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October 2019 Semi-supervised inference: General theory and estimation of means
Anru Zhang, Lawrence D. Brown, T. Tony Cai
Ann. Statist. 47(5): 2538-2566 (October 2019). DOI: 10.1214/18-AOS1756

Abstract

We propose a general semi-supervised inference framework focused on the estimation of the population mean. 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. 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.

Estimators are proposed along with corresponding confidence intervals for the population mean. Theoretical analysis on both the asymptotic distribution and $\ell_{2}$-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 simple, transparent form of the estimator lends confidence to the perception that its asymptotic improvement over the ordinary sample mean also nearly holds even for moderate size samples. 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.

Citation

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Anru Zhang. Lawrence D. Brown. T. Tony Cai. "Semi-supervised inference: General theory and estimation of means." Ann. Statist. 47 (5) 2538 - 2566, October 2019. https://doi.org/10.1214/18-AOS1756

Information

Received: 1 August 2017; Revised: 1 August 2018; Published: October 2019
First available in Project Euclid: 3 August 2019

zbMATH: 07114921
MathSciNet: MR3988765
Digital Object Identifier: 10.1214/18-AOS1756

Subjects:
Primary: 62F10, 62J05
Secondary: 62F12, 62G08

Rights: Copyright © 2019 Institute of Mathematical Statistics

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Vol.47 • No. 5 • October 2019
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