Linear regression with interval censored data (Q1807121)

Consider the linear model \[ Y_i=\beta^{\text{tr}}Z_i+\varepsilon_i,\quad 1\leq i\leq n, \] where \(Z_i\) are known covariate vectors and \(\beta\) is an unknown parameter vector. We are interested in estimating \(\beta\) from the current status data such that \[ X_i=(\delta_i,T_i,Z_i),\quad 1\leq i\leq n, \] are observed instead of the usual \((Y_i,Z_i)\), where \(\delta_i=I_{\{Y_i\leq T_i\}}\) for some censoring times \(T_i\). This observation scheme is also called the binary choice model and Case 1 interval censoring. We assume that the errors \(\{\varepsilon_i\}\) are independent of \(\{Z_i,T_i\}\) and are i.i.d. variables with a common distribution \(F(t)\). \(M\)-estimators for the regression coefficients are derived. Asymptotic consistency and normality of the \(M\)-estimators are obtained via an exponential inequality for \(U\)-statistics. Asymptotically efficient estimators are provided under mild conditions.