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

Abstract

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.

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

“…The probability that a standard normal random variable is less than a real value is known as the cumulative standard normal distribution, which is mathematically given by where is the standard normal probability density function of a random variable The function cannot be expressed in a closed form; therefore, there are numerous proposed approximations for that developed in the literature (see previous study [1]). Also, many works are interested to find upper bounds for [2] and lower bounds for [3]. There are two main interesting functions related to .…”

“…, À ∞ < t < ∞ is the standard normal probability density function of a random variable X: The function Φ x ð Þ cannot be expressed in a closed form; therefore, there are numerous proposed approximations for Φ x ð Þ that developed in the literature (see previous study [1]). Also, many works are interested to find upper bounds for Φ x ð Þ [2] and lower bounds for Φ x ð Þ [3]. There are two main interesting functions related to Φ x ð Þ.…”