To respect the conditionality principle, it may be necessary to consider conditioning on an approximate ancillary, as finding an exact ancillary is not always straightforward. Since several approximate ancillaries exist, a comparison of the properties of two commonly used approximate ancillaries, the affine ancillary (Efron and Hinkley, 1978) and the likelihood ancillary (Barndorff-Nielsen, 1980), is carefully carried out for Butler’s (2004) and Pedersen’s (1981) examples where the parameter of interest is scalar. The inference drawn in both examples show similar outcomes. For inference conditional on the likelihood ancillary, the (unconditional) parametric simulation of the signed root likelihood ratio statistic works well even for small sample sizes, giving highly accurate approximation to exact conditional inference, as good as those obtained by more complicated analytic procedures. For inference conditional on the affine ancillary, we argue that to obtain good approximations it is necessary to consider some form of stratification of the bootstrap simulation (Hinkley and Schechtman, 1987) as the range of the estimator of the parameter of interest depends on the value of the affine ancillary. Our results are different to Butler’s (2004), as on that paper it was not realized that the range of the estimator of the parameter of interest depended on the value of the affine ancillary, as it was suggested for Pedersen’s example in Barndorff-Nielsen (1980).
Focus will also be given to models that present multidimensional ancillary statistics. Of particular interest is the location-scale family, where the ancillary statistic can be of a very high dimension, as its dimension is often very close to the sample size n. But also, examples where there are nuisance parameters will be treated. Solutions to conditional inference are offered, whether they are SPA or based on bootstrap methods for the extreme value model and the Weibull regression. Cox and Reid’s (1987) solution is introduced and applied to these models. Given the Cox and Reid’s solution, a quantity relating it to r∗ is derived (DiCiccio and Young, Properties of parametric bootstrap procedures based on signed roots of likelihood ratio statistics, unpublished paper) which could facilitate the computations required to respect Fisher’s conditional principle. An exact solution to conditional inference is given using MCMC techniques, namely Metroplis within Gibbs that could be applied to the joint conditional density of the pivots.