In many survival studies, observation on the occurrence of the event of interest (called a failure) may be prevented by the previous occurrence of another event (called a censoring event). We assume the random censorship model in which the censoring time is independent of the survival time. Considering least squares local linear and local quadratic approximations to the Nelson-Aalen estimator of the cumulative hazard function, the estimators of the linear coefficients are called the local linear and local quadratic estimators of the hazard rate, respectively. The asymptotic normal distributions of the local linear and local quadratic estimators are established. Performance of the proposed estimators is illustrated by simulations. We compare the proposed estimators with the kernel estimator and the Jiang and Doksum (2003) estimator by means of the estimated MISE, and find that the local quadratic estimator behaves favorably.