This is the log-likelihood function that is passed to optim for likelihood maximization.
pgam.likelihood(par, y, x, offset, fperiod, env = parent.frame())vector of parameters to be optimized
observed time series which is the response variable of the model
observed explanatory variables for parametric fit
model offset. Just like in GLM
vector of seasonal factors to be passed to pgam.par2psi
the caller environment for log-likelihood value to be stored
List containing log-likelihood value, optimum linear predictor and the gamma parameters vectors.
Log-likelihood function of hyperparameters \(\omega\) and \(\beta\) is given by
$$\log L\left(\omega,\beta\right)=\sum_{t=\tau+1}^{n}{\log \Gamma\left(a_{t|t-1}+y_{t}\right)-\log y_{t}!-\cr
\log \Gamma\left(a_{t|t-1}\right)+a_{t|t-1}\log b_{t|t-1}-\left(a_{t|t-1}+y_{t}\right)\log \left(1+b_{t|t-1}\right)}$$
where \(a_{t|t-1}\) and \(b_{t|t-1}\) are estimated as it is shown in pgam.filter.
Harvey, A. C., Fernandes, C. (1989) Time series models for count data or qualitative observations. Journal of Business and Economic Statistics, 7(4):407--417
Harvey, A. C. (1990) Forecasting, structural time series models and the Kalman Filter. Cambridge, New York
Junger, W. L. (2004) Semiparametric Poisson-Gamma models: a roughness penalty approach. MSc Dissertation. Rio de Janeiro, PUC-Rio, Department of Electrical Engineering.