On HPM Approximations for the Cumulative Normal Distribution Function

V K Shchigolev

doi:10.13189/ujcmj.2019.070102

Abstract

In this paper, some new approximations to the cumulative distribution function of the standard normal distribution via the He's homotopy perturbation method are proposed. There are several methods which provide an approximation of the integral in the formula for the cumulative distribution function by different numerical methods. For the same purpose, we first establish a differential equation of the second order that the cumulative distribution function satisfied subjected with the certain initial conditions. Then we apply the Homotopy Perturbation Method to solve the Cauchy problem for the governing equation. As well known, the result of solving an equation by this method and the convergence rate greatly depend on the choice of homotopy applied. Therefore, we consider two cases in this work. In one case, we construct the homotopy from the idea of simplicity. In the next case, we just follow the procedure of the general approach proposed early. As a result, we obtain several approximations which can be are easily calculated and are better than some other approximations. Numerical comparison shows that our approximations are very accurate.

Highlights

IntroductionThe importance of the normal distribution in many areas of science (for example, in mathematical statistics and statistical physics) follows from the central limit theorem of probability theory
The importance of the normal distribution in many areas of science follows from the central limit theorem of probability theory
We apply the He’s homotopy perturbation method (HPM) in order to obtain the approximations for the cumulative normal distribution function

Summary

Introduction

The importance of the normal distribution in many areas of science (for example, in mathematical statistics and statistical physics) follows from the central limit theorem of probability theory. As noted in [4], the solution for (6) is similar, qualitatively, to a hyperbolic tangent because when x tends to ±∞, the derivative Φ (x) tends to zero, by symmetry, Φ(x) tends to the same constant on both directions It is why the authors of [4] made the conclusion that the first approach of the HPM method contains a hyperbolic term. We apply the He’s HPM in order to obtain the approximations for the cumulative normal distribution function. For this purpose , we consider Φ(x) as the unknown function to be determine from solving a certain differential equation. 2π where the prime stands for the derivative with respect to x

Approximation by HPM

The case of trivial homotopy

The case of improved homotopy

A one-parameter tuning of improved homotopy

Conclusions