Accurate directional inference for vector parameters

We consider statistical inference for a vector-valued parameter of interest in a regular asymptotic model with a finite-dimensional nuisance parameter. We use highly accurate likelihood theory to derive a directional test, in which the $p$-value is obtained by one-dimensional numerical integration. This extends the results of Davison et al. (2014) for linear exponential families to nonlinear parameters of interest and to more general models. Examples and simulations provide comparisons with the likelihood ratio test and adjusted versions of the likelihood ratio test. The directional approach gives extremely accurate inference, even in high-dimensional settings where the likelihood ratio versions can fail catastrophically.