Introduction to the theory of regular exponential families

Preface The first version of these notes appeared in 1970 when I lectured on exponential families on the basis of some notes written by Ole Barndorff - Nielsen and Anders Holst Andersen. A Danish version appeared in 1976 in connection with a graduate course in statistics and the present version is essentially a translation of this. In the meantime the excellent book by Barndorff - Nielsen (1978) has appeared. The present notes differ in the choice of material in that I have concentrated On the algebraic aspects,of the regu- lar models and then added the classical application of the Neyman Pearson theory as well as the asymptotic theory, without which it is difficult to apply any of the models. I have benefitted from a visit to Lund (1972) and to Goteborg (1976) where I presented the topic in a series of lectures. I welcome the opportunity to thank Gunnar Blom and Peter Jagers for the invitation and for hospitality during the visits. Another main source of inspiration has been the lecture notes by Per Martin - LOf (1970) who has developed the exponential families and their statistical analysis from the point of view of stati- stical mechanics. This point of view has not been adopted here. A short account can be found in Per Martin - Löf (1974). The lecture notes by Søren Tolver Jensen (1978) have helped me clarify the formulation of the case of a singular covariance ma- trix, see section 7. Anders Hald and Steffen Lauritzen have carefully read the manu- script in one of its previous versions with the result that some formulations were improved and others corrected. Vivi Nielsen has typed the final manuscript. Thus it will be apparent that I have borrowed all the material from other sources and have had experts to check the presentation. My only contributions if any thus lies in the initiative plus the remaining errors. Copenhagen, February 1979 Søren Johansen