Higher-order asymptotics provide accurate approximations for use in parametric statistical modelling. In this thesis, we investigate using higher-order approximations in two-specific settings, with a particular emphasis on the tangent exponential model.

The first chapter introduces first-order asymptotic theory and reviews key concepts such as sufficiency, significance, and exponential families. We then discuss higher-order approximations, which have been studied by many authors. The literature is rich with examples demonstrating the limitations of first-order methods when applied to models with many nuisance parameters and showcasing the increased accuracy of higher-order approximations.

The second chapter concerns collision assessment of space objects. Satellite conjunctions involving `near misses' are becoming increasingly likely. A common approach to risk analysis involves the computation of the collision probability, but this has been regarded as having some counter-intuitive properties, and its interpretation has been debated. We formulate an approach to satellite conjunction based on a simple statistical model and discuss inference on the miss distance between the two objects, for linear and non-linear motion. We point out that the usual collision probability estimate can be badly biased, but highly accurate inference on the miss distance is possible using the tangent exponential model. The ideas are illustrated with case studies and Monte Carlo results that show its excellent performance.

In the third chapter we study statistics used to test hypotheses concerning parameters on the boundary of their domain. These often have non-standard limiting distributions, which may be poor finite-sample approximations even when the sample size is very large. We distinguish soft and hard boundary problems, discuss elementary approached to both and describe an approach to small-sample approximation based on the tangent exponential model. Numerical results show that the approach can give much improved approximations, even in small samples.

We finish the thesis with ideas for future research in the field of particle physics, and including some preliminary results.