Penalized maximum likelihood estimation for generalized linear point processes

A framework of generalized linear point process models (glppm) much akin to glm for regression is developed where the intensity depends upon a linear predictor process through a known function. In the general framework the parameter space is a Banach space. Of particular interest is when the intensity depends on the history of the point process itself and possibly additional processes through a linear filter, and where the filter is parametrized by functions in a Sobolev space. We show two main results. First we show that for a special class of models the penalized maximum likelihood estimate is in a finite dimensional subspace of the parameter space -- if it exists. In practice we can find the estimate using a finite dimensional glppm framework. Second, for the general class of models we develop a descent algorithm in the Sobolev space. We conclude the paper by a discussion of additive model specifications.