Bounds on the Bivariate Normal Distribution Function

Abstract

Inequalities are presented for the upper bivariate-normal tail probability Lðh; k; rÞ for use in bounding the bivariate probability integral Fðh; k; rÞ. These are relatively simple and more widely applicable than existing bounds with similar performance, and have superior performance if jrj is small or if Fðh; k; rÞ is very large. The upper bound is tight when Fðh; k; rÞ is large, and has a simple form when h ¼ k.

Energy landscape for large average submatrix detection problems in Gaussian random matrices

“…The claim now follows from the factZ − Z 1 is normal with mean zero and variance (k − 1)/k, and that The following result from [39] makes these ideas precise.…”

“…Thus, conditional on A, the event {ρZ + 1 − ρ 2 Z > x} is roughly the same as {Z > θx} with θ = (1 − ρ)/(1 + ρ). The following result from [39] makes these ideas precise.…”

The Bivariate Normal Copula

“…Note also that for many existing approximations and bounds the diagonal u = v may be considered a worst case, cf. (Willink 2004). Mee & Owen (1983) elaborated on the so-called conditional approach proposed by Pearson (1901b).…”

“…His lower bound is too weak (even negative) on the diagonal. A recent overview on known bounds, and derivation of some new ones, is provided by Willink (2004). We will present some of his bounds below in more general context.…”

“…The bounds (5.3) and (5.6) have been discussed, without explicit computation of the maximum error, by Willink (2004). The maximum error of (5.5) is difficult to grasp analytically.…”