Parametric bootstrap under model mis-specification

Under model correctness, highly accurate inference on a scalar interest parameter in the presence of a nuisance parameter can be achieved by several routes, among them considering the bootstrap distribution of the signed root likelihood ratio statistic. The context of model mis-specification is considered and inference based on a robust form of the signed root statistic is discussed in detail. Stability of the distribution of the statistic allows accurate inference, outperforming that based on first-order asymptotic approximation, by considering the bootstrap distribution of the statistic under the incorrectly assumed distribution. Comparisons of this simple approach with alternative analytic and non-parametric inference schemes are discussed.