  We extend the model used in Gardiner et al. (2002) and Polverejan et al.
(2003) through deriving an explicit expression for the joint probability
density function of hospital charge and length of stay (LOS) under a general
class of conditions. Using this joint density function, we can apply the full
maximum likelihood method (FML) to estimate the effect of covariates on charge
and LOS. By FML, the endogeneity issues arisen from the dependence between
charge and LOS can be efficiently resolved. As an illustrative example, we
apply our method to real charge and LOS data sampled from New York State
Statewide Planning and Research Cooperative System 2013 (SPARCS 2013). We
compare our fitting result with the fitting to the marginal LOS data generated
by the widely used Phase-Type model, and conclude that our method is more
efficient in fitting.
