  We obtain two theorems extending the use of a saddlepoint approximation to
multiparameter problems for likelihood ratio-like statistics which allow their
use in permutation and rank tests and could be used in bootstrap
approximations. In the first, we show that in some cases when no density
exists, the integral of the formal saddlepoint density over the set
corresponding to large values of the likelihood ratio-like statistic
approximates the true probability with relative error of order $1/n$. In the
second, we give multivariate generalizations of the Lugannani--Rice and
Barndorff-Nielsen or $r^*$ formulas for the approximations. These theorems are
applied to obtain permutation tests based on the likelihood ratio-like
statistics for the $k$ sample and the multivariate two-sample cases. Numerical
examples are given to illustrate the high degree of accuracy, and these
statistics are compared to the classical statistics in both cases.
