In some semiparametric survival models with time dependent coefficients, a closed-form solution for coefficients estimates does not exist. Therefore, they have to be estimated by using approximate numerical methods. Due to the complicated forms of such estimators, it is too hard to extract their sampling distributions. In such cases, one usually uses the asymptotic theory to evaluate properties of the estimators. In this paper, first the model is introduced and a method is proposed, by using the Taylor expansion and kernel methods, to estimate the model. Then, the consistency and asymptotic normality of the estimators are established. The performance of the model and estimating procedure are evaluated by a heavy simulation study as well. Finally, the proposed model is applied on a real data set on heart disease patients in one of the Mashhad hospitals.

In this article, we propose two non-parametric estimators for the past entropy based on length-biased data, and the strong consistency of the proposed estimators is proved. In addition, some simulations are conducted to evaluate the performance of the proposed estimators. Based on the results, we show that they have better performance in a different region of the probability distribution for length-biased random variables.