The LAN property is proved in the statistical model based on discrete-time observations of a solution to a Lévy driven SDE. The proof is based on a general sufficient condition for a statistical model based on discrete observations of a Markov process to possess the LAN property, and involves substantially the Malliavin calculus-based integral representations for derivatives of log-likelihood of the model.
Cox proportional hazards model is considered. In Kukush et al. (2011), Journal of Statistical Research, Vol. 45, No. 2, 77–94 simultaneous estimators $\lambda _{n}(\cdot )$ and $\beta _{n}$ of baseline hazard rate $\lambda (\cdot )$ and regression parameter β are studied. The estimators maximize the objective function that corrects the log-likelihood function for measurement errors and censoring. Parameter sets for $\lambda (\cdot )$ and β are convex compact sets in $C[0,\tau ]$ and ${\mathbb{R}}^{k}$, respectively. In present paper the asymptotic normality for $\beta _{n}$ and linear functionals of $\lambda _{n}(\cdot )$ is shown. The results are valid as well for a model without measurement errors. A way to compute the estimators is discussed based on the fact that $\lambda _{n}(\cdot )$ is a linear spline.
In this paper we define the consistent criteria of hypotheses such as the probability of any kind of errors is zero for given criteria. We prove necessary and sufficient conditions for the existence of such criteria.