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Free exponential families as kernel families

Bryc W.

Demonstratio Mathematica

2009

Vol. 42, nr 3

657-672

Free exponential families have been previously introduced as a special case of the q-exponential family. We show that free exponential families arise also from the approach analogous to the definition of exponential families by using the Cauchy-Stieltjes kernel 1/(1 - Qx) instead of the exponential kernel exp(Qx). We use this approach to re-derive some known results and to study further similarities with exponential families and reproductive exponential models.