On multidimensional expansions of distributions of sums by the saddle point method

Many problems of applied statistics, in particular, hypotheses testing, contain no information about true distributions of the statistics under consideration. Therefore, limit distributions of those statistics are used, which may give inadequate results in actual practice. Problems on the refinement of the distributions thus arise. In the present paper, we extend the results of H. Daniels, R. Lugannani, and S. Rice to the multidimensional case. We prefer to use the Lugannani-Rice method because it yields approximations which are as good as those of other forms and do not depend on variation of statistics. Translated fromStatisticheskie Metody Otsenivaniya i Proverki Gipotez, pp. 195–202, Perm, 1993.