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Half-Normal Distribution: Ordinary Differential Equations

  • Hilary I. OkagbueEmail author
  • Oluwole A. Odetunmibi
  • Sheila A. Bishop
  • Pelumi E. Oguntunde
  • Abiodun A. Opanuga
Conference paper
First Online:

Abstract

In this chapter, homogenous ordinary differential equations (ODES) of different orders were obtained for the probability density function, quantile function, survival function inverse survival function, hazard function and reversed hazard functions of half-normal distribution. This is possible since the aforementioned probability functions are differentiable. Differentiation and modified product rule were used to obtain the required ordinary differential equations, whose solutions are the respective probability functions. The different conditions necessary for the existence of the ODEs were obtained and it is almost in consistent with the support that defined the various probability functions considered. The parameters that defined each distribution greatly affect the nature of the ODEs obtained. This method provides new ways of classifying and approximating other probability distributions apart from half-normal distribution considered in this chapter. In addition, the result of the quantile function can be compared with quantile approximation using the quantile mechanics.

Keywords

Differential calculus Half-normal distribution Hazard function Inverse survival function Quantile function Quantile mechanics Reversed hazard function Survival function 
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Notes

Acknowledgements

This work was supported by Covenant University, Ota, Nigeria.

References

  1. 1.
    G. Steinbrecher, W.T. Shaw, Quantile mechanics. Euro. J. Appl. Math. 19(2), 87–112 (2008)MathSciNetCrossRefGoogle Scholar
  2. 2.
    H.I. Okagbue, M.O. Adamu, T.A. Anake, Quantile approximation of the chi-square distribution using the quantile mechanics, in Lecture Notes in Engineering and Computer Science: Proceedings of the World Congress on Engineering and Computer Science, 25–27 Oct 2017 (San Francisco, U.S.A., 2017), pp. 477–483Google Scholar
  3. 3.
    H.I. Okagbue, M.O. Adamu, T.A. Anake, Solutions of chi-square quantile differential equation, in Lecture Notes in Engineering and Computer Science: Proceedings of the World Congress on Engineering and Computer Science, 25–27 Oct 2017 (San Francisco, U.S.A., 2017), pp. 813–818Google Scholar
  4. 4.
    Y. Kabalci, On the Nakagami-m inverse cumulative distribution function: closed-form expression and its optimization by backtracking search optimization algorithm. Wireless Pers. Comm. 91(1), 1–8 (2016)CrossRefGoogle Scholar
  5. 5.
    W.P. Elderton, Frequency Curves and Correlation (Charles and Edwin Layton, London, 1906)zbMATHGoogle Scholar
  6. 6.
    N. Balakrishnan, C.D. Lai, Continuous Bivariate Distributions, 2nd edn. (Springer, New York, London, 2009)zbMATHGoogle Scholar
  7. 7.
    N.L. Johnson, S. Kotz, N. Balakrishnan, Continuous Univariate Distributions, vol. 2. 2nd edn (Wiley, 1995)Google Scholar
  8. 8.
    N.L. Johnson, S. Kotz, N. Balakrishnan, Continuous Univariate Distributions (Wiley, New York, 1994). ISBN: 0-471-58495-9Google Scholar
  9. 9.
    H. Rinne, Location scale distributions, linear estimation and probability plotting using MATLAB (2010)Google Scholar
  10. 10.
    H.I. Okagbue, P.E. Oguntunde, A.A. Opanuga, E.A. Owoloko, Classes of ordinary differential equations obtained for the probability functions of Fréchet distribution, in Lecture Notes in Engineering and Computer Science: Proceedings of the World Congress on Engineering and Computer Science, 25–27 Oct 2017 (San Francisco, U.S.A., 2017), pp. 186–191Google Scholar
  11. 11.
    H.I. Okagbue, P.E. Oguntunde, P.O. Ugwoke, A.A. Opanuga, Classes of ordinary differential equations obtained for the probability functions of exponentiated generalized exponential distribution, in Lecture Notes in Engineering and Computer Science: Proceedings of the World Congress on Engineering and Computer Science, 25–27 Oct 2017 (San Francisco, U.S.A., 2017), pp. 192–197Google Scholar
  12. 12.
    H.I. Okagbue, A.A. Opanuga, E.A. Owoloko, M.O. Adamu, Classes of ordinary differential equations obtained for the probability functions of cauchy, standard cauchy and log-cauchy distributions, in Lecture Notes in Engineering and Computer Science: Proceedings of the World Congress on Engineering and Computer Science, 25–27 Oct 2017 (San Francisco, U.S.A., 2017), pp. 198–204Google Scholar
  13. 13.
    H.I. Okagbue, S.A. Bishop, A.A. Opanuga, M.O. Adamu, Classes of ordinary differential equations obtained for the probability functions of Burr XII and Pareto distributions, in Lecture Notes in Engineering and Computer Science: Proceedings of the World Congress on Engineering and Computer Science, 25–27 Oct 2017 (San Francisco, U.S.A., 2017), pp. 399–404Google Scholar
  14. 14.
    H.I. Okagbue, M.O. Adamu, E.A. Owoloko, A.A. Opanuga, Classes of ordinary differential equations obtained for the probability functions of Gompertz and Gamma Gompertz distributions, in Lecture Notes in Engineering and Computer Science: Proceedings of the World Congress on Engineering and Computer Science, 25–27 October, 2017 (San Francisco, U.S.A., 2017), pp. 405–411Google Scholar
  15. 15.
    H.I. Okagbue, M.O. Adamu, A.A. Opanuga, J.G. Oghonyon, Classes of ordinary differential equations obtained for the probability functions of 3-parameter Weibull distribution, in Lecture Notes in Engineering and Computer Science: Proceedings of the World Congress on Engineering and Computer Science, 25–27 Oct 2017 (San Francisco, U.S.A., 2017), pp. 539–545Google Scholar
  16. 16.
    H.I. Okagbue, A.A. Opanuga, E.A. Owoloko, M.O. Adamu, Classes of ordinary differential equations obtained for the probability functions of exponentiated Fréchet distribution, in Lecture Notes in Engineering and Computer Science: Proceedings of the World Congress on Engineering and Computer Science, 25–27 Oct 2017 (San Francisco, U.S.A., 2017), pp. 546–551Google Scholar
  17. 17.
    H.I. Okagbue, M.O. Adamu, E.A. Owoloko, S.A. Bishop, Classes of ordinary differential equations obtained for the probability functions of Half-Cauchy and power Cauchy distributions, in Lecture Notes in Engineering and Computer Science: Proceedings of the World Congress on Engineering and Computer Science, 25–27 Oct 2017 (San Francisco, U.S.A., 2017), pp. 552–558Google Scholar
  18. 18.
    H.I. Okagbue, P.E. Oguntunde, A.A. Opanuga, E.A. Owoloko, Classes of ordinary differential equations obtained for the probability functions of exponential and truncated exponential distributions, in Lecture Notes in Engineering and Computer Science: Proceedings of the World Congress on Engineering and Computer Science, 25–27 Oct 2017 (San Francisco, U.S.A., 2017), pp. 858–864Google Scholar
  19. 19.
    H.I. Okagbue, O.O. Agboola, P.O. Ugwoke, A.A. Opanuga, Classes of Ordinary differential equations obtained for the probability functions of exponentiated Pareto distribution, in Lecture Notes in Engineering and Computer Science: Proceedings of the World Congress on Engineering and Computer Science, 25–27 Oct 2017 (San Francisco, U.S.A., 2017), pp. 865–870Google Scholar
  20. 20.
    H.I. Okagbue, O.O. Agboola, A.A. Opanuga, J.G. Oghonyon, Classes of ordinary differential equations obtained for the probability functions of Gumbel distribution, in Lecture Notes in Engineering and Computer Science: Proceedings of the World Congress on Engineering and Computer Science, 25–27 Oct 2017 (San Francisco, U.S.A., 2017), pp. 871–875Google Scholar
  21. 21.
    H.I. Okagbue, O.A. Odetunmibi, A.A. Opanuga, P.E. Oguntunde, Classes of ordinary differential equations obtained for the probability functions of half-normal distribution, in Lecture Notes in Engineering and Computer Science: Proceedings of the World Congress on Engineering and Computer Science, 25–27 Oct 2017 (San Francisco, U.S.A., 2017), pp. 876–882Google Scholar
  22. 22.
    H.I. Okagbue, M.O. Adamu, E.A. Owoloko, E.A. Suleiman, Classes of ordinary differential equations obtained for the probability functions of Harris extended exponential distribution, in Lecture Notes in Engineering and Computer Science: Proceedings of the World Congress on Engineering and Computer Science, 25–27 Oct 2017 (San Francisco, U.S.A., 2017), pp. 883–888Google Scholar
  23. 23.
    H.I. Okagbue, M.O. Adamu, T.A. Anake, Ordinary differential equations of the probability functions of weibull distribution and their application in ecology. Int. J. Engine. Future Tech. 15(4), 57–78 (2018)Google Scholar
  24. 24.
    A. Pewsey, Large-sample inference for the general half-normal distribution. Comm. Stat. Theo. Meth. 31(7), 1045–1054 (2002)MathSciNetCrossRefGoogle Scholar
  25. 25.
    A. Pewsey, Improved likelihood based inference for the general half-normal distribution. Comm. Stat. Theo. Meth. 33(2), 197–204 (2004)MathSciNetCrossRefGoogle Scholar
  26. 26.
    A.G. Nogales, P. Perez, Unbiased estimation for the general half-normal distribution. Comm. Stat. Theo. Meth. 44(7), 3658–3667 (2015)MathSciNetCrossRefGoogle Scholar
  27. 27.
    J.J. Duarte Sanchez, W.W. da Luz Freitas, G.M. Cordeiro, The extended generalized half-normal distribution. Braz. J. Prob. Stat. 30(3), 366–384 (2016)MathSciNetCrossRefGoogle Scholar
  28. 28.
    R.R. Pescim, C.G.B. Demétrio, G.M. Cordeiro, E.M.M. Ortega, M.R. Urbano, The beta generalized half-normal distribution. Comput. Stat. Data Anal. 54(4), 945–957 (2010)MathSciNetCrossRefGoogle Scholar
  29. 29.
    K. Cooray, M.M.A. Ananda, A generalization of the half-normal distribution with applications to lifetime data. Comm. Stat. Theo. Methods 37(9), 1323–1337 (2008)MathSciNetCrossRefGoogle Scholar
  30. 30.
    W.J. Huang, N.C. Su, A study of generalized normal distributions. Comm. Stat. Theo. Methods 46(11), 5612–5632 (2017)MathSciNetCrossRefGoogle Scholar
  31. 31.
    A.W. Kemp, The discrete half-normal distribution, in Advances in Mathematical and Statistical Modeling (Birkhäuser, Boston), pp. 353–360CrossRefGoogle Scholar
  32. 32.
    N.M. Olmos, H. Varela, H.W. Gómez, H. Bolfarine, An extension of the half-normal distribution. Stat. Papers 53(4), 875–886 (2012)MathSciNetCrossRefGoogle Scholar
  33. 33.
    G.M. Cordeiro, R.R. Pescim, E.M.M. Ortega, The Kumaraswamy generalized half-normal distribution for skewed positive data. J. Data Sci. 10(2), 195–224 (2012)MathSciNetGoogle Scholar
  34. 34.
    T. Ramires, E.M. Ortega, G.M. Cordeiro, G. Hamedani, The beta generalized half-normal geometric distribution. Studia Scient. Math. Hunga. 50(4), 523–554 (2013)zbMATHGoogle Scholar
  35. 35.
    A. Alzaatreh, K. Knight, On the gamma half-normal distribution and its applications. J. Modern Appl. Stat. Methods 12(1), 103–119 (2013)CrossRefGoogle Scholar
  36. 36.
    W. Gui, An alpha half-normal slash distribution for analyzing non negative data. Comm. Stat. Theo. Methods 44(22), 4783–4795 (2015)MathSciNetCrossRefGoogle Scholar
  37. 37.
    L.M. Castro, H.W. Gómez, M. Valenzuela, Epsilon half-normal model: properties and inference. Comput. Stat. Data Anal. 56(12), 4338–4347 (2012)MathSciNetCrossRefGoogle Scholar
  38. 38.
    G.M. Cordeiro, E.M.M. Ortega, G.O. Silva, The exponentiated generalized gamma distribution with application to lifetime data. J. Stat. Comput. Simul. 81(7), 827–842 (2011)MathSciNetCrossRefGoogle Scholar
  39. 39.
    B.Y. Murat, J. Ahad, F.Z. Dogru, O. Arslan, The generalized half-t distribution. Stat. Interf. 10(4), 727–734 (2017)MathSciNetCrossRefGoogle Scholar
  40. 40.
    G.M. Cordeiro, M. Alizadeh, R.R. Pescim, E.M. Ortega, The odd log-logistic generalized half-normal lifetime distribution: properties and applications. Comm. Stat. Theo. Methods 46(9), 4195–4214 (2017)MathSciNetCrossRefGoogle Scholar
  41. 41.
    D.O. Cahoy, Minkabo, Inference for three-parameter M-Wright distributions with applications. Model. Assist. Stat. Appl. 12(2), 115–125 (2017)Google Scholar
  42. 42.
    A.G. Glen, L.M. Leemis, D.J. Luckett, Survival distributions based on the incomplete gamma function ratio, in Proceedings Winter Simulation Conference, Article. Number 7822105 (2017)Google Scholar
  43. 43.
    C.Y. Chou, H.R. Liu, Properties of the half-normal distribution and its application to quality control. J. Industr. Technol. 14(3), 4–7 (1998)Google Scholar
  44. 44.
    G. Lang, The difference between wages and wage potentials: earnings disadvantages of immigrants in Germany. J. Econ. Inequal. 3(1), 21–42 (2005)CrossRefGoogle Scholar
  45. 45.
    F.P. Schoenberg, R. Peng, J. Woods, On the distribution of wildfire sizes. Environmetrics 14(6), 583–592 (2003)CrossRefGoogle Scholar

Copyright information

© Springer Nature Singapore Pte Ltd. 2019

Authors and Affiliations

  • Hilary I. Okagbue
    • 1
    Email author
  • Oluwole A. Odetunmibi
    • 1
  • Sheila A. Bishop
    • 1
  • Pelumi E. Oguntunde
    • 1
  • Abiodun A. Opanuga
    • 1
  1. 1.Department of MathematicsCovenant UniversityOtaNigeria

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