TITLE:
The Bivariate Normal Integral via Owen’s T Function as a Modified Euler’s Arctangent Series
AUTHORS:
Janez Komelj
KEYWORDS:
Owen’s T Function, Bivariate Normal Integral, Euler’s Arctangent Series, Recursion, R Package Phi2rho
JOURNAL NAME:
American Journal of Computational Mathematics,
Vol.13 No.4,
October
19,
2023
ABSTRACT: The Owen’s T function is
presented in four new ways, one of them as a series similar to the Euler’s
arctangent series divided by 2π, which is its majorant series. All
possibilities enable numerically stable and fast convergent computation of the bivariate normal integral with simple
recursion. When tested computation on a random sample of one million
parameter triplets with
uniformly distributed components and using double precision arithmetic, the
maximum absolute error was 3.45 × 10-16.
In additional testing, focusing on cases with correlation coefficients close to
one in absolute value, when the computation may be very sensitive to small
rounding errors, the accuracy was retained.
In rare potentially critical cases, a simple adjustment to the computation
procedure was performed—one potentially critical computation was
replaced with two equivalent non-critical ones. All new series are suitable for
vector and high-precision computation, assuming they are supplemented with appropriate efficient and accurate
computation of the arctangent and standard normal cumulative
distribution functions. They are implemented by the R package Phi2rho,
available on CRAN. Its functions allow vector arguments and are ready to work
with the Rmpfr package, which enables the use of arbitrary precision
instead of double precision numbers. A special test with up to 1024-bit
precision computation is also presented.