Advertisement

Half-Cauchy and Power Cauchy Distributions: Ordinary Differential Equations

  • Hilary I. OkagbueEmail author
  • Muminu O. Adamu
  • Patience I. Adamu
  • Sheila A. Bishop
  • Ezinne C. Erondu
Conference paper

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-Cauchy and power Cauchy distributions. 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-Cauchy and power Cauchy distributions 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-Cauchy distribution Hazard function Inverse survival function Power Cauchy distribution Quantile function Quantile mechanics Reversed hazard function Survival function 

Notes

Acknowledgements

This work was supported by Covenant University, Nigeria.

References

  1. 1.
    G. Steinbrecher, W.T. Shaw, Quantile mechanics. Euro. J. Appl. Math. 19(2), 87–112 (2008)Google Scholar
  2. 2.
    H.I. Okagbue, M.O. Adamu, T.A. Anake, Quantile approximation of the Chi-square distribution using the quantile mechanics, in Proceedings of the World Congress on Engineering and Computer Science 2017, 25–27 October 2017, San Francisco, U.S.A. Lecture Notes in Engineering and Computer Science (2017), pp. 477–483Google Scholar
  3. 3.
    H.I. Okagbue, M.O. Adamu, T.A. Anake, Solutions of Chi-square quantile differential equation, in Proceedings of the World Congress on Engineering and Computer Science 2017, 25–27 October 2017, San Francisco, U.S.A. Lecture Notes in Engineering and Computer Science (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. Wirel. Pers. Commun. 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 Proceedings of the World Congress on Engineering and Computer Science 2017, 25–27 October 2017, San Francisco, U.S.A. Lecture Notes in Engineering and Computer Science (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 Proceedings of the World Congress on Engineering and Computer Science 2017, 25–27 October 2017, San Francisco, U.S.A. Lecture Notes in Engineering and Computer Science (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 Proceedings of the World Congress on Engineering and Computer Science 2017, 25–27 October 2017, San Francisco, U.S.A. Lecture Notes in Engineering and Computer Science (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 Proceedings of the World Congress on Engineering and Computer Science 2017, 25–27 October 2017, San Francisco, U.S.A. Lecture Notes in Engineering and Computer Science (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 Proceedings of the World Congress on Engineering and Computer Science 2017, 25–27 October 2017, San Francisco, U.S.A. Lecture Notes in Engineering and Computer Science (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 Proceedings of the World Congress on Engineering and Computer Science 2017, 25–27 October 2017, San Francisco, U.S.A. Lecture Notes in Engineering and Computer Science (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 Proceedings of the World Congress on Engineering and Computer Science 2017, 25–27 October 2017, San Francisco, U.S.A. Lecture Notes in Engineering and Computer Science (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 Proceedings of the World Congress on Engineering and Computer Science 2017, 25–27 October 2017, San Francisco, U.S.A. Lecture Notes in Engineering and Computer Science (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 Proceedings of the World Congress on Engineering and Computer Science 2017, 25–27 October 2017, San Francisco, U.S.A. Lecture Notes in Engineering and Computer Science (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 Proceedings of the World Congress on Engineering and Computer Science 2017, 25–27 October 2017, San Francisco, U.S.A. Lecture Notes in Engineering and Computer Science (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 Proceedings of the World Congress on Engineering and Computer Science 2017, 25–27 October 2017, San Francisco, U.S.A. Lecture Notes in Engineering and Computer Science (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 Proceedings of the World Congress on Engineering and Computer Science 2017, 25–27 October 2017, San Francisco, U.S.A. Lecture Notes in Engineering and Computer Science (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 Proceedings of the World Congress on Engineering and Computer Science 2017, 25–27 October 2017, San Francisco, U.S.A. Lecture Notes in Engineering and Computer Science (2017), pp. 883–888Google Scholar
  23. 23.
    N.G. Polson, J.G. Scott, On the half-Cauchy prior for a global scale parameter. Bayes. Anal. 7(4), 887–902 (2012)MathSciNetCrossRefGoogle Scholar
  24. 24.
    M.W. Shaw, Simulation of population expansion and spatial pattern when individual dispersal distributions do not decline exponentially with distance. Proc. R. Soc. B 259, 243–248 (1995)CrossRefGoogle Scholar
  25. 25.
    H.J. Kim, On the ratio of two folded normal distributions. Commun. Stat. Theory Methods 35(6), 965–977 (2006)MathSciNetCrossRefGoogle Scholar
  26. 26.
    S. Psarakis, J. Panaretoes, The folded t distribution. Commun. Stat. Theory Methods 19(7), 2717–2734 (1990)MathSciNetCrossRefGoogle Scholar
  27. 27.
    A. Diédhiou, On the self-decomposability of the half-Cauchy distribution. J. Math. Anal. Appl. 220(1), 42–64 (1998)MathSciNetCrossRefGoogle Scholar
  28. 28.
    L. Bondesson, On the infinite divisibility of the half-Cauchy and other decreasing densities and probability functions on the nonnegative line. Scand. Actua. J. 1987(3), 225–247 (1987)MathSciNetCrossRefGoogle Scholar
  29. 29.
    E. Jacob, K. Jayakumar, On half-Cauchy distribution and process. Int. J. Statist. Math. 3(2), 77–81 (2012)zbMATHGoogle Scholar
  30. 30.
    I. Ghosh, The Kumaraswamy-half-Cauchy distribution: properties and applications. J. Stat. Theo. Appl. 13(2), 122–134MathSciNetCrossRefGoogle Scholar
  31. 31.
    G.M. Cordeiro, A.J. Lemonte, The beta-half-Cauchy distribution. J. Probab. Stat. Art. no. 904705 (2011)Google Scholar
  32. 32.
    G.M. Cordeiro, M. Alizadeh, T.G. Ramires, E.M. Ortega, The generalized odd half-Cauchy family of distributions: properties and applications. Commun. Stat. Theory Methods 46(11), 5685–5705 (2017)MathSciNetCrossRefGoogle Scholar
  33. 33.
    E. Paradis, S.R. Baillie, W.J. Sutherland, Modeling large-scale dispersal distances. Ecol. Model. 151(2–3), 279–292 (2002)CrossRefGoogle Scholar
  34. 34.
    B. Rooks, A. Schumacher, K. Cooray, The power Cauchy distribution: derivation, description, and composite models, in NSF-REU Program Reports (2010)Google Scholar
  35. 35.
    G. Venter, Transformed beta and gamma distributions and aggregate losses. Proc. Casualty Act. Soc. 156–193 (1983)Google Scholar
  36. 36.
    M.H. Tahir, M. Zubair, G.M. Cordeiro, A. Alzaatreh, M. Mansoor, The Poisson-X family of distributions. J. Stat. Comput. Simul. 86(14), 2901–2921 (2016)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Nature Singapore Pte Ltd. 2019

Authors and Affiliations

  • Hilary I. Okagbue
    • 1
    Email author
  • Muminu O. Adamu
    • 2
  • Patience I. Adamu
    • 1
  • Sheila A. Bishop
    • 1
  • Ezinne C. Erondu
    • 1
  1. 1.Department of MathematicsCovenant UniversityOtaNigeria
  2. 2.Department of MathematicsUniversity of LagosAkokaNigeria

Personalised recommendations