Bivariate Tail Probability Approximation

Music City Center 

We generalize saddlepoint approximations for tail areas of cumulative distribution functions of multivariate random variables by extending the univariate Lugannani and Rice saddlepoint tail approximation to multiple dimensions. The proposed approximation is derived by parts. The resulting approximation uses a multivariate Gaussian approximation to the distribution of the signed roots of the log-likelihood statistic. As in the univariate case, the next correction terms involve the difference between reciprocals of the signed root of the likelihood statistics and the analogous Wald statistics. This approximation also uses the curvature of the boundary of the tail region in terms of signed roots of likelihood ratio statistics. The separate versions of the approximation extended to lattice variables and conditional distributions are also provided. Numerical comparisons with other methodologies show better agreement with the exact distribution. We discuss the practical importance of the approximation for multiple dimensions, providing applications for sufficient statistics and statistical multivariate inference.

Saddlepoint approximation

Multivariate Gaussian approximation

Complex analysis

Curvature of the boundary of tail region 

IMS