  We extend known saddlepoint tail probability approximations to multivariate
cases, including multivariate conditional cases. Our approximation applies to
both continuous and lattice variables, and requires the existence of a cumulant
generating function. The method is applied to some examples, including a real
data set from a case-control study of endometrial cancer. The method contains
less terms and is easier to implement than existing methods, while showing an
accuracy comparable to those methods.
