Properties of the Libby-Novick Beta Distribution with Application

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Zafar Iqbal, Muhammad Rashad, Muhammad Hanif

Abstract

The beta distribution is one of the most popular probability distributions with applications to real life data. In this paper, an extension of this distribution called the Libby-Novick Beta distribution (LNBD) which is believed to provide greater flexibility to model scenarios involving skew-normal data than original one. Analytical expressions for various mathematical properties along with characterization based on one truncated moment. The estimation of LNBD's parameters is undertaken using the method of maximum likelihood estimation. For illustration and performance evaluation of LNBD two real-life data sets adapted from earlier studies are used.

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References

  1. A. Azzalini, A Class of Distributions Which Includes the Normal Ones, Scand. J. Stat. 12(2) (1985), 171-178.
  2. A. K. Sheikh, M. Ahmad, Z. Ali, Some remarks on the hazard functions of the inverted distributions, Reliab. Eng. 19 (1987), 255-261.
  3. A. P. Prudnikov, Y. A. Brychkov, and O.I. Marichev, Integrals and series. Vol. 1. Gordon and Breach Science Publishers, Amsterdam, (1986).
  4. A. Réyni, Probability Theory, Dover Publications, New York, 1970.
  5. A. Telcs, W. Glanzel, and A. Schubert, Characterization and Statistical Test using Truncated Expectations for a Class of Skew Distributions. Math. Soc. Sci. 10 (1985), 169-178.
  6. A. W Marshall, and I. Olkin, A new method for adding a parameter to a family of distributions with application to the exponential and weibull families. Biometrika. 84 (1997), 641-652.
  7. B.D. Sharma, I.J. Taneja, Entropy of type (α, β) and other generalized measures in information theory, Metrika. 22 (1975), 205-215.
  8. B. D. Sharma, P. Mittal, New non-additive measure of relative information. J. Comb. Inform. Syst. Sci. 2 (1977), 122-133.
  9. C. Alexander, G. M. Cordeiro, E. M. M. Ortega, J. M. Sarabia, Generalized beta-generated distributions, Comput. Stati. Data Anal. 56 (2012), 1880-1897.
  10. C.-D. Lai, G. Jones, Beta Hazard Rate Distribution and Applications, IEEE Trans. Rel. 64 (2015), 44-50.
  11. C. E. Shannon, A mathematical theory of communication, SIGMOBILE Mob. Comput. Commun. Rev. 5 (2001), 3-55.
  12. C. Tsallis, Possible generalization of Boltzmann-Gibbs statistics, J. Stat. Phys. 52 (1988), 479-487.
  13. D. Chotikapanich, Modeling Income Distributions and Lorenz Curves, Springer, NY, 2008.
  14. D. Desai, V. Mariappan, and M. Sakhardanda, Nature of Reverse Hazard Rate: An Investigation. Int. J. Perform. Eng. 7(2) (2011), 165-171.
  15. D. L. Libby, M. R. Novick, Multivariate Generalized Beta Distributions with Applications to Utility Assessment, J. Educ. Stat. 7 (1982), 271-294.
  16. D. Salomon, Data Compression, Springer, New York, 1998.
  17. D. N. Shanbagh, The Characterization for Exponential and Geometric Distributions. J. Amer. Stat. Assoc. 65(331) (1970), 1256-1259.
  18. S. T. Dara, Recent advances in moment distributions and their hazard rate. Ph.D. thesis, National College of Business Administration and Economics, Lahore, Pakistan, (2012).
  19. E. Boekee, and A. C. J. Van Der Lubbe, The R-norm information measure. Inform. Control. 45 (1980), 136-155.
  20. E. R. Barlow, W. A. Marshall, and F. Proschan, Properties of Probability Distribution with Monotone Hazard Rate. Ann. Math. Stat. 34(2) (1963), 375-389.
  21. E. M. Ghitany, The Monotonicity of the Reliability Measures of the Beta Distribution. Appl. Math. Lett. 17 (2004), 1277-1283.
  22. F. Farmoye, C. Lee and O. Olumolade, The beta-weibull distribution. J. Stat. Theory Appl. 4 (2005), 121-136.
  23. G. M. Cordeiro, L. H. de Santana, E. M. M. Ortega, R. R. Pescim, A New Family of Distributions: Libby-Novick Beta, Int. J. Stat. Probab. 3 (2014), 63-80.
  24. G. G. Hamedani, On certain generalized gamma convolution distributions II. Technical Report, No. 484, MSCS, Marquette University, 2013.
  25. G. M. Cordeiro, M. de Castro, A new family of generalized distributions. J. Stat. Comput. Simul. 81(7) (2011), 883-898.
  26. A. K. Gupta, S. Nadarajah, Handbook of beta distribution and its applications. Marcel Dekker, New York, 2004.
  27. I. S. Gradshteyn, I. M. Ryzhik, Table of integrals, series and products. 7th ed. Academic Press, Diego, 2004.
  28. J. Havrda, F. S. Charvat, Quantification method of classification processes: Concept of structural-entropy, Khbernetika. 3(1967), 30-35.
  29. J. Navarro, A. Guillamon, M.C. Ruiz, Generalized mixture in reliability modeling: Applications to the construction of bathtub shaped hazard model and the study of systems. Appl. Stoch. Models Bus. Ind. 25(3) (2009), 323-337.
  30. J. N. Kapur, Generalized entropy of order α and type β. Math. Seminar, 4 (1967), 78-94.
  31. M. Ahsanullah, G. G. Hamedani, Characterizations of certain continuous univariate distributions based on the conditional distribution of generalized order statistics. Pak. J. Stat. 28 (2012), 253-258.
  32. M. Ahsanullah, Characterizations of Univariate Continuous Distributions. Atlantis Press, Paris, France 2017.
  33. M. Ahsanullah, M. Shakil, M. B. Golam Kibria, Characterizations of Continuous Distributions by Truncated Moment. J. Mod. Appl. Stat. Meth. 1(15) (2016), 316-331.
  34. M. A. Awad, J. A. Alawneh, Application of Entropy to life-Time Model. IMA J. Math. Control Inform. 4(1987), 143-147.
  35. M. C. Jones, Families of distributions arising from distributions of order statistics. Test, 13(2004), 1-43.
  36. G.M. Giorgi, S. Nadarajah, Bonferroni and Gini indices for various parametric families of distributions, METRON. 68 (2010), 23-46.
  37. M. H. Barakat, H. Y. Abdelkader, Computing the moments of order statistics from non-identical random variables. Stat. Meth. Appl. 13 (2004), 15-26.
  38. M. I. Kamien, N.L. Schwartz, A generalized hazard rate, Econ. Lett. 5(3) (1980), 245-249.
  39. M. Mitra, S. K. Basu, On some properties of the bathtub failure rate family of life distributions. Microelectron. Reliab. 36(5) (1996), 679-684.
  40. M. R. Fazlollah, An Introduction to Information Theory. McGraw-Hill, New York, 1961.
  41. M. M. Ristić, B. V. Popović, S. Nadarajah, Libby and Novick's generalized beta exponential distribution, J. Stat. Comput. Simul. 85 (2015), 740-761.
  42. N. Eugene, C. Lee, Felix Famoye, Beta-normal distribution and its application. Commun. Stat. - Theory Meth. 31 (2002), 497-512.
  43. N. Pushkarna, J. Saran, and R. Tiwari, Bonferroni and Gini Indices and Recurrence Relation for Moments Progressive Type-II Right Censored Order Statistics from Marshall-Olkin Exponential Distribution. J. Stat. Theory Appl. 12(3) (2013), 306-320.
  44. P. T. Gia, N. Turkkan, Determine of the Beta Distribution form its Lorenz Curve. Math. Comput. Model. 16(2) (1992), 73-84.
  45. R Core Team, R A language and environment for statistical computing. Austria, Vienna: R Foundation for Statistical Computing, 2013.
  46. R. D. Gupta, D. Kundu, Generalized exponential distributions, Aust NZ J Stat. 41 (1999), 173-188.
  47. R. S. Varma, Generalizations of Renyi's entropy of order α. J. Math. Sci. 1 (1966), 34-48.
  48. S. A. Hasnain, Z. Iqbal, M. Ahmad, One Exponentiated Moment Exponential Distribution. Pak. J. Stat. 31(2) (2015), 267-280.
  49. S. Arimoto, Information theoretical considerations on estimation theory. Inform. Control, 73 (1971), 181-190.
  50. S. Kullback, Information Theory and Statistics, Wiley, NY, 1959.
  51. S.M. Sunoj, P.G. Sankaran, S.S. Maya, Characterizations of Life Distributions Using Conditional Expectations of Doubly (Interval) Truncated Random Variables, Commun. Stat. - Theory Meth. 38 (2009) 1441-1452.
  52. S. Nadarajah, A. K. Gupta, The beta Fréchet distribution. Far East Theor. Stat. 14 (2004),15-24.
  53. S. Nadarajah, S. Kotz, The beta exponential distribution. Reliab. Eng. Syst. Safe. 91 (2006), 689-697.
  54. S. Nadarajah, S. Kotz, The beta Gumbel distribution. Math Probl. Eng. 10 (2004), 323-332.
  55. S. Pundir, S. Arora, K. Jain, Bonferroni Curve and the related Statistical Inference. Stat. Probab. Lett. 75(2005), 140-150.
  56. T. M. Cover, J. A. Thomas, Elements of Information Theory, John Wiley, New York, 1991.
  57. U. N. Nair, G. P. Sankaran, Properties of a Mean Residual Life Function Arising from Renewal Theory. Naval Res. Logist. 57 (2010), 373-379.
  58. V. Mameli, M. Musio, A new generalization of the skew-normal distribution: the beta skew-normal. Commun. Stat. - Theory Meth. 42 (2013) 2229-2244.
  59. V. Mameli, The Kumaraswamy skew-normal distribution, Stat. Probab. Lett. 104 (2015), 75-81.
  60. V. Mameli, Two generalizations of the skew-normal distribution and two variants of McCarthys Theo- rem. Doctoral dissertation, Cagliari University, Italy, 2012.
  61. V. Dardanoni, A. Forcina, Inference for Lorenz Curve orderings. Econ. J. 2 (1999), 49-75.
  62. W. F. Sharpe, Investments, Prentice Hall, Englewood Cliffs, 1985.