Two Interval Upper-Bound Q-Function Approximations with Applications

Perić, Zoran; Marković, Aleksandar; Kontrec, Nataša; Nikolić, Jelena; Petković, Marko D.; Jovanović, Aleksandra

doi:10.3390/math10193590

Cited by 4 publications

(6 citation statements)

References 35 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Although

{normalΦ}_{L - PER 2} (x)

is narrower than

{normalΦ}_{L - PER 1} (x)

for

0.4993 \leq x \leq 1.7250

(see a previous study [12]), we found that the proposed lower bound remains closer than

{normalΦ}_{L - PER 2} (x)

over the same interval. For

0 < x < 1.7

, the absolute error associated with Bercu (BE) bounds is very small compared to the other bounds.…”

Section: Numerical Comparisons and Conclusionsupporting

confidence: 43%

“…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 [7]. Their lower bound is given as follows:

{normalΦ}_{L - PER 1} (x) \leq Φ (x),

where

{normalΦ}_{L - PER 1} (x) = 1 - \min \{q_{1} (x), q_{2} (x)\}

q_{1} (x) = \frac{π ϕ_{X} (x)}{2 x + \sqrt{{((), π - 2)}^{2} x^{2} + 2 π}}

and

q_{2} (x) = \frac{4 ϕ_{X} (x)}{3 x + \sqrt{x^{2} + 8}}

…”

Section: Overview Of Some Bounds For φXmentioning

confidence: 99%

“…Their lower bound is given as follows:

{normalΦ}_{L - PER 1} (x) \leq Φ (x),

where

{normalΦ}_{L - PER 1} (x) = 1 - \min \{q_{1} (x), q_{2} (x)\}

q_{1} (x) = \frac{π ϕ_{X} (x)}{2 x + \sqrt{{((), π - 2)}^{2} x^{2} + 2 π}}

and

q_{2} (x) = \frac{4 ϕ_{X} (x)}{3 x + \sqrt{x^{2} + 8}}

. Peric et al [12] proved that

q_{1} (x) \leq q_{2} (x)

for

x \leq (4 - π) / \sqrt{2 (π - 3) (3 π - 8)} ≅ 1.3514

. In addition, they gave another lower bound by combining the lower bound of Abreu [9] with their above lower bound.…”

Section: Overview Of Some Bounds For φXmentioning

confidence: 99%

“…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) = |{normalΦ}_{s - t} (x) - Φ (x)|

, where

s

stands for

U

L

and

t

stands for

RW

[7],

AL

[8],

AB

[9],

MJ

[10],

PE

[3],

BE

[11],

PER 1

[12], and

EI

(Proposed bounds as given in Section 3).…”

Section: Numerical Comparisons and Conclusionmentioning

confidence: 99%

See 3 more Smart Citations

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

Eidous

2023

Math Methods in App Sciences

View full text Add to dashboard Cite

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.

show abstract

“…Although

{normalΦ}_{L - PER 2} (x)

is narrower than

{normalΦ}_{L - PER 1} (x)

for

0.4993 \leq x \leq 1.7250

(see a previous study [12]), we found that the proposed lower bound remains closer than

{normalΦ}_{L - PER 2} (x)

over the same interval. For

0 < x < 1.7

, the absolute error associated with Bercu (BE) bounds is very small compared to the other bounds.…”

Section: Numerical Comparisons and Conclusionsupporting

confidence: 43%

“…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 [7]. Their lower bound is given as follows:

{normalΦ}_{L - PER 1} (x) \leq Φ (x),

where

{normalΦ}_{L - PER 1} (x) = 1 - \min \{q_{1} (x), q_{2} (x)\}

q_{1} (x) = \frac{π ϕ_{X} (x)}{2 x + \sqrt{{((), π - 2)}^{2} x^{2} + 2 π}}

and

q_{2} (x) = \frac{4 ϕ_{X} (x)}{3 x + \sqrt{x^{2} + 8}}

…”

Section: Overview Of Some Bounds For φXmentioning

confidence: 99%

“…Their lower bound is given as follows:

{normalΦ}_{L - PER 1} (x) \leq Φ (x),

where

{normalΦ}_{L - PER 1} (x) = 1 - \min \{q_{1} (x), q_{2} (x)\}

q_{1} (x) = \frac{π ϕ_{X} (x)}{2 x + \sqrt{{((), π - 2)}^{2} x^{2} + 2 π}}

and

q_{2} (x) = \frac{4 ϕ_{X} (x)}{3 x + \sqrt{x^{2} + 8}}

. Peric et al [12] proved that

q_{1} (x) \leq q_{2} (x)

for

x \leq (4 - π) / \sqrt{2 (π - 3) (3 π - 8)} ≅ 1.3514

. In addition, they gave another lower bound by combining the lower bound of Abreu [9] with their above lower bound.…”

Section: Overview Of Some Bounds For φXmentioning

confidence: 99%

“…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) = |{normalΦ}_{s - t} (x) - Φ (x)|

, where

s

stands for

U

L

and

t

stands for

RW

[7],

AL

[8],

AB

[9],

MJ

[10],

PE

[3],

BE

[11],

PER 1

[12], and

EI

(Proposed bounds as given in Section 3).…”

Section: Numerical Comparisons and Conclusionmentioning

confidence: 99%

See 2 more Smart Citations

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

Eidous

2023

Math Methods in App Sciences

View full text Add to dashboard Cite

show abstract

“…The cumulative distribution function (CDF) and its various forms (see Table 1 in the appendix) have significant applications in a wide range of fields [30], such as engineering [14,27,29], Mathematics [21], Statistics [4] computer science [15], diffusion theory [5], communication theory [26], physics [6], and chemistry [24]. Due to the mathematical complexity of the integral form, various approximations have been presented in the literature [19,11].…”

Section: Introductionmentioning

confidence: 99%

New Mathematical Approximations for the Cumulative Normal Distribution Function

Etesami,

Madadi,

Keynia

et al. 2024

Preprint

View full text Add to dashboard Cite

The normal cumulative distribution function and its derivatives, such as the error function, the Q-function, and the Mills ratio, are widely used in engineering, mathematics, statistics, computer science, diffusion theory, communication theory, physics, and chemistry. However, their non-closed form nature has led to the development of new approximations with varying levels of accuracy and complexity. These new approximations are often more accurate; nevertheless, they can also be more complex, which may limit their practical utility. In this article, a new approach for approximating is proposed. which combines Taylor series expansion and logistic function to create an initial approximation, to enhance the accuracy of the initial approximation, the Hunter-Prey Optimization algorithm is utilized to minimize both the maximum absolute error and the mean absolute error, leading to a significantly more precise approximation. Furthermore, this algorithm is employed to enhance the accuracy of other existing approximations introduced by researchers. The results showed that the improved approximations have much higher accuracy. To show the effectiveness of the new findings of this article, two case studies with applications are presented.

show abstract

Novel approximation of $$Q$$-function using Gauss quadrature rule

Goel,

Gupta

2023

Sādhanā

View full text Add to dashboard Cite

Two Interval Upper-Bound Q-Function Approximations with Applications

Cited by 4 publications

References 35 publications

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

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

New Mathematical Approximations for the Cumulative Normal Distribution Function

Novel approximation of $$Q$$-function using Gauss quadrature rule

Contact Info

Product

Resources

About