A New One-term Approximation to the Standard Normal Distribution

This paper deals with a new simple one-term approximation to the cumulative distribution function (cdf) of the standard normal distribution which does not have a closed-form representation. The accuracy of the proposed approximation has been evaluated by using the maximum absolute error (MAE), which was about 0.0016 and this accuracy is sufficient for most practical applications. To compare the proposed approximation with some of the existing one-term approximations, the MAE measure was considered. In addition, the difference between the exact and approximation of the normal cdf was computed for specific values of the normal random variable.


Introduction
Perhaps the most important and most widely used continuous probability distribution in statistics is the normal distribution. The main reasons are due to its importance in statistical inference and the existence of the central limit theorem. In addition, it plays a significant role in many applications in scientific fields, like engineering, genetics, psychology, biology, medicine, financial risk management, hydrology, mechanics, physics, reliability, and others.
The use of normal distribution often involves computing the area under its probability curve. Suppose that a random variable follows a probability density function (pdf) of the standard normal distribution, then its pdf and cdf are, respectively, defined as: As it is well-known, a closed-form solution for the integral in equation (1) Polya, 1949). Even though the simple ones may not be very accurate, they are nevertheless useful as accuracy has to be gauged vis-à-vis simplicity.
Because of the symmetry of the standard normal distribution, we get ( ) = 1 − (− ), which implies that it is sufficient to approximate * ( ).
This paper presents a simple approximation with one-term for ( ) when ≥ 0. The idea of approximating ( ) presented here is similar to that used in Polya (1949); Aludaat and Alodat (2008) and Eidous and Al-Salman (2016).
On the other hand, the MAE of the presented approximation is less than the MAE of the other one-term approximation approaches in the previously mentioned works.
The remaining part of this paper is presented as: In Section 2, a brief literature review on one-term approximation functions is given. A discussion on our proposed one-term approximation is provided in Section 3. Section 4 illustrates the performance comparisons between our proposed approximation with many other existing one-term approximations. Finally, section 5 provides a short conclusion on our work.

Literature review
There are many existing approximation approaches of normal density introduced in the literature. However, in our work, we are interested in the approximations that are of one-term form. This section presents a historical review of different one-term approximations for the normal cdf.
The following table gives some typical one-term approximation formulas of ( ) when ≥ 0 and the corresponding MAE of each formula.  The above approximation approaches are simple due to using one-term to evaluate ( ). Several other approximation approaches for ( ) that used more than one term have been reported in the literature (see for instance Eidous and Abu-Shareefa, 2020).

The proposed method
The main purpose for this paper is to present a new formula to approximate the cdf of standard normal distribution ( ), which is defined as following: Due to Johnson et al. (1994), the Ploya's approximation represents an upper bound for G(z), i.e., .
So, the conjectured by Aludaat and Alodat (2008) that their upper bound approximation is sharper than Polya's one is not accurate. However, it is still true that their approximation has a lower MAE compared to Polya's one. Now to approximate ( ), we know that  Table 1. Suppose that = 9 ( ), it follows that approximation of the inverse cdf is given by: Next, setting ′( )equal to zero and solving with respect to , we get two solutions = 0.600925 and = 1.96739 Plugging these values in ( ), we find that ( ) is equal to 0.00162038 and −0.00162702 at = 0.600925 and = 1.96739 respectively. Hence, the MAE of 9 ( ) is 0.00162702. By comparing this MAE with the one's in the others approaches, we can see that this MAE is the smallest comparing with the MAE of 1 ( ), 2 ( ), ..., 8 ( ) that are stated in Table (1), i.e., 9 ( ) gives higher accuracy approximation.

Comparison
In this section, we have carried out a comparison between our approximation 9 ( ) and the other one-term approximation approaches 1 ( ), 2 ( ), ..., 8 ( ) that are presented in Table ( 1). This comparison has been done for the values of between 0.2 to 4.2 with step 0.4. The results of the different approximation approaches are given in Table (2) below. The corresponding error between the exact ( ) and each of the different approximations is also provided to compare the accuracy of the approaches easily. From Table (2), we can see that our approximation 9 ( ) is the best one for 1.4 ≤ ≤ 3. In this case, the corresponding MAE of 9 ( ) is less than that of the other approximation approaches.

Table (2):
The exact value of ( ) and the corresponding values of its one-term approximation using the different approaches 1 ( ), 2 ( ), ..., 8 ( )and 9 ( ). The differences between ( ) and the other approximation approaches are given in the parentheses.
Our code is available upon request

Conclusion
In this paper, we have proposed a new one-term approximation for the cumulative distribution function of standard normal distribution. We have compared our approximation with many other existing one-term approximations, the MAE of the proposed approximation is less than that of all the existing approximations stated in this paper.