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.