# Accurate RMM-Based Approximations for the CDF of the Normal Distribution

@article{Shore2005AccurateRA, title={Accurate RMM-Based Approximations for the CDF of the Normal Distribution}, author={Haim Shore}, journal={Communications in Statistics - Theory and Methods}, year={2005}, volume={34}, pages={507 - 513} }

Abstract A variation of the RMM error distribution, used to model the exponential distribution, has recently been applied to derive a three-parameter approximation for the standard normal CDF, with a maximum absolute error of order (10)−5. In this short communication, a simple modification enhances the accuracy to the order of (10)−6. Another RMM-based approximation, based on the original RMM error distribution, achieves an absolute maximum error of (10)−7. The simplicity of the new non…

## 14 Citations

A Sharp Polya-Based Approximation to the Normal CDF

- Mathematics, Computer Science
- 2016

We introduce a closed form approximation to the cumulative distribution function of the standard normal variable involving only five explicit constants with an approximation error of 5.79 x 10^{-6}…

New approximations for standard normal distribution function

- MathematicsCommunications in Statistics - Theory and Methods
- 2019

Abstract This article proposes nine new approximations for the standard normal cumulative distribution function In addition, it collects most of the approximations existing in the literature. The…

A Simple Approximation to the Area Under Standard Normal Curve

- Mathematics
- 2014

Of all statistical distributions, the standard normal is perhaps the most popular and widely used. Its use often involves computing the area under its probability curve. Unlike many other statistical…

APPROXIMATIONS TO THE NORMAL DISTRIBUTION FUNCTION AND AN EXTENDED TABLE FOR THE MEAN RANGE OF THE NORMAL VARIABLES

- Mathematics
- 2008

This article presents a formula and a series for approx¬imating the normal distribution function. Over the whole range of the normal variable z, the proposed formula has the greatest absolute error…

Very simply explicitly invertible approximations of normal cumulative and normal quantile function

- Mathematics
- 2014

For the normal cumulative distribution function: Φ(x) we give the new approximation 2**(-22**(1-41**(x/10))) for any x>0, which is very simple (with only integer constants and operations and / and…

Response modeling methodology

- Computer Science
- 2011

This overview of response modeling methodology details the motivation that led to the development of RMM, explains RMM core concepts, and introduces RMM basic model and variations.

Profit Maximizing Warranty Period with Sales Expressed by a Demand Function

- MathematicsQual. Reliab. Eng. Int.
- 2007

The problem of determining the optimal warranty period, assumed to coincide with the manufacturer's lower specification limit for the lifetime of the product, is addressed and a general solution is derived using Response Modeling Methodology (RMM) and a new approximation for the standard normal cumulative distribution function.

Approximations to the Normal Probability Distribution Function using Operators of Continuous-valued Logic

- Computer Science, MathematicsActa Cybern.
- 2018

It is demonstrated here that application of the averaging Dombi conjunction operator to two symmetric Sigmoid fuzzy membership functions results in a function that is identical with Tocher’s approximation to the standard normal probability distribution function.

Sample-Size Determination

- Mathematics
- 2008

Sample data may be collected with different objectives in different scenarios. In some cases, one may wish to collect enough observations that would guarantee minimal prespecified reliability for the…

A Fairly Accurate Approximation to the Area Under Normal Curve

- MathematicsCommun. Stat. Simul. Comput.
- 2009

A new approximation to the cumulative distribution function of standard normal distribution is presented that outperforms other such approximations available in literature and is fairly accurate with minimum accuracy of seven decimal digits.

## References

SHOWING 1-10 OF 11 REFERENCES

Response Modeling Methodology (RMM): Current distributions, transformations, and approximations as special cases of the RMM error distribution

- Mathematics
- 2004

Recently a new Response Modeling Methodology (RMM) has been introduced, which models the relationship between a response and the affecting factors, assuming only that this relationship is monotone…

CALCULATING NORMAL PROBABILITIES

- Mathematics
- 1995

This is probably the best known example of an integral that cannot be evaluated in terms of elementary functions. In this note we develop an elementary approximation to P(a) which arises in a natural…

General control charts for variables

- Mathematics
- 2000

When the distribution of the monitoring statistic used in statistical process control is non-normal, traditional Shewhart charts may not be applicable. A common practice in such cases is to normalize…

Continuous Univariate Distributions.

- Mathematics, Economics
- 1995

Continuous Univariate Distributions.1-2Characterizations of Univariate Continuous DistributionsCharacterizations of Univariate Continuous DistributionsDictionary and Classified Bibliography of…

Response modeling methodology (RMM)-On the relationship between the RMM model and the assumption of normal (or lognormal) errors. Under review

- 2004

Response modeling Methodology (RMM)-current statistical distributions, transformations and approximations as special cases of RMM

- Commun. Statist. Theory Meth
- 2004

RESPONSE MODELING METHODOLOGY (RMM)—EXPLORING THE PROPERTIES OF THE IMPLIED ERROR DISTRIBUTION

- Engineering
- 2002

ABSTRACT Modeling efforts in engineering and the sciences often attempt to describe the relationship between a response and some external affecting factor (or a linear combination of factors), where…

Introduction to Reliability Mathematics

- 1986

Handbook of Mathematical Functions. National Bureau of Standards

- 1970