Semi-supervised learning using copula-based regression and model averaging

The available data in semi-supervised learning usually consists of relatively small sized labeled data and much larger sized unlabeled data. How to effectively exploit unlabeled data is the key issue. In this paper, we write the regression function in the form of a copula and marginal distributions, and the unlabeled data can be exploited to improve the estimation of the marginal distributions. The predictions based on different copulas are weighted, where the weights are obtained by minimizing an asymptotic unbiased estimator of the prediction risk. Error-ambiguity decomposition of the prediction risk is performed such that unlabeled data can be exploited to improve the prediction risk estimation. We demonstrate the asymptotic normality of copula parameters and regression function estimators of the candidate models under the semi-supervised framework, as well as the asymptotic optimality and weight consistency of the model averaging estimator. Our model averaging estimator achieves faster convergence rates of asymptotic optimality and weight consistency than the supervised counterpart. Extensive simulation experiments and the California housing dataset demonstrate the effectiveness of the proposed method.