Counterfactual model is put forward to discuss the causal inference in the directed acyclic graph and its corresponding identifiability is thus studied with the ancillary information based on conditional independence. It is shown that the assumption of ignorability can be expanded to the assumption of replaceability, under which the causal effects are identifiable.
The nonlinear wavelet estimator of regression function with random design is constructed. The optimal uniform convergence rate of the estimator in a ball of Besov spaceB 3 p,q is proved under quite general assumpations. The adaptive nonlinear wavelet estimator with near-optimal convergence rate in a wide range of smoothness function classes is also constructed. The properties of the nonlinear wavelet estimator given for random design regression and only with bounded third order moment of the error can be compared with those of nonlinear wavelet estimator given in literature for equal-spaced fixed design regression with i.i.d. Gauss error.
This paper considers local median estimation in fixed design regression problems. The proposed method is employed to estimate the median function and the variance function of a heteroscedastic regression model. Strong convergence rates of the proposed estimators are obtained. Simulation results are given to show the performance of the proposed methods.
