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

TL;DR

This paper develops a sharper upper bound for the standard normal distribution function, improving upon Polya's bound, with demonstrated higher accuracy through numerical comparisons.

Contribution

It introduces a new, more precise upper bound for the standard normal distribution function that outperforms existing bounds in accuracy.

Findings

The new bound is tighter than existing bounds.

Numerical comparisons confirm improved accuracy.

The bound is suitable for approximation purposes.

Abstract

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