This research contributes to the development of accurate approximation techniques for small samples sizes. It shows that high order approximations may be extended to robust testing and should be used when the sample size is small.

In the parametric framework, high order approximations for classical test statistics have been developed under the strict assumption that the distribution of the observations exactly follows the model. Since this assumption is not satisfied in reality, robust tests are actually required and the need for high order approximations to their test statistics motivated our work.

We begin with a survey of the literature on high order expansions for tests and we focus in particular on the Bartlett correction.

The second part of this research contains results which give high order expansions for robust statistics. We begin by deriving high order expansions to a robust test stastistic and its expectation which apply to a general parametric model. We then focus on linear regression models and propose formulas to compute robust p-values. In addition we also examine robust quantiles.

In the third part, we carry out a wide numerical study, showing that for real data sets our formulas are reliable both with respect to small sample sizes and departures from the assumptions of the model.

Finally, we provide the user with routines in S-Plus.