Second order ancillary: A differential view from continuity

TL;DR

This paper explores second order approximate ancillaries in statistical inference, using a differential approach based on quantile functions to improve likelihood-based methods and confidence calculations.

Contribution

It introduces a differential geometric perspective on second order ancillaries, extending their construction to vector parameters and demonstrating their properties through examples.

Findings

Conditions hold in a restricted sense for vector cases

Methodology can generate exact or approximate ancillaries

Applications include nonlinear regression and complex examples

Abstract

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.