pgam.filter: Estimation of the conditional distributions parameters of the level

Description

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.

Usage

pgam.filter(w, y, eta)

Arguments

running estimate of discount factor $\omega$ of a Poisson-Gamma model

$n$ length vector of the time series observations

eta

full linear or semiparametric predictor. Linear predictor is a trivial case of semiparameric model

Value

A list containing the time varying parmeters of the priori and posteriori conditional distribution is returned.

Details

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.

References

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.