Accurate approximation to sampling distributions

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I aim to develop accurate approximations to sampling distributions of various statistical models, which would be applicable to samples of any practical size (in many cases, the usual Normal approximation works well only with unrealistically large samples). This is done by expanding these distributions in inverse powers of the sample size, and finding the first few terms of such expansions. The technique requires large amount of symbolic computation, and the corresponding program development. The results are used for constructing accurate confidence intervals for values of unknown parameters of these models, which is one of the primary tasks of Statistics. The project will also involve a more theoretical study of convergence of these expansions.