DISCRIMINATING BETWEEN WEIBULL AND LOG-NORMAL DISTRIBUTIONS BASED ON KULLBACK-LEIBLER DIVERGENCE

Year 2012, Issue: 16, 44 - 54, 02.12.2014

Ali Akbar Bromıdeh

Abstract

DISCRIMINATING BETWEEN WEIBULL AND LOG-NORMAL DISTRIBUTIONS BASED ON KULLBACK-LEIBLER DIVERGENCE

Abstract

The Weibull and Log-Normal distributions are frequently used in reliability to analyze lifetime (or failure time) data. The ratio of maximized likelihood (RML) has been extensively used in choosing between the two distributions. The Kullback-Leibler information is a measure of uncertainty between two densities. We examine the use of Kullback-Leibler Divergence (KLD) in discriminating either the Weibull or Log-Normal distribution. An advantage of the KLD is that it incorporates entropy of each model. We explain the applicability of the KLD by a real data set and the consistency of the KLD with the RML is established. 

Keywords

Model discrimination Weibull distribution Log-Normal distribution Kullback-Leibler divergence Ratio of maximized likelihood

References

• Abernethy, R., (2006), "The New Weibull Handbook: Reliability and Statistical Analysis for Predicting Life, Safety, Supportability, Risk, Cost and Warranty Claims," 5th Edition, Barringer & Associates.
• Bain, L.J., Engelhardt, M. (1980), "Probability of correct selection of Weibull versus gamma based on likelihood ratio," Comm. Statist. Ser. A. 9, 375-381.
• Blishke, W.R. and Murthy, D.N.P. (2000) "Reliability Modeling, prediction and optimization," Wiley Interscience Publications.
• Burnham, K. P., and Anderson, D. R. (2002), "Model selection and multimodel inference: a practical information-theoretic approach," 2nd eds., Springer, New York.
• Cohen, A.C., Whitten, B.J. (1988), "Parameter estimation in reliability and life span models," Marcel Dekker Inc.
• Crow and Shimitzu, (1988), "Log-Normal Distributions, Theory and Applications," Marcel Dekker Inc., New York.
• Dey, A. K. and Kundu, D. (2009), "Discriminating among the Log-Normal, Weibull and Generalized Exponential distributions," IEEE Transactions on Reliability , vol. 58, no. 3, 416-424.
• Dumonceaux, R., and Antle, C.E., (1973), "Discrimination between the log-normal and the Weibull distributions, Technometrics 15 (4), 923-926.
• Fearn, D.H. and Nebenzahl, E. (1991), "On the maximum likelihood ratio method of deciding between the Weibull and Gamma distributions", Communications in Statistics - Theory and Methods, vol. 20, 579-593.
• Gross, A.J., and Clark, V.A. (1975), "Survival distribution: reliability application in the biomedical science," John Wiley & sons Inc.
• Gupta, R.D., Kundu, D., (2003), "Discriminating between Weibull and generalized exponential distributions," Computational Statistics & Data Analysis 43, 179-196.
• Lieblein, J, and Zelen, M. (1956), "Statistical investigation of the fatigue life of deep groove ball bearings," J. Res. Nat. Bar. Stand., 57, 273-316.
• Lawless, J.F., (1982), "Statistical Models and Methods for Lifetime Data," Wiley, New York.
• Meeker and Escobar (1998), "Statistical Methods for Reliability Data, New York: John Wiley and Sons.
• Mohd Saat, N. Z.; Jemain, A. A. and Al-Mashoor, S. H. (2008), "A Comparison of Weibull and Gamma Distributions in Application of Sleep Spnea," Asian Journal of Mathematics and Statistics, 1 (3), 132-138.
• Kundu, D. and Manglick, A. (2004), ""Discriminating between the Weibull and Log- Normal distributions", Naval Research Logistics, vol. 51, 893-905.
• Kundu, D. and Manglick, A. (2005), "Discriminating between the Log-Normal and gamma distributions", Journal of the Applied Statistical Sciences, vol. 14, 175-187, 2005.
• Kundu D., Gupta, R. D. and Manglick, A. (2005), "Discriminating between the Log- Normal and generalized exponential distributions," Journal of Statistical Planning and Inference. v127. 213-227.
• Pascual, F.G. (2005), "Maximum likelihood estimation under misspecied Log-Normal and Weibull distributions", Communications in Statistics - Simulation and Computations vol. 34, No. 3, 503 - 524.
• Pasha, G. R., Shuaib Khan, M. and Pasha, Ahmed Hesham, (2006), "Discrimination Between Weibull and Log-Normal Distributions For Lifetime data", Journal of Research (Science), Bahauddin Zakariya University, Multan, Pakistan. Vol. 17,No.2, April, pp. 103- 114.
• Thoman, D.R., Bain, L.J. and Antie, C.E. (1969), "Inferences on the parameter of the weibull distribution," Thechnometrics, 11, 445-460.