  Second order approximate ancillaries have evolved as the primary ingredient
for recent likelihood development in statistical inference. This uses quantile
functions rather than the equivalent distribution functions, and the intrinsic
ancillary contour is given explicitly as the plug-in estimate of the vector
quantile function. The derivation uses a Taylor expansion of the full quantile
function, and the linear term gives a tangent to the observed ancillary
contour. For the scalar parameter case, there is a vector field that integrates
to give the ancillary contours, but for the vector case, there are multiple
vector fields and the Frobenius conditions for mutual consistency may not hold.
We demonstrate, however, that the conditions hold in a restricted way and that
this verifies the second order ancillary contours in moderate deviations. The
methodology can generate an appropriate exact ancillary when such exists or an
approximate ancillary for the numerical or Monte Carlo calculation of
$p$-values and confidence quantiles. Examples are given, including nonlinear
regression and several enigmatic examples from the literature.
