  The asymptotically efficient online learning problem is investigated for
stochastic censored regression models, which arise from various fields of
learning and statistics but up to now still lacks comprehensive theoretical
studies on the efficiency of the learning algorithms. For this, we propose a
two-step online algorithm, where the first step focuses on achieving algorithm
convergence, and the second step is dedicated to improving the estimation
performance. Under a general excitation condition on the data, we show that our
algorithm is strongly consistent and asymptotically normal by employing the
stochastic Lyapunov function method and limit theories for martingales.
Moreover, we show that the covariances of the estimates can achieve the
Cramer-Rao (C-R) bound asymptotically, indicating that the performance of the
proposed algorithm is the best possible that one can expect in general. Unlike
most of the existing works, our results are obtained without resorting to the
traditionally used but stringent conditions such as independent and identically
distributed (i.i.d) assumption on the data, and thus our results do not exclude
applications to stochastic dynamical systems with feedback. A numerical example
is also provided to illustrate the superiority of the proposed online algorithm
over the existing related ones in the literature.
