Predictive densities and prediction limits based on predictive likelihoods

The notion of predictive likelihood stems from the fact that in the prediction problem there are two unknown quantities to deal with: the future observation and the model parameter. Since, according to the likelihood principle, all the evidence is contained in the joint likelihood function, a predictive likelihood for the future observation is obtained by eliminating the nuisance quantity, namely the unknown model parameter. This paper focuses on the profile predictive likelihood and on some modified versions obtained by mimicking the solutions proposed to improve the profile (parametric) likelihood. These predictive likelihoods are evaluated by studying how well they generate prediction intervals. In particular, we find that, at least in some specific applications, these solution usually improve on those ones based on the plug-in procedure. However, the associated predictive densities and prediction limits do not correspond to the optimal frequentist solutions already described in the literature.