Length Biased Burr- XII Distribution: Properties and Application

Authors

  • Abd-Elfattah A. M Department of MathematicalStatistics, Institute of Statistical Studies & Research, Cairo University,
  • Mahdy .M Department of Statistics, Mathematics and Insurance,Benha University, Egypt.
  • Ismail .G Ministry of Insurance and Pensions

Keywords:

Burr XII distribution, Length Biased, Survival Function, Order Statistics, Renei Entropy, Maximum Likelihood Method

Abstract

Burr XII distribution is widely applicable in reliability and life testing problems in engineering as well as in survival analysis. The concept of weighted and size- biased sampling and length biased  distribution due to importance of this kind of distributions and it's application  in many fields such as medicine, ecology, reliability and human populations. In this paper, length biased Burr XII  distribution is proposed and studied. Different properties of this new distribution are discussed such as the density function and its behaviour , moments, hazard , survival functions and order statistics. The parameters of this new distribution are estimated by maximum likelihood method . The observed information matrix is derived. Finally, we provide a simulation and real data analysis to see how the new model is applicable in practice.

References

. A. Azzalini, “A class of distributions which includes the normal ones,” Scandinavian Journal of Statistics, vol.12, pp.171-178, .1985

. A.L. Al-Saiari et al. , “Marshall-Olkin Extended Burr Type XII Distribution, ” International Journal of Statistics and Probability, vol.3, no.1, pp.78-84,.2014

. A.W. Lawis, “The burr distribution as a general parametric family in survivorship and reliability theory applications, ” Ph dissertation, Chapel Hill,. 1981

. A.A. Yusuf et al. , “ The geometric inverse burr Distribution: Model, Properties and Simulation,” Journal of Mathematics,vol.1, no.2, pp.83-83,. 2015

. B.Efrom, “Logistic regression survival analysis and the Kaplan-Meier curve,” Journal of the American Statistical Association, vol.83, no.402, pp.414-425, . 1988

. C. R. Rao, “ On discrete distributions arising out of methods of ascertainment,” Classical and Contagious Discrete Distributions (G. P. Patil ed.), Calcutta, India: Pergamon Press and Statistical Publishing Society, pp. 320–332,. 1965

. F, A.Bhatti,et al. ,“On generalized log burr –XII distribution , ” Pakistan Journal of Statistics and Operation Research, vol.14, no.3, pp.615-643,. 2018

. G.P.Patil, “Weighted distributions, ” Encyclopaedia of Environmetrics, vol.4, pp.2369-2377, . 2002

. H. M. Reyad, et al. ,“ The length-biased weighted frechet distribution: Properties and estimation,” International Journal Statistics and Applied Mathematics, vol.3, no.1, pp.189-200, . 2018

. I, W. Burr, “ Cumulative frequency functions, ” Annals of Mathematical Statistics, vol.13, pp.215-232,. 1942

. K. A., Amir and J.A. Reshi,“ Structural Properties of length biased Beta distribution of first kind, ” American Journal of Engineering Research, vol.2, pp.1-6,.2013

. K. Abed Al- kadim and N. A. Hussein,“ New proposed length biased weighted exponential and Reyleigh distribution with applications, ” Mathematical theory and Modeling, vol.4, no.7, pp.137-152,. 2014

. K.K. Das and T.D. Roy,“ On some length –biased weighted Weibull distribution, ” Advances in Applied Science Research, vol.2,no.5,pp.465-475,. 2011

. K. K. Das and T.D. Roy,“ Applicability of Length Biased Weighted Generalized Rayleigh Distribution, ” Advances in Applied Science Research,vol.2, no.4, pp.320-327,. 2011

. K. Pearson,“ Contributions to the mathematical theory of evolution. II. Skew variations in homogeneous material, ” Royal Society of London, vol.186, pp. 343-414,. 1895

. L. Handique and S. Chakraborty,“ A new four parameter extension of Burr-XII Distribution- its properties and applications, ” 2017 , https://hal.archives-ouvertes.fr/hal-01560542

. M. A. Ahmed,“ The new form Libby-Novick distribution, ” Communications in Statistics, pp.1-17,. 2019

. M,G. Bader and A.M . Priest,“ Statistical aspects of fiber and bundle strength in hybrid composites. In: Hayashi T, et al. (Eds.) ,” Progress in Science in Engineering Composites.vol. 4, pp.1129-1136,.1982

. M.E. Mead,“ A new generalization of burr XII distribution,” Journal of Statistics: Advances in Theory and Applications,vol.12, no.2, pp. 53-73,. 2014

. N. Nanuwong and W.Bodhisuwan,“ Length biased beta Pareto distribution and its structural properties with application,” Journal of Mathematics and Statistics, vol.10, no.1, pp. 49-57,. 2014

. N.L. Johnson et al ,“ Continuous univariate distributions,” New York: John Wiley and Sons.1995

. P.F. Paranaíba et al ,“ The Beta Burr-XII Distribution with application to lifetime data,” Computational Statistics and Data Analysis, vol. 55, pp.1118-1136,. 2011

. R.A. Fisher,“ The effects of methods of ascertainment upon the estimation of frequencies,” Annals of Eugenics,vol.6, pp.13-25,. 1934

. R. B. Arellano-Valle and A. Azzalini ,“ On the unification of families of skew-normal distributions,” Scandinavian Journal of Statistics, vol.33, pp.561-574,. 2006

. R. L. Smith and J.C. Naylor,“ A comparison of Maximum likelihood and Bayesian estimators for the three parameter Weibull distribution,” Applied Statistics , vol.36, no.3, pp.358-369,. 1987

. T, H, Abouelmagd et al ,“ The extended burr XII distribution with variable shapes for the hazard rate,”.Pak.j.stat.res, vol.13, no.3, pp.687-698,. 2017

. W.O. Meeker, and L,A, Escobra ,“ Statistical methods for reliability,” New York: John Wiley,.1998

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Published

2021-02-25

How to Cite

A. M, A.-E., .M, M., & .G, I. (2021). Length Biased Burr- XII Distribution: Properties and Application. International Journal of Sciences: Basic and Applied Research (IJSBAR), 56(1), 222–244. Retrieved from https://www.gssrr.org/index.php/JournalOfBasicAndApplied/article/view/12165

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Vol. 56 No. 1 (2021)

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