A non-stationary Cox model (Q2711679)

Let \(S\) be the starting time, \(T\) the failure of some phenomenon, \(Z(u)\) a vector of covariates at time \(u\), and \(X=T-S>0\) the duration of the phenomenon. The authors use the following model of the conditional (by \(S\) and \(Z\)) hazard rate of failure: NEWLINE\[NEWLINE\lambda_{X|S,Z}(x;s,Z(s+x))\lambda_{X|S}(x,s)\exp(\beta^TZ(s+x)),NEWLINE\]NEWLINE where \(\beta\) is a vector of unknown parameters and the baseline hazard function \(\lambda_{X|S}\) is considered as unknown. The difference of this model from the Cox one is that \(\lambda_{X|S}\) depends on the start time \(S\), i.e. the model is non-stationary. The problem is to estimate \(\lambda_{X|S}\) and \(\beta\) by censored data. NEWLINENEWLINENEWLINEThe authors use kernel smoothing and a local likelihood approach to derive the estimators. It is demonstrated that they are consistent and asymptotically normal. The estimator for \(\beta\) is efficient. Results of Monte-Carlo studies are presented.