III-posed problems with counts, the composite link model and penalized likelihood

Certain data sets with distributions or counts can be interpreted as indirect observations of latent distributions or (time) series of counts., The structure of such data matches elegantly with the composite link model (CLM). The parameters can be estimated with iteratively re-weighted linear regression. Unfortunately, the estimating equations generally are singular or severely ill-conditioned. An effective solution is to impose smoothness on the solution, by penalizing the likelihood with a roughness measure. The optimal smoothing parameter is found efficiently by minimizing Akaike's Information Criterion (AIC). Several applications are presented.