Nonparametric tests for multi-parameter M-estimators

We consider likelihood ratio like test statistics based on M-estimators for multi-parameter hypotheses for some commonly used parametric models where the assumptions on which the standard test statistics are based are not justified. The nonparametric test statistics are based on empirical exponential families and permit us to give bootstrap methods for the tests. We further consider saddlepoint approximations to the tail probabilities used in these tests. This generalizes earlier work of Robinson et al. (2003) in two ways. First, we generalize from bootstraps based on resampling vectors of both response and explanatory variables to include bootstrapping residuals for fixed explanatory variables, resulting in a surprising result for the weighted resampling. Second, we obtain a theorem for tail probabilities under weak conditions providing essential justification for the approximation to bootstrap results for both cases. We use as examples linear regression, non-linear regression and generalized linear models under models with independent and identically distributed residuals or vectors of observations, giving numerical illustrations of the results.