New Tightness Lower and Upper Bounds for the Standard Normal
Distribution Function and Related Functions
Abstract
Most researches interested in finding the bounds of the cumulative
standard normal distribution Φ(x) are not tight for all positive values
of the argument x. This paper mainly proposes new simple lower and upper
bounds for Φ(x). Over the whole range of the positive argument x, the
maximum absolute difference between the proposed lower bound and Φ(x) is
less than 3×〖10〗^(-4), while it is less than 4.8×〖10〗^(-4)
between the proposed upper bound and Φ(x). 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.