New Refinements for the Error Function with Applications in Diffusion Theory

Math Methods in App Sciences

2023

Most researches interested in finding the bounds of the cumulative standard normal distribution are not tight for all positive values of the argument . This paper mainly proposes new simple lower and upper bounds for . Over the whole range of the positive argument , the maximum absolute difference between the proposed lower bound and is less than , while it is less than between the proposed upper bound and . Numerical comparisons have been made between the proposed bounds and some of the other existing bounds, which showed that the proposed bounds are more compact than most alternative bounds found in the literature.

“…With regard to the previous lower and upper bounds of

Φ (x)

, the common limitation to almost all of them is that they do not show a tight lower and upper bound for all values of the argument

x > 0

. For example, the numerical results in Section 4 of this paper showed that the bounds of Bercu [11] (i.e.

{normalΦ}_{L - BE} (x)

and

{normalΦ}_{U - BE} (x)

) are very tight for small values of

x

, but it is less sharp for large values of

x

.…”

Section: Overview Of Some Bounds For φXmentioning

confidence: 90%

“…Bercu [11] gave the following lower and upper bounds for

Φ (x)

when

0 \leq x \leq 6.248

{normalΦ}_{L - BE} (x) \leq Φ (x) \leq {normalΦ}_{U - BE} (x),

where

{normalΦ}_{L - BE} (x) = \frac{1}{2} + \frac{β_{1} (x / \sqrt{2})}{\sqrt{π}}

and

{normalΦ}_{U - BE} (x) = \frac{1}{2} + \frac{β_{2} (x / \sqrt{2})}{\sqrt{π}}

. Also,

β_{1} (normalt) = \frac{110 {normalt}^{3} + 210 t}{39 {normalt}^{4} + 180 {normalt}^{2} + 210}

and

β_{2} (normalt) = \frac{113400 t}{29 t^{8} - 660 t^{6} + 1260 t^{4} + 37800 t^{2} + 113400} .

As noted by Bercu [11], the upper bound

{normalΦ}_{U - BE} (x)

is valid only for

0 \leq x \leq 6.248

. Peric et al [12] suggested a new lower bound for

Φ (x)

based on the lower bound of Boyd [6] and the lower bound of Ruskai and Werner […”

Section: Overview Of Some Bounds For φXmentioning

confidence: 99%

New tightness lower and upper bounds for the standard normal distribution function and related functions

Math Methods in App Sciences

2023

“…Also, let ℎ 𝐿−𝑡 (𝑥) be the absolute error function between the lower bound of Φ(𝑥) and the exact Φ(𝑥). That is, ℎ 𝑠−𝑡 (𝑥) = |Φ 𝑠−𝑡 (𝑥) − Φ(𝑥)|, where 𝑠 stands for 𝑈 or 𝐿 and 𝑡 stands for 𝑅𝑊 (Roskai and Werner, 2000), 𝐴𝐿 (Alzer, 2010), 𝐴𝐵 (Abreu, 2012), 𝑀𝐽 (Mastin and Jaillet, 2013), 𝑃𝐸 (Peric et al, 2019), 𝐵𝐸 (Bercu, 2020) and 𝐸𝐼 (Proposed bounds as given in Section 3). The absolute error associated with these bounds becomes very small as the size of 𝑥 increases.…”

Section: Numerical Comparisons and Conclusionmentioning

confidence: 99%

New Tightness Lower and Upper Bounds for the Standard Normal Distribution Function and Related Functions

2022

Preprint

Most researches interested in finding the bounds of the cumulative standard normal distribution Φ(x) are not tight for all positive values of the argument x. This paper mainly proposes new simple lower and upper bounds for Φ(x). Over the whole range of the positive argument x, the maximum absolute difference between the proposed lower bound and Φ(x) is less than 3×〖10〗^(-4), while it is less than 4.8×〖10〗^(-4) between the proposed upper bound and Φ(x). Numerical comparisons have been made between the proposed bounds and some of the other existing bounds, which showed that the proposed bounds are more compact than most alternative bounds found in the literature.

“…The Q-function and error function are widely used in Bit Error Rate (BER) analysis of communication systems (see Trigui et al, 2021 and the references theirin). Due to the relationship between Φ(𝑥) and the above functions along with the Mills' ratio (Fan, 2013), it makes sense that the results would find application in the broader fields such as statistical computations, applied statistics, mathematical models in biology, mathematical physics and diffusion theory (see Bercu, 2020 andLipoth et al, 2022). Eidous and Abu-Shareefa (2020) have reviewed 45 approximations for Φ(𝑥) reported in literature from 1945 to 2019.…”

Section: Introductionmentioning

confidence: 99%

“…𝐴𝐵(Abreu, 2012),𝑁𝐸 (Neumann, 2013), 𝑌𝐴(Yang et al, 2018), 𝐵𝐸(Bercu, 2020) and 𝑃𝑂 (Polya, 1949) (see Section 2). The error function for the proposed upper bound is ℎ r~( 𝑥) = Φ r~( 𝑥) − Φ(𝑥) (see Section 3).…”

mentioning

confidence: 99%

Improvements of Polya Upper Bound for Cumulative Standard Normal Distribution and Related Functions

2022

Preprint

Although there is extensive literature on the upper bound for cumulative standard normal distribution Φ(χ), there are relatively not sharp for all values of the interested argument χ. The aim of this paper is to establish a sharp upper bound for Φ(χ), in the sense that its maximum absolute difference from Φ(χ) is less than 5.785 x10-5 for all values of χ ≥ 0. The established bound improves the well-known Polya upper bound and it can be used as an approximation for Φ(χ) itself with very satisfactory accuracy. Numerical comparisons between the proposed upper bound and some other existing upper bounds have been achieved, which shows that the proposed bound is tighter than alternative bounds found in the literature.