Isotonic Regression And Reducing Variance Estimation For Nonparametric Models

The semi-parametric models are very flexible because they contain both the parametric component and the non-parametric component. So they are widely used in many fields. But response variables are monotonously related to covariates in practice. Ordinary non-parametric approaches could not guarantee the monotonicity of estimator. Non-parametric models with the monotone variables have been received a lot of concern for a long time. We usually assumes that the relationship is smoothing between variables which could not guarantee the monotonicity of estimator. To overcome this problem, the monotone regression estimation has been used. To avoid the question which caused by the general non-parametric estimation, the monotone regression estimation is available. This method makes the selecting of the smooth parameter automatically and totally dependent on the observed data. In this paper, we study the case that some linear covariates are not observed, ancillary variables are available to rectify the models. We consider the semi-parametric monotone regression models. Then we obtain the asymptotic distributions of the estimator under a certain conditions. We show the finite-sample properties of the estimators by the simulation.varying-coefficient models are stable semi-parametric models. Varying-coefficient was widely used in many fields such as biology,medical and economics. There is a large literature on the estimation of varying-coefficient models. Some of them focus on the improvements of the traditional approaches. But very few variance reduction methods were in print so far. Most of this was achieved by limiting the distribution of the measurement error. Local linear regression is frequently used in reality among others. One of the best approaches in the case of estimate the varying-coefficient models is local linear estimator. For the applications of the local linear estimation to the varying-coefficient models, the reducing variance estimator for varying-coefficient models is proposed which based on the local linear estimation. In purpose of improving the result of the local linear estimator, we study the reducing variance estimation in this paper. The resulting estimator has smaller asymptotic conditional variance than the local linear estimator but retains the same asymptotic conditional bias, table as the local linear estimator. Simulation shows the optimizing effect of the reducing variance estimation by comparing the result of the two approaches.