Second Order Ancillary is Rotation Based

The present work concerns the second order ancillary determined by the tangent directions V = ∂y/∂θ|{y0; θ̂(y0)} at the observed data y 0 for fixed pivotal value, by using the structural approach of Fraser (1986). We show that for scalar parameter case, such an approximate ancillary is well defined and can be described as a rotation. These second order ancillaries are particularly useful in providing access to higher order approximations of p-values and conditional density of the model in a neighborhood of the observed data. More precisely, Fraser (1988) revealed that accurate inference can obtained by using only the tangent directions to a second order ancillary, thus avoiding its exact specification. Our approach uses V vectors as introduced by Fraser & Reid (1995, 2001), which require a coordinate-by-coordinate pivotal variable, always available.