Tests based on Monte Carlo simulations conditioned on maximum likelihood estimates of nuisance parameters

A method to eliminate nuisance parameters in statistical inference is to condition on sufficient statistics for those parameters. In many situations it is not possible to find appropriate sufficient statistics on which to condition, the Behrens-Fisher problem being a well known example. In this paper a general procedure, based on stochastic simulations, is given for how to deal with nuisance parameters in such cases. The technique is based on adjusting the parameters in the simulations so that the observed maximum likelihood estimates of the nuisance parameters is obtained. The method is shown to have good properties when applied to the Behrens-Fisher problem, and is also shown to work well when comparing means in more than two groups with possibly unequal variances. The simulation procedure has the advantage that it is rather general, and can easily be applied to other problems.