Transcription of 18 The Exponential Family and Statistical Applications
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18 The Exponential Family and Statistical Applications
18 The Exponential Family and Statistical ApplicationsThe Exponential Family is a practically convenient and widely used uni ed Family of distributionson nite dimensional Euclidean spaces parametrized by a nite dimensional parameter to the case of the real line, the Exponential Family contains as special cases most of thestandard discrete and continuous distributions that we use for practical modelling, such as the nor-mal, Poisson, Binomial, Exponential , Gamma, multivariate normal, etc. The reason for the specialstatus of the Exponential Family is that a number of important and useful calculations in statisticscan be done all at one stroke within the framework of the Exponential Family .
Clearly, something very interesting is going on. We started with a basic density in a speciflc form, namely, f(xj¾)=e· (¾)T x ¡ˆ h(x), and then we found that the joint density and the density of the relevant one dimensional statistic P n i=1 X 2 in that joint density, are once again densities of exactly that same general form.
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