In this paper we propose a semiparametric model to fit medical cost data with a proportion of zero cost values. In our model, the unknown cumulative cost is defined to be a function of the failure time to account for the correlation between the cost and the failure time. The nonparametric nature of the cost function allows full flexibility in matching the reality. Local likelihood estimation is proposed to estimate the unknown accumulative cost functions and the related parameters, and their asymptotic properties are investigated as well. Simulation studies are performed to illustrate our models and proposed methods.