Question

The representation of random variable in a exponential family given by f_{y}(y;\theta ,\phi )= exp(A[y\theta -\gamma (\theta )]/\phi+\tau (y,\phi /A) ) is not unique.Show that if the density Y can be represented in the from f_{y}(y;\theta ,\phi )= exp(A[y\theta -\gamma (\theta )]/\phi+\tau (y,\phi /A) ) with natural parameter \theta then it can also be expressed in a similar way with natural parameter \theta ^{\ast }=\theta/2 by using \gamma ^{*}(\theta )=\gamma (2\theta ^{*})/2 and A ^{*}=2A

Answer 1

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