Repetitive Structures of Power Type. Discrete Sample Spaces
The dominating type of statistic occurring in well known models is additive in an abstract sense, whether this be a counting function such as the number of timescertain events occur, the maximum of the observations in a sample or the frequency table (histogram). The fundamental property that these statistics have in common can be expressed by assuming them to take values in an Abelian semigroup. In fact one can show that any statistic which depends symmetrically on the observations and is algebraically transitive, must be of this type. Thus it seems very worthwhile to devote a large part of the present piece of work to the study of extremal families with such statistics. The models rightly deserve the name of exponential families, where ‘exponential’ refers to the fact that they are determined by multiplicative homorphisms of the additive semigroup. To us this seems more natural than the classical approach, where vector spaces have been used. We show in section 4 that most results about estimation in exponential families carry over to the more general notion, in a more streamlined fashion, since usual problems with existence of maximum likelihood estimates are avoided. The first section is a compact description of the basic semigroup theory needed.
KeywordsExtreme Point Exponential Family Semi Group Local Limit Theorem Repetitive Structure
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