On a Logistic Approximation to the Normal and Other Cumulative Distribution Functions
DOI:
https://doi.org/10.32871/rmrj1402.01.08Keywords:
logistic approximation, cumulative distribution functions, normal distribution, chi-square distribution, estimation of parametersAbstract
This paper proposes an approach that maintains the simplicity of various
approximations as well as having a closed- form algebraic inverse yet easily generalizable to the approximation of other cumulative distribution functions. The proposed logistic approximation to the normal cumulative distribution function has a maximum error of 0.000560, lower than the maximum error reported by Aludaat et al (2007). Furthermore, the logistic approximation is more flexible in that it can be used to approximate the cumulative distribution functions of other distribution. A Chi-square approximation was also investigated and reported a maximum error of 0.00494.
References
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Shorack, G.R., Wellner J.A. (1986) Empirical Processes with Applications to Statistics, Wiley.
Tyurin, I.S. (2009). “On the accuracy of the Gaussian approximationâ€. Doklady Mathematics 80 (3): 840-843.
Van der Vaart, A. W. (1998), Asymptotic Statistics. Cambridge Series in Probabilistic Mathematics.
Vapnik, V.N. and Chervonenkis, A. Ya (1971). On uniform convergence of the frequencies of events to their probabilities. Theor. Prob. Appl. 16, 264-280.
Yeo, In-Kwon and Johnson, Richard (2000). A new family of power transformations to improve normality or symmetry. Biometrika, 87, 954-959.
Cook, R. D. & Weisberg, S. (1999). Applied Regression Including Computing and Graphics, New York: Wiley.
Cardano, G. (1663). Opera Omnia Hieronymi Cardani, Mediolanensis: Ars Magna (The Great Art), Lyons, Europe.
Craig, W. & Hogg (2000), An Introduction to Mathematical Statistics, (Wiley and Sons, New York)
Dudley, R. M. (1999). “Uniform Central Limit Theoremsâ€, Cambridge University Press. ISBN 0 521 46102.
Durrett, R. (1991). Probability: Theory and Examples. Pacific Grove, CA: Wadsworth & Brooks/Cole.
Esseen, C. G. (1956). “A moment inequality with an application to the central limit theoremâ€. Skand. Aktuarietidskr. 39: 160–170.
Feller, W. (1972). An Introduction to Probability Theory and Its Applications, Volume II (2nd ed.). New York: John Wiley & Sons.
Graybill, J.(1987) AnIntroductoryCourseinMathematical Statistics (Wiley Series, New York)
Huber, P. (1985). Projection pursuit. The annals of Statistics, 13(2):435 – 475.
Johnson, R and Wichern (2000) Applied Multivariate Statistical Analysis (Wiley and Sons, New York)
Manoukian, E. B. (1986). Modern Concepts and Theorems of Mathematical Statistics. New York: Springer-Verlag.
Serfling, R. J. (1980). Approximation Theorems of Mathematical Statistics. New York: John Wiley & Sons.
Shevtsova, I. G. (2007). “Sharpening of the upper bound of the absolute constant in the Berry– Esseen inequalityâ€. Theory of Probability and its
Applications 51 (3): 549–553.
Shevtsova, I. G. (2008). “On the absolute constant in the Berry-Esseen inequalityâ€. The Collection of Papers of Young Scientists of the Faculty of Computational Mathematics and Cybernetics Theory of Probability
and its Applications (5): 101-110.
Shiganov, I.S. (1986). “Refinement of the upper bound of a constant in the remainder term of the central limit theoremâ€. Journal of Soviet mathematics 35: 109–115.
Shorack, G.R., Wellner J.A. (1986) Empirical Processes with Applications to Statistics, Wiley.
Tyurin, I.S. (2009). “On the accuracy of the Gaussian approximationâ€. Doklady Mathematics 80 (3): 840-843.
Van der Vaart, A. W. (1998), Asymptotic Statistics. Cambridge Series in Probabilistic Mathematics.
Vapnik, V.N. and Chervonenkis, A. Ya (1971). On uniform convergence of the frequencies of events to their probabilities. Theor. Prob. Appl. 16, 264-280.
Yeo, In-Kwon and Johnson, Richard (2000). A new family of power transformations to improve normality or symmetry. Biometrika, 87, 954-959.
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Published
2014-06-30
How to Cite
Padua, R. N., Borres, M. S., & Barabat, E. O. (2014). On a Logistic Approximation to the Normal and Other Cumulative Distribution Functions. Recoletos Multidisciplinary Research Journal, 2(1). https://doi.org/10.32871/rmrj1402.01.08
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