Confidence Intervals From Monte Carlo Tests
Journal of the American Statistical Association
It is argued that confidence sets can be derived from Monte Carlo tests by exploiting the equivalence between confidence estimation and hypothesis testing. The approach benefits from the wide applicability of this class of tests and the high level accuracy that is passed on to the confidence sets. The main problem is whether the confidence sets are simple enough to be of practical interest. A conservative general approximation is given, but most of the paper deals with exact methods in
... methods in one-parameter situations. We work with statistics that allow stochastic representations that are almost surely monotone in terms of the parameter of interest. Simulated samples can then be adjusted by varying the parameter and keeping random drawings fixed. By making some selected fractile of such Monte Carlo samples equal to the observed statistic, a critical point with exact confidence level can be determined. A simple theory to compare the Monte Carlo uncertainty to the uncertainty contributed by the data is developed. The main application is to models belonging to the one-parameter exponential class. Other examples considered are locationand scale models, the correlation coefficient, the size parameter in hypergeometric experiments, binomials and rank statistics. The choice of sampling technique is delicate and strongly problem dependent. Exact results, with the exception of pure location-scale models, are rarely possible when there are nuisance parameters, but it is hoped that the basic approach may turn out to be a way to obtain accurate, approximate results under many circumstances.