Half-Normal Distribution: Ordinary Differential Equations
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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 functionNotes
Acknowledgements
This work was supported by Covenant University, Ota, Nigeria.
References
- 1.G. Steinbrecher, W.T. Shaw, Quantile mechanics. Euro. J. Appl. Math. 19(2), 87–112 (2008)MathSciNetCrossRefGoogle Scholar
- 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.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.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.W.P. Elderton, Frequency Curves and Correlation (Charles and Edwin Layton, London, 1906)zbMATHGoogle Scholar
- 6.N. Balakrishnan, C.D. Lai, Continuous Bivariate Distributions, 2nd edn. (Springer, New York, London, 2009)zbMATHGoogle Scholar
- 7.N.L. Johnson, S. Kotz, N. Balakrishnan, Continuous Univariate Distributions, vol. 2. 2nd edn (Wiley, 1995)Google Scholar
- 8.N.L. Johnson, S. Kotz, N. Balakrishnan, Continuous Univariate Distributions (Wiley, New York, 1994). ISBN: 0-471-58495-9Google Scholar
- 9.H. Rinne, Location scale distributions, linear estimation and probability plotting using MATLAB (2010)Google Scholar
- 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.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.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.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.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.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.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.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.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.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.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.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.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.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.A. Pewsey, Large-sample inference for the general half-normal distribution. Comm. Stat. Theo. Meth. 31(7), 1045–1054 (2002)MathSciNetCrossRefGoogle Scholar
- 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.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.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.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.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.W.J. Huang, N.C. Su, A study of generalized normal distributions. Comm. Stat. Theo. Methods 46(11), 5612–5632 (2017)MathSciNetCrossRefGoogle Scholar
- 31.A.W. Kemp, The discrete half-normal distribution, in Advances in Mathematical and Statistical Modeling (Birkhäuser, Boston), pp. 353–360CrossRefGoogle Scholar
- 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.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.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.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.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.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.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.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.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.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.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.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.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.F.P. Schoenberg, R. Peng, J. Woods, On the distribution of wildfire sizes. Environmetrics 14(6), 583–592 (2003)CrossRefGoogle Scholar