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

Abstract

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.