The second order ancillary is rotation based

This paper concerns the approximate ancillary needed for higher order asymptotic likelihood inference. The existence of an exact or approximate ancillary is the key to such inference. The exact ancillary is, however, only available for transformation model and an approximate ancillary is typically very difficult to find. Fraser (1988) revealed that the high order inference can be obtained by using only the tangent directions to the second order ancillary, thus avoiding its specification. The present work discuses the second order ancillary that has the observed contour tangent to the sensitivity directions v = ∂ y / ∂ θ | { y 0 ; θ ^ ( y 0 ) } corresponding to a pivot and derived by keeping the pivotal variable fixed at the observed fitted value; these directions are always available (Fraser and Reid, 1995, 2001). For scalar parameter case, such an approximate ancillary is well defined and can be described as a rotation. Our approach provides insights into the second order ancillary outlined by the sensitivity directions in the vector parameter case.