A new very simply explicitly invertible approximation for the standard normal cumulative distribution function

Jessica Lipoth; Yoseph Tereda; Simon Michael Papalexiou; Raymond J. Spiteri; Jessica Lipoth; Yoseph Tereda; Simon Michael Papalexiou; Raymond J. Spiteri

doi:10.3934/math.2022648

AIMS Mathematics

2022, Volume 7, Issue 7: 11635-11646. doi: 10.3934/math.2022648

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Research article

A new very simply explicitly invertible approximation for the standard normal cumulative distribution function

1.
Department of Electrical and Computer Engineering, College of Engineering, University of Saskatchewan, 57 Campus Dr, Canada
2.
Department of Computer Science, College of Arts and Science, University of Saskatchewan, 176 Thorvaldson Building, 110 Science Place, Canada
3.
Department of Civil Engineering, University of Calgary, Canada

Received: 11 June 2021 Revised: 23 December 2021 Accepted: 10 April 2022 Published: 15 April 2022
MSC : 65D10, 62J02

This paper proposes a new very simply explicitly invertible function to approximate the standard normal cumulative distribution function (CDF). The new function was fit to the standard normal CDF using both MATLAB's Global Optimization Toolbox and the BARON software package. The results of three separate fits are presented in this paper. Each fit was performed across the range $ 0 \leq z \leq 7 $ and achieved a maximum absolute error (MAE) superior to the best MAE reported for previously published very simply explicitly invertible approximations of the standard normal CDF. The best MAE reported from this study is 2.73e–05, which is nearly a factor of five better than the best MAE reported for other published very simply explicitly invertible approximations.
- normal distribution,
- cumulative distribution function,
- optimization
Citation: Jessica Lipoth, Yoseph Tereda, Simon Michael Papalexiou, Raymond J. Spiteri. A new very simply explicitly invertible approximation for the standard normal cumulative distribution function[J]. AIMS Mathematics, 2022, 7(7): 11635-11646. doi: 10.3934/math.2022648

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Abstract

This paper proposes a new very simply explicitly invertible function to approximate the standard normal cumulative distribution function (CDF). The new function was fit to the standard normal CDF using both MATLAB's Global Optimization Toolbox and the BARON software package. The results of three separate fits are presented in this paper. Each fit was performed across the range $ 0 \leq z \leq 7 $ and achieved a maximum absolute error (MAE) superior to the best MAE reported for previously published very simply explicitly invertible approximations of the standard normal CDF. The best MAE reported from this study is 2.73e–05, which is nearly a factor of five better than the best MAE reported for other published very simply explicitly invertible approximations.

References

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