The priori and posteriori conditional distributions of the level is gamma and their parameters are estimated through this recursive filter. See Details for a thorough description.
pgam.filter(w, y, eta)A list containing the time varying parmeters of the priori and posteriori conditional distribution is returned.
running estimate of discount factor \(\omega\) of a Poisson-Gamma model
\(n\) length vector of the time series observations
full linear or semiparametric predictor. Linear predictor is a trivial case of semiparameric model
Washington Leite Junger wjunger@ims.uerj.br and Antonio Ponce de Leon ponce@ims.uerj.br
Consider \(Y_{t-1}\) a vector of observed values of a Poisson process untill the instant \(t-1\). Conditional on that, \(\mu_{t}\) has gamma distribution with parameters given by $$a_{t|t-1}=\omega a_{t-1}$$ $$b_{t|t-1}=\omega b_{t-1}\exp\left(-\eta_{t}\right)$$ Once \(y_{t}\) is known, the posteriori distribution of \(\mu_{t}|Y_{t}\) is also gamma with parameters given by $$a_{t}=\omega a_{t-1}+y_{t}$$ $$b_{t}=\omega b_{t-1}+\exp\left(\eta_{t}\right)$$ with \(t=\tau,\ldots,n\), where \(\tau\) is the index of the first non-zero observation of \(y\).
Diffuse initialization of the filter is applied by setting \(a_{0}=0\) and \(b_{0}=0\). A proper distribution of \(\mu_{t}\) is obtained at \(t=\tau\), where \(\tau\) is the fisrt non-zero observation of the time series.
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.
pgam, pgam.likelihood, pgam.fit, predict.pgam