A New Lifetime Distribution with Properties, Characterizations, Validation Testing and Different Estimation Methods
Abstract
A new lifetime distribution is proposed and its properties are studied. The new density function has a heavy right skew tails with different shapes. The new failure rate function can be “constant”, “bathtub (U)”, “increasing-constant”, “decreasing-constant”, “upside down” and “upside down-U”. Complexity of the most integrals related to statistical properties is solved and numerically analyzed. Simple type copula is presented. Numerical calculations for analyzing the skewness and kurtosis are presented. Different estimation methods such as the maximum likelihood estimation method, Cramer-von-Mises estimation method, ordinary least square estimation method, weighted least square estimation method, Anderson Darling estimation method, right tail Anderson Darling estimation method and left tail-Anderson Darling estimation method are considered. Numerical simulation studies are performed. An example of environmental real data set is employed to compare the estimation methods. Another example is presented to measure importance and flflexibility of the new model. Using the validation approach proposed by Bagdonavicius and Nikulin (2011) for censored data, we propose the construction of modifified chi-square goodness-of-fifit tests for the new model. Based on the maximum likelihood estimators on initial data, the modifified statistics recover the information lost while grouping data and follow chi-square models. All elements of the modifified criteria tests are given explicitly. Numerical example from simulated samples and four real data sets have been analyzed to illustrate the feasibility of the modifified test.References
Abouelmagd, T. H. M., Hamed, M. S., Handique, L., Goual, H., Ali, M. M., Yousof, H. M. and Korkmaz, M. C. (2019). A new class of distributions based on the zero truncated Poisson distribution with properties and applications, Journal of Nonlinear Sciences & Applications, 12(3), 152-164.
Abouelmagd, T. H. M., Hamed, M. S., Hamedani, G. G., Ali, M. M., Goual, H., Korkmaz, M. C. and Yousof, H. M. (2019). The
zero truncated Poisson Burr X family of distributions with properties, characterizations, applications, and validation test. Journal of Nonlinear Sciences and Applications, 12(5), 314-336.
Abdul-Moniem, I. B. (2016). Exponentiated Nadarajah and Haghighi exponential distribution. International Journal of Mathematical Analysis and Applications, 2, 68-73.
Ahmed, A., Muhammed, H. Z. and Elbatal, I. (2015). A new class of extension exponential distribution. International Journal of Applied Mathematical Sciences, 8, 13–30.
Bagdonaviˇcius, V. and Nikulin, M. (2011). Chi-squared goodness-of-fit test for right censored data, International Journal of Applied Mathematics and Statistics, vol. 24, 30-50.
Bagdonaviˇcius, V., Levuliene, R., J. and Nikulin, M., (2013). Chi-squared goodness-of-fit tests for parametric accelerated failure time models, Communications in Statistics- Theory and Methods, Volume: 42 Issue: 15, 2768-2785.
Bourguignon, M., Lima, M. C. S., Leao,J., Nascimento, A. D. C., Pinho, L. G. B. and Cordeiro, G. M. (2015). A new generalized gamma distribution with applications. American Journal of Mathematical and Management Sciences, 34, 309–342.
Cordeiro, G. M. and Castro, M. (2011). A new family of generalized distributions. Journal of statistical computation and simulation, 81(7), 883-898.
Crowder, M. J., Kimber A. C., Smith R. L and Sweeting, T. J. (1991). Statistical analysis of reliability data, CHAPMAN &HALL/CRC
Dias, C. R. B., Alizadeh, M. and Cordeiro, G. M. (2018). The beta Nadarajah-Haghighi distribution. Hacettepe Journal of Mathematics and Statistics, 47, 1302 13203.
El Damcese, M. A. and Ramadan, D. A. (2015). Studies on properties and estimation problems for modified extension of exponential distribution. International Journal of Computer Applications, 125.
Elgohari, H. and Yousof, H. M. (2020). New Extension of Weibull Distribution: Copula, Mathematical Properties and Data Modeling. Statistics, Optimization & Information Computing, 8(4), 972-993. https://doi.org/10.19139/soic-2310-5070-1036
Elsayed, H. A. H. and Yousof, H. M. (2019). The Burr X Nadarajah Haghighi distribution: statistical properties and application to the exceedances of flood peaks data. Journal of Mathematics and Statistics, 15, 146-157.
John P. Klein, Melvin L. Moeschberger (1997), Survival Analysis :Techniques for Censored and Truncated Data , Statistics for Biology and Health
Galambos, J. and Kotz, S. Characterizations of probability distributions. A unified approach with emphasis on exponential and related models, Lecture Notes in Mathematics, p.675. Springer, Berlin (1978).
Gl¨anzel, W., A characterization theorem based on truncated moments and its application to some distribution families, Mathematical Statistics and Probability Theory (Bad Tatzmannsdorf, 1986), Vol. B, Reidel, Dordrecht, (1987), 75–84.
Gl¨anzel, W., Some consequences of a characterization theorem based on truncated moments, Statistics: A Journal of Theoretical and Applied Statistics, 21 (4), (1990), 613–618.
Gl¨anzel, W, Telcs, A, Schubert, A. Characterization by truncated moments and its application to Pearson-type distributions, Z. Wahrsch. Verw. Gebiete 66, 173-182 (1984).
Gl¨anzel, W. and Hamedani, G.G. Characterizations of the univariate distributions, Studia Scien. Math. hung., 37, 83-118 (2001).
Hamedani, G.G., On certain generalized gamma convolution distributions II, Technical Report, No. 484, MSCS, Marquette University, 2013.
Goual, H. and Yousof, H. M. (2020). Validation of Burr XII inverse Rayleigh model via a modified chi-squared goodness-of-fit test. Journal of Applied Statistics, 47(3), 393-423.
Goual, H., Yousof, H. M. and Ali, M. M. (2019a). Validation of the odd Lindley exponentiated exponential by a modified goodness of fit test with applications to censored and complete data. Pakistan Journal of Statistics and Operation Research, 15(3), 745-771.
Goual, H., Yousof, H. M. and Ali, M. M. (2020b). Lomax inverse Weibull model: properties, applications, and a modified Chi-squared goodness-of-fit test for validation. Journal of Nonlinear Sciences & Applications, 13(6), 330-353.
Gupta, R. D. and Kundu, D. (1999). Generalized exponential distributions. Australian & New Zealand Journal of Statistics, 41(2), 173-188.
Ibrahim, M. (2020b). The generalized odd Log-logistic Nadarajah Haghighi distribution: statistical properties and different methods of estimation. Journal of Applied Probability and Statistics, 15(2), 61-84.
Ibrahim, M., Altun, E. and Yousof, H. M. (2020a). A new distribution for modeling lifetime data with different methods of estimation and censored regression modeling. Statistics, Optimization & Information Computing, 8(2), 610-630.
Ibrahim, M., Altun, E., Goual, H., and Yousof, H. M. (2020b). Modified goodness-of-fit type test for censored validation under a new Burr type XII distribution with different methods of estimation and regression modeling. Eurasian Bulletin of Mathematics, 3(3), 162-182.
Ibrahim, M., Yadav, A. S., Yousof, H. M., Goual, H. and Hamedani, G. G. (2019). A new extension of Lindley distribution: modified validation test, characterizations and different methods of estimation. Communications for Statistical Applications and Methods, 26(5), 473-495.
Kim, J.H. and Jeon, Y. Credibility theory based on trimming, Insur. Math. Econ. 53(1), 36-47 (2013).
Kotz, S. and Shanbhag, D.N. Some new approach to probability distributions, Adv. Appl. Probab. 12, 903-921 (1980).
Lemonte, A. J. (2013). A new exponential type distribution with constant, decreasing, increasing, upside-down bathtub and bathtubshaped failure rate function. Computational Statistics & Data Analysis, 62, 149–170.
Lemonte, A. J., Cordeiro, G. M. and Arenas, G. M. (2016). A new useful threeparameter extension of the exponential distribution. Statistics, 50, 312–337.
Lee, E.T. and Wang, J. W. (2003) Statistical Methods for Survival Data Analysis, third ed. Wiley, New York.
Lima, S. R. L. (2015). The half normal generalized family and kummaraswamy Nadarajah Haghighi distribution. Master thesis, Universidade Federal de Pernumbuco.
Nadarajah, S. and Kotz, S. (2006). The beta exponential distribution. Reliability engineering & system safety, 91(6), 689-697.
Ortega, E. M. M., Lemonte, A. J., Silva, G. O. and Cordeiro, G. M. (2015). New flexible models generated by gamma random variables for lifetime modeling. Journal of Applied Statistics, 42, 2159–2179.
Vedo, V. T., Nascimento, A. D. C., Miranda, W. R. Lima, M. C. S. Pinho, L. G. B. and Cordeiro, G. M. (2016). Some computational and theoretical aspects of the exponentiated generalized Nadarajah-Haghighi distribution. Chilean Journal of Statistics, forthcoming.
Ravi, V. and Gilbert, P. D. (2009). BB: An R package for solving a large system of nonlinear equations and for optimizing a highdimensional nonlinear objective function, J. Statist. Software, 32(4).
Rezaei, S., Sadr, B. B., Alizadeh, M., & Nadarajah, S. (2017). Topp–Leone generated family of distributions: Properties and
applications. Communications in Statistics-Theory and Methods, 46(6), 2893-2909.
Sedmak, D. D., Meineke, T. A., Knechtges, D. S. and Anderson, J. (1989). Prognostic significance of cytokeratin-positive breast cancer metastases. Modern pathology: an official journal of the United States and Canadian Academy of Pathology, Inc, 2(5), 516-520.
Tsuang, M. T. and Woolson, R. F. (1977). Mortality in patients with schizophrenia, mania, depression and surgical conditions. The British Journal of Psychiatry, 130, 162–166.
Lemonte, A. J. (2013). A new exponential type distribution with constant, decreasing, increasing, upside-down bathtub and bathtubshaped failure rate function. Computational Statistics & Data Analysis, 62, 149-170.
Nadarajah, S.and Haghighi, F. (2011). An extension of the exponential distribution. Statistics. 45, 543–558.
Shehata, W. A. and Yousof, H. M. (2021). A novel two-parameter Nadarajah-Haghighi extension: properties, copulas, modeling real data and different estimation methods. Statistics, Optimization & Information Computing, forthcoming.
Yadav, A. S., Goual, H., Alotaibi, R. M., Ali, M. M. and Yousof, H. M. (2020). Validation of the Topp-Leone-Lomax model via a modified Nikulin-Rao-Robson goodness-of-fit test with different methods of estimation. Symmetry, 12(1), 57.
Yousof, H. M. and Korkmaz, M. C. (2017). Topp-Leone Nadarajah-Haghighi distribution: mathematical properties and applications, International Journal of Applied Mathematics. Journal of Statisticians: Statistics and Actuarial Sciences, 2, 119-128.
Yousof, H. M., Korkmaz, M. C. Hamedani G. G. (2017). The odd Lindley Nadarajah-Haghighi distribution. J. Math. Comput. Sci., 7, 864-882.
- Authors retain copyright and grant the journal right of first publication with the work simultaneously licensed under a Creative Commons Attribution License that allows others to share the work with an acknowledgement of the work's authorship and initial publication in this journal.
- Authors are able to enter into separate, additional contractual arrangements for the non-exclusive distribution of the journal's published version of the work (e.g., post it to an institutional repository or publish it in a book), with an acknowledgement of its initial publication in this journal.
- Authors are permitted and encouraged to post their work online (e.g., in institutional repositories or on their website) prior to and during the submission process, as it can lead to productive exchanges, as well as earlier and greater citation of published work (See The Effect of Open Access).
