Introduction

In this chapter we illustrate the breadth of application of higher order asymptotics by presenting a variety of examples, most of which have appeared in the published literature. In contrast to the earlier chapters, the emphasis is on the methods for higher order approximation, with the data treated as mainly illustrative. Section 7.2 outlines a problem of calibration in normal linear regression, and the two succeeding sections outline higher order approximation for a variance components setting and for dependent data, respectively. Sections 7.5 and 7.6 concern a problem of gamma regression; we compare Bartlett correction to Skovgaard's multivariate adjustment to the likelihood ratio statistic, and indicate the use of Laplace approximation for Bayes inference. In Section 7.7 we consider if it is worthwhile to apply higher order approximation to partial likelihood. The final section concerns use of a constructed exponential family to find the distribution of interest for a randomisation test.

Calibration

Table 7.1 shows measurements of the concentration of an enzyme in human blood plasma. The true concentration x is obtained using an accurate but expensive laboratory method, and the measured concentration y is obtained by a faster and less expensive automatic method. The goal is to use the observed data pairs to estimate values of the true concentration based on further measurements using the less expensive method. This is an example of a calibration problem: we have a model for E(y|x) that depends on some unknown parameters, and use a sample of pairs (x1, y1),…,(xn, yn) to estimate these parameters.