model is the maximum partial likelihood (MPL) procedure, which applies well whether or not the covariates are time-dependent. However, when time-dependent covariates being involved, the consecutive observation of the covariates is required in the sense that every subject must have its covariates observed at all the failure time prior to its own failure. In the other words, let $T_i<T_j$ be the observed failure time for two different subjects $i$ and $j$, then for the dying later subject $j$, its covariate $Z_j$ must have values observed at both of $T_i$ and $T_j$. Otherwise, the MPL procedure won’t work. Although some approximation methods were proposed to relax consecutive observation requirement as discussed in [@andersen1992repeated], the MPL procedure can’t be applied effectively to the medical cost datasets, like the New York State’s Statewide Planning and Research Cooperative System (SPARCS), or the National Inpatient Sample (NIS), where it is important to find the hazard rate of being discharged given the accumulative cost while only the total cost for each inpatient observed at the discharge time is available.

In this paper, we shall propose a novel estimation procedure for the model , which can generate consistent estimates for parameters and the baseline hazard in the 