Bounds on the Bivariate Normal Distribution Function

Abstract

Inequalities are presented for the upper bivariate-normal tail probability L(h, k, ρ) for use in bounding the bivariate probability integral Φ(h, k, ρ). These are relatively simple and more widely applicable than existing bounds with similar performance, and have superior performance if |ρ| is small or if Φ(h, k, ρ) is very large. The upper bound is tight when Φ(h, k, ρ) is large, and has a simple form when h = k.

Key Words:

• Inequalities
• Correlation
• Truncated normal distribution
• Stochastic order

Mathematics Subject Classification:

• Primary 60E15
• Secondary 62E99, 62H99

Acknowledgment

The author is indebted to C. S. Withers for helpful comments in the preparation of this manuscript.