# pgam.likelihood: Likelihood function to be maximized In pgam: Poisson-Gamma Additive Models

## Description

This is the log-likelihood function that is passed to optim for likelihood maximization.

## Usage

 1 pgam.likelihood(par, y, x, offset, fperiod, env = parent.frame()) 

## Arguments

 par vector of parameters to be optimized y observed time series which is the response variable of the model x observed explanatory variables for parametric fit offset model offset. Just like in GLM fperiod vector of seasonal factors to be passed to pgam.par2psi env the caller environment for log-likelihood value to be stored

## Details

Log-likelihood function of hyperparameters ω and β is given by

\log L≤ft(ω,β\right)=∑_{t=τ+1}^{n}{\log Γ≤ft(a_{t|t-1}+y_{t}\right)-\log y_{t}!-\cr \log Γ≤ft(a_{t|t-1}\right)+a_{t|t-1}\log b_{t|t-1}-≤ft(a_{t|t-1}+y_{t}\right)\log ≤ft(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.

## Value

List containing log-likelihood value, optimum linear predictor and the gamma parameters vectors.

## Note

This function is not intended to be called directly.

## Author(s)

Washington Leite Junger wjunger@ims.uerj.br and Antonio Ponce de Leon ponce@ims.uerj.br

## 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.

pgam, pgam.filter, pgam.fit