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We obtain two theorems extending the use of a saddlepoint approximation tomultiparameter problems for likelihood ratio-like statistics which allow theiruse in permutation and rank tests and could be used in bootstrapapproximations. In the first, we show that in some cases when no densityexists, the integral of the formal saddlepoint density over the setcorresponding to large values of the likelihood ratio-like statisticapproximates the true probability with relative error of order $1/n$. In thesecond, we give multivariate generalizations of the Lugannani--Rice andBarndorff-Nielsen or $r^*$ formulas for the approximations. These theorems areapplied to obtain permutation tests based on the likelihood ratio-likestatistics for the $k$ sample and the multivariate two-sample cases. Numericalexamples are given to illustrate the high degree of accuracy, and thesestatistics are compared to the classical statistics in both cases.

Saddlepoint approximations for likelihood ratio like statistics with applications to permutation tests