Approximation to the cumulative normal distribution using hyperbolic tangent based functions

다국어 초록 (Multilingual Abstract)

This paper presents a method for approximation of the standard
normal distribution by using hyperbolic tangent based functions.
The presented approximate formula for the cumulative distribution depends on one
numerical coefficient only, and its accuracy is admissible.
Furthermore, in some particular cases, closed forms of inverse
formulas are derived.

Numerical results of the present method are
compared with those of an existing method.

다국어 초록 (Multilingual Abstract)

This paper presents a method for approximation of the standard
normal distribution by using hyperbolic tangent based functions.
The presented approximate formula for the cumulative distribution depends on one
numerical coefficient only, and its accuracy is admissible.
Furthermore, in some particular cases, closed forms of inverse
formulas are derived.

Numerical results of the present method are
compared with those of an existing method.

참고문헌 (Reference)

1 R. G. Kerridge, "Yet another series for the normal integral" 63 : 401-407, 1976

2 D. Elliott, "Sigmoidal transformations and the trapezoidal rule" 40 : E77-E137, 1999

3 R. M. Norton, "Pocket-calculator approximation for areas under the standard normal curve" 43 : 24-26, 1989

4 S. Pr¨ossdorf, "On an integral equation of the first kind arising from a cruciform crack problem, Integral equations and inverse problems" Longman Sci. Tech. 235 : 210-219, 1991

5 T. W. Sag, "Numerical evaluation of high-dimensional integrals" 18 : 245-253, 1964

6 R. J. Bagby, "Calculating normal probabilities" 102 (102): 46-48, 1995

7 P. A. P. Moran, "Caculation of the normal distribution function" 67 : 675-676, 1980

8 E. Page, "Approximation to the cumulative normal function and its inverse for use on a pocket calculator" 26 : 75-76, 1977

9 H. Hamaker, "Approximating the cumulative normal distribution and its inverse" 27 : 76-77, 1978

10 J. D. Vedder, "An invertible approximation to the normal-distribution function" 16 : 119-123, 1993

11 W. Bryc, "A uniform approximation to the right normal tail integral" 127 (127): 365-374, 2002

12 J. T. Lin, "A simpler logistic approximation to the normal tail probability and its inverse" 39 (39): 255-257, 1990

13 G. R.Waissi, "A sigmoid approximation to the standard normal integral" 77 : 91-95, 1996

14 R. G. Hart, "A closed approximation related to the error function" 20 : 600-602, 1966