We can formulate density estimation as a count regression; For “nice” distributions this will be the same as where the target is estimating the correct Poisson intensity for any given small region of the state space. (e.g. Gu93 or EiMa96)

In the nonparametric case this can be your basic functional data problem, although I’m not quite sure how the integrates-to-unity thing is handled. If it’s not, probably it’s linear point process regression/ Possibly we ignore the requirement to integrate to unity unless we need it (as in mixture models)

TBD: which precisely are the nice distributions?
I suspect it will be necessary that the samples viewed as a spatial point process
are *simple* which means, e.g. no atoms.
Other criteria?
See Scho05 for this niceness question in a regression setting.

## Questions

- What does general functional regression get me here? (see Ch21 of Ramsay and Silverman)
- Can I use point process estimation theory to improve density estimation? After all, normal point-process estimation claims to be an un-normalised vesion of density estimation. Lies11 draws some parallels there, esp. with mixture models.

## Refs

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