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This is mainly a reference request. There must be some generalizations of the concept of Fisher information for discrete (say, integer-valued) parameters, and of related results such as the Cramer-Rao bound (or information inequality). I have just never seen them.

Are there any good references, to the concept(s) itself, or to interesting applications?

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  • $\begingroup$ This paper on cardinality estimation uses the Cramér-Rao inequality in a context where the parameter is integer-valued (page 9, proposition 2.9). However, they seem to do some kind of transformation to transform their estimator $\hat \xi$ into $\xi^*$ to make it differentiable. I'm not sure I understand what they're doing exactly, and an answer to your question would probably help… $\endgroup$ – Ted Jun 16 '17 at 16:01

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