Documentation
This Excel sheet contains various codes for the standard cumulative normal
distribution in dimensions 1 and 2 through series developments. The code
would result in 'arbitrary precision' for better environments than Excel
(and i originally did that in Maple) and they are up to machine precision
in Excel.
Codes are in cdfN1 and cdfN2_series as fcts cdfN_Marsaglia, cdfN_Marsaglia_0
and cdfN2_as_sum:
For dim=1 these solution are given by Marsaglia and series are stopped if
adding more terms does no longer change the summation, while for arguements
beyond 7 an asymptotics from Abramowitz & Stegun is used (their series can
be shown to have a closed form). In cdfN_Marsaglia functions are 'Taylored'
around 0, ... , 7 and should finish after 20 steps with Excel's exactness.
In cdfN_Marsaglia_0 the series is given in Zero, so it might need up to
140 steps, the tail error may be estimated through the Gamma distribution
as 1-GAMMA(-1/2+n,1/2*x^2)/GAMMA(-1/2+n) for n steps.
For dim 2 and lower correlation Marsaglia's method can be used to write
down recursions for the Taylor series while for higher correlations it is
better to use a method of Vasicek (which i modified a bit), both more or
less are recursions for the incomplete Gamma function involved. For both
the expansion is stopped by machine precision (and the cdfN1 used).
AVt, Jan 2005