摘要：

Over the past decade or so, model averaging has attracted more and more attention and is regarded as a much better tool to solve model uncertainty than model selection, which helps reduce the risk of choosing a ``poor" model. And model averaging can effectively improve the predictive ability of the model. Most of the current related work mainly focuses on the model averaging of linear models, semiparametric models and nonparametric models based on conditional mean regression. In this paper, we develop the model averaging theories of several nonparametric models under smooth or robust loss functions. The smooth loss functions include the squared error loss, and the robust loss functions include the quantile loss and the expectile loss.

First, we consider a regression model with the mean being a varying-coefficient model and the variance being multiplicative heteroscedasticity and introduce a model averaging approach that uses the B-spline smoothing method to estimate unknown coefficient functions and obtains the estimators of unknown parameters in both the mean and variance functions of the model by the pseudo maximum likelihood method. We propose a weight criterion for this model to choose the optimal weight. We complement the multiplicative heteroscedastic structure to the varying-coefficient model with the random error being heteroscedastic and improve the information utilization of covariates in the model. This improves the prediction accuracy of the model averaging estimator of the mean part. The resulting model averaging estimator is proved to have asymptotic optimality in terms of minimizing the squared error under some regular conditions and model misspecification . 

Second, we propose a jackknife model averaging(JMA) procedure that chooses the weights by minimizing a leave-one-out cross-validation criterion function for mixed-data kernel-weighted spline quantile regressions that contain both continuous and categorical regressors when all candidate models are potentially misspecified. Compared with the conditional mean regression, the quantile regression serves as a robust alternative and shows a lot more information about the conditional distribution of a response variable. We estimate each candidate model by the tensor-product polynomial splines weighted categorical kernel functions. We demonstrate the JMA estimator is asymptotically optimal in terms of minimizing the out-of-sample quantile prediction error. 

Third, we propose a jackknife model averaging(JMA) method for the quantile single-index coefficient model, which is widely used in statistics. We estimate parameters in each candidate model by the local polynomial smoothing method. Under model misspecification, the model averaging estimator is proved to be asymptotically optimal in terms of minimizing out-of-sample quantile prediction error. 

Finally, we introduce a model averaging estimator for additive expectile prediction. Expectile prediction is widely used for modeling data with heterogeneous conditional distribution. We approximate each unknown function  by the B-spline smoothing method. The resulting model averaging estimator is shown to have asymptotic optimality in terms of minimizing out-of-sample expectile prediction error under some regular conditions and model misspecification.