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We saw that in order to prove a distribution is a part of a exponential family we should be able to write it in this form: f(y;θ)=exp(a(y)b(θ)+c(θ)+d(y)

And to prove that:

  1. All statistics T are linearly independent
  2. The interior of the parameter set is non empty.

For example I am working on the Beta family and I proved everything except the 2). I can not understand how one can prove that the interior of parameter set is non empty?

Anna
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1 Answers1

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There is some confusion about the notion:

(i) For a density of the form $$f(x;\theta) = \exp\{a(\theta)^\text{T}b(x)+c(\theta)+d(x)\}$$ to be an exponential family the only constraint is that the support of the density does not depend on $\theta$.

(ii) The representation of this density is minimal if

  1. the components of $b(\cdot)$ are linearly independent
  2. the components of $a(\cdot)$ are linearly independent

(iii) For a minimal representation, the exponential family is full-rank if the set of the $a(\theta)$'s (when $\theta$ varies in the original parameter set) contains an open set. Otherwise it is curved.

Xi'an
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