Quantal models: a review with additional methodological development

Analysis of quantal models is a particular aspect of the general problem of investigating multimodality. The distinction is that the spacings between modes are integral multiples of some unspecified fundamental unit and that the number of modes is not defined. Such semi-structured models arise in a wide variety of contexts such as biology, cosmology, archaeology and molecular physics. This paper presents a brief review of their historical development in such areas as an aid to their recognition in other contexts as well as giving guidance to their analysis from the statistical viewpoint. The available methodology for their analysis is collated into a coherent and self-contained account, establishing various optimality properties under particular parametric distributional assumptions. An illustrative power study shows how dependence on sample size and failure of assumptions such as underlying distribution, origin of measurements and independence affect the power of various analyses. These aspects are illustrated by an example from developmental biology.