Semi-supervised Inference: General Theory and Estimation of Means
Anru Zhang, Lawrence D. Brown, and T. Tony Cai
We propose a general semi-supervised inference framework focused on the estimation of the population mean. We consider both the ideal semi-supervised setting where infinitely many unlabeled samples are available, as well as the ordinary semi-supervised setting in which only a finite number of unlabeled samples is available. As usual in semi-supervised settings, there exists an unlabeled sample of covariate vectors and a labeled sample consisting of covariate vectors along with real-valued responses (“labels"). Otherwise the formulation is 'assumption-lean" in that no major conditions are imposed on the statistical or functional form of the data. Estimators are proposed along with corresponding confidence intervals for the population mean. Theoretical analysis on both the asymptotic behavior and l2-risk for the proposed procedures are given. Surprisingly, the proposed estimators, based on a simple form of the least squares method, outperform the ordinary sample mean. The method is further extended to a nonparametric setting, in which the oracle rate can be achieved asymptotically. The proposed estimators are further illustrated by simulation studies and a real data example involving estimation of the homeless population.