Introduction

The last two decades have seen a rapid growth in the application of differential geometry in statistics. Efron (1975) stimulated much of this research with his definition of statistical curvature and he showed that curvature has serious consequences for statistical inference.

Many papers on the application of differential geometry in statistics go straight into defining all the necessary tools, such as the metric and an affine connection on a manifold, and show how they can be used in statistical analysis. The emphasis is predominantly on asymptotic theory and applications are mainly in estimation, information loss and higher-order efficiency. Barndorff-Nielsen, Cox and Reid (1986) give a very accessible account of the relevant ideas in differential geometry and also provide a historical overview; see also Amari (1985) and Okamoto, Amari and Takeuchi (1991) provide a brief recent summary of main achievements.

This chapter is concerned with the effects of curvature on hypothesis testing. However, our approach differs from the differential geometrical approach mentioned above in a number of ways. First, we give a global analysis, i.e. for the whole of the sample and parameter space, not merely in a neighbourhood of a fixed point such as the true parameter value. Secondly, the analysis is valid for all sample sizes, not just asymptotically. Finally, we are concerned with hypothesis testing, which is less common, and use the partitioning of the sample space into critical region and acceptance region to illustrate our arguments graphically, rather than solely investigating the (analytic) properties of test statistic(s).