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A generalized linear point process is specified in terms of an intensity thatdepends upon a linear predictor process through a fixed non-linear function. Wepresent a framework where the linear predictor is parametrized by a Banachspace and give results on Gateaux differentiability of the log-likelihood. Ofparticular interest is when the intensity is expressed in terms of a linearfilter parametrized by a Sobolev space. Using that the Sobolev spaces arereproducing kernel Hilbert spaces we derive results on the representation ofthe penalized maximum likelihood estimator in a special case and the gradientof the negative log-likelihood in general. The latter is used to develop adescent algorithm in the Sobolev space. We conclude the paper by extensions tomultivariate and additive model specifications. The methods are implemented inthe R-package ppstat.

Penalized maximum likelihood estimation for generalized linear point processes