An asymptotic formula and a series expansion for bivariate Normal tail probability

Abstract

Abstract This work presents a new asymptotic formula for the bivariate Normal tail probability. It only requires the larger threshold to grow indefinitely, but otherwise has no restriction on how the thresholds grow. The correlation parameter can change and possibly depend on the thresholds. The formula is applicable regardless of Salvage condition. Asymptotically, it reduces to Ruben’s formula and Hashorva’s formula under the corresponding conditions, and therefore can be considered a generalisation. Under a mild condition, it satisfies Plackett’s identity on the derivative with respect to correlation parameter. Motivated by the asymptotic formula, a series expansion in terms of the derivatives of univariate Mill’s ratio is also obtained for the exact tail probability, whose terms can be calculated recursively. Based on the series expansion, a simple procedure is developed for general numerical computation by suitable redefinition of parameters. Examples are presented to illustrate the theoretical findings.