Some New Insights about the Accelerated Failure Time Model.

The semiparametric linear model is an important alternative to the Cox proportional hazards model for censored survival outcomes. In this dissertation, we provide some new insights for the parameter estimators and their asymptotic properties in the semiparametric linear model with censored data.In Chapter 2, we have shown that in a linear regression model, where the outcome variable is subject to right censoring and the error distribution is unspecified, the intercept parameter is consistent and asymptotically normal when the support of some covariates with nonzero coefficients is unbounded. This holds even with finite follow-up times. In a practical setting, it makes the prediction of survival time possible under a linear regression model when the covariate range is wide. Without the commonly assumed regularity condition of bounded covariates, we have also shown that the slope estimators obtained by solving the Gehan-weighted rank based estimating equation are consistent and asymptotically normal, which provides a crucial condition for the asymptotic properties of the intercept estimator.In Chapter 3, we have proposed a new approach to estimate the slope parameters in the semiparametric linear model by directly maximizing the log likelihood function in a sieve space, in which the log hazard function of the error term is approximated by B-splines. The maximization can be achieved through the gradient-based search algorithm over the sieve space. The resulting slope estimators have been shown to be consistent and asymptotically normal. In addition, the limiting covariance matrix of the proposed estimators reaches the semiparametric efficiency bound and can be estimated nicely by inverting either the information matrix based on the efficient score function of the regression parameters or the observed information matrix for all parameters including the "nuisance" parameters for estimating the log hazard function.