goodness-of-fit test, Journal of Quality Technology 27, 154161, 1995.
• Comtet, L. Advanced Combinatorics (D. Reidel Publishing Co., Dordrechet, 1974).
• Cordeiro, G.M. and de Castro, M. A new family of generalized distributions, Journal of Statistical Computation and Simulation 81, 883893, 2011.
• Cordeiro, G.M. and Lemonte, A.J. The beta-half Cauchy distribution, Journal of Probability and Statistics Article ID 904705, 18 pages, 2011.
• Dahiya, R.C., Staneski, P.G. and Chaganty, N.R. Maximum likelihood estimation of parameters of the truncated Cauchy distribution, Communications in Statistics Theory and Methods 30, 17371750, 2001.
• Duncan, A.J. Quality Control and Industrial Statistics (Irwin Homewood, USA, 1974).
• Eugene, N., Lee, C. and Famoye, F. Beta-normal distribution and its applications, Communications in StatisticsTheory and Methods 31, 497512, 2002.
• Forbes, C., Evans, M., Hastings, N. and Peacock, B. Statistical Distributions, fourth edition (Wiley, New York, 2011).
• Ghosh, I. The Kumaraswamy-half Cauchy distribution: Properties and applications, Journal of Statistical Theory and Applications 13, 122134, 2014.
• Glänzel, W. A characterization theorem based on truncated moments and its ap- plication to some distribution families, In: Mathematical Statistics and Probability Theory, Volume B (Reidel, Dordrecht, pp. 7584, 1987).
• Gradshteyn, I.S. and Ryzhik, I.M. Table of Integrals, Series, and Products, Sixth edition (Academic Press, San Diego, 2000).
• Gross, A.J. and Clark, V.A. Survival distributions: Reliability applications in the biomedical sciences (Wiley, New York, 1975).
• Hamedani, G.G. and Ghosh, I. Kumaraswamy-half-Cauchy distribution: Charac- terizations and related results, International Journal of Statistics and Probability 4, 94100, 2015.
• Jacob, E. and Jayakumar, K. On half-Cauchy distribution and process, International Journal of Statistika and Mathematika 3, 7781, 2012.
• Johnson, N.L., Kotz, S. and Balakrishnan, N. Continuous Univariate Distributions, Volume 1, Second edition (Wiley, New York, 1994).
• Kravchuk, O.Y. Rank test for location optimal for hyperbolic secant distribution, Communications in StatisticsTheory and Methods 34, 16171630, 2005.
• Kravchuk, O.Y. and Pollett, P.K. Hodges-Lehmann scale estimator for Cauchy dis- tribution, Communications in StatisticsTheory and Methods 41, 36213632, 2012.
• Krishnamoorthy, K. Handbook of Statistical Distributions with Applications (Chapman and Hall, London, 2006).
• Lee, C., Famoye, F. and Alzaatreh, A. Methods for generating families of continuous distribution in the recent decades, WIRs on Computational Statistics 5, 219238, 2013.
• Manoukian, E.B. and Nadeau, P. A note on hyperbolic secant distribution, The American Statistician 42, 7779, 1988.
• Marshall, A.N. and Olkin, I. A new method for adding a parameter to a family of distributions with applications to the exponential and Weibull families, Biometrika 84, 641652, 1997.
• Nadarajah, S. and Kotz, S. A truncated Cauchy distribution, International Journal of Mathematical Education in Science and Technology 37, 605608, 2006.
• Nadarajah, S. and Kotz, S. R programs for computing truncated dstributions, Journal of Statistical Software 16, 605608, 2006.
• Nadarajah, S. and Kotz, S. Programs in R for computing truncated Cauchy distribu- tion, Quality Technology and Quality Management 4, 407412, 2007.
• Ohakwe, J. and Osu, B. The existence of the moments of the Cauchy distribution under a simple transformation of dividing with a constant, Journal of Theoretical Mathematics and Application 1, 2735, 2011.
• Proschan, F. Theoretical explanation of observed decreasing failure rate, Technometrics 5, 375383, 1963.
• Prudnikov, A.P., Bruychkov, Y.A. and Marichev, O.I. Integral and Series, Volume 3 (Gordon and Breach, New York, 1986).
• R Development Core Team R- A Language and Environment for Statistical Com- puting (R Foundation for Statistical Computing, Austria, Vienna, 2009).
• Rider, P.R. Generalized Cauchy distribution, Annals of Mathematical Statistics 9, 215223, 1957.
• Rooks, B., Schumacher, A. and Cooray, K. The power Cauchy distribution: deriva- tion, description, and composite models, NSF-REU Program Reports, 2010. Available from http://www.cst.cmich.edu/mathematics/research/REU_and_LURE.shtml
• Tahir, M.H., Cordeiro, G.M., Alizadeh, M., Mansoor, M., Zubair, M. and Hamedani, G.G. The odd generalized exponential family of distributions with applications, Journal of Statistical Distributions and Applications 2, Article 1, 2015.
• Vrbik, J. Sampling distribution of ML estimators: Cauchy example, Mathematica Journal 13, 1319, 2011.
• Wright, E.M. The asymptotic expansion of the generalized hypergeometric function, Journal of London Mathemnatical Society 10, 286293, 1935.