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Comparison of Three Weibull Extensions; Generalized Gamma, Exponentiated Weibull, and Odd Weibull Distributions
Nonhle Channon Mdziniso, Kahadawala Cooray
Pages - 18 - 30     |    Revised - 31-10-2020     |    Published - 01-12-2020
Volume - 8   Issue - 2    |    Publication Date - December 2020  Table of Contents
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KEYWORDS
Exponentiated Odd Weibull Distribution, Simulation Study, Vuong Test, Empirical Distribution Function, Shapiro-Wilk W-test.
ABSTRACT
In this paper, three-parameter Weibull extensions: generalized gamma (GG), exponentiated Weibull (EW), and Odd Weibull (OW) distributions, which are capable of modeling data that exhibit five major hazard shapes, are compared using Vuong, empirical distribution function, and Shapiro-Wilk W- tests. The goal of this work is to compare the GG, EW, and OW distribution using better model selection criteria under general conditions as addressed by the Voung test, in addition to the tests considered. Our simulation study and graphical analysis show that the OW distribution is different from both GG and EW distributions even though all three distributions have five common hazard shapes. An example using voltage data is also considered to illustrate applications of the three distributions. Our comparative results led us to develop a six-parameter generalized distribution which is an extension of the GG, EW, and OW distributions. A four-parameter sub-model of this distribution, the exponentiated Odd Weibull distribution which has applications in modeling lifetime data, was studied in our recent publication.
C. Cox and M. Matheson. “A comparison of the generalized gamma and exponentiated Weibull distributions.” Statistics in Medicine, vol. 33, pp. 3772-3780, 2014.
D. Lacey, A. Nguyen, and K. Cooray. “Bathtub and unimodal hazard flexibility classification of parametric lifetime distributions.” NSF-REU Program Reports 2014. Available: https://www.cmich.edu/colleges/cst/math/Pages/REU-and-LURE.aspx.
E.S. Pearson and O.H. Hartley. Biometrika tables for statisticians. vol. 2 London: Biometrika Trust, 1976.
E.W. Stacy. “A generalization of the gamma distribution.” The Annals of Mathematical Statistics, vol. 33, pp. 1187-1192, 1962.
G.S. Mudholkar, D.K. Srivastava, and M. Freimer. “The exponentiated Weibull family: a reanalysis of the bus-motor-failure data”. Technometrics, vol. 37, pp. 436-445, 1995.
H. Jiang, M. Xie, and L.C. Tang. “On the Odd Weibull Distribution.” Proceedings of the Institution of Mechanical Engineers, Part O: Journal of Risk and Reliability, 2008, pp. 583- 594.
K. Cooray. “A study of moments and likelihood estimators of the odd Weibull distribution.” Statistical Methodology, vol. 26, pp. 72-83, 2015.
K. Cooray. “Analyzing grouped, censored, and truncated data using the Odd Weibull family.” Communications in Statistics - Theory and Methods, vol. 41, pp. 2661-2680, 2012.
K. Cooray. “Generalization of the Weibull distribution: the odd Weibull family.” Statistical Modelling, vol. 6, pp. 265-277, 2006.
M. Matheson, A. Munôz, and C. Cox (2017). “Describing the flexibility of the generalized gamma and related distributions.” Journal of Statistical Distributions and Applications, 4(15). Available: https://doi.org/10.1186/s40488-017-0072-5 [Nov. 1, 2020].
N.C. Mdziniso and K. Cooray. “Parametric analysis of renal failure data using the exponentiated Odd Weibull distribution.” International Journal of Statistics in Medical Research, vol. 7, pp. 96-105, 2018.
Q.H. Vuong. “Likelihood ratio tests for model selection and non-nested hypotheses.” Econometrica, vol. 57, pp. 307-333, 1989.
R.B. D’Agostino and M.A. Stephens. Goodness-of-fit techniques. New York: Marcel Dekker, 1986.
S. Nadarajah, G.M. Cordeiro, and E.M.M. Ortega. “General results for the beta-modified Weibull distribution.” Journal of Statistical Computation and Simulation. pp. 1 – 22, 2011. Available: https://doi.org/10.1080/00949651003796343.
S.S. Shapiro and M.B. Wilk. “An analysis of variance test for normality (complete samples).” Biometrika, vol. 52, pp. 591-611, 1965.
Dr. Nonhle Channon Mdziniso
Department of Mathematical and Digital Sciences, Bloomsburg University of Pennsylvania, Bloomsburg, PA 17815-1301 - United States of America
nmdziniso@bloomu.edu
Dr. Kahadawala Cooray
Department of Statistics, Actuarial and Data Sciences, Central Michigan University, Mount Pleasant MI 48859 - United States of America