## The Annals of Statistics

- Ann. Statist.
- Volume 47, Number 5 (2019), 2538-2566.

### Semi-supervised inference: General theory and estimation of means

Anru Zhang, Lawrence D. Brown, and T. Tony Cai

#### 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.

#### Article information

**Source**

Ann. Statist., Volume 47, Number 5 (2019), 2538-2566.

**Dates**

Received: August 2017

Revised: August 2018

First available in Project Euclid: 3 August 2019

**Permanent link to this document**

https://projecteuclid.org/euclid.aos/1564797856

**Digital Object Identifier**

doi:10.1214/18-AOS1756

**Mathematical Reviews number (MathSciNet)**

MR3988765

**Zentralblatt MATH identifier**

07114921

**Subjects**

Primary: 62F10: Point estimation 62J05: Linear regression

Secondary: 62F12: Asymptotic properties of estimators 62G08: Nonparametric regression

**Keywords**

Confidence interval efficiency estimation of mean limiting distribution semi-supervised inference

#### Citation

Zhang, Anru; Brown, Lawrence D.; Cai, T. Tony. Semi-supervised inference: General theory and estimation of means. Ann. Statist. 47 (2019), no. 5, 2538--2566. doi:10.1214/18-AOS1756. https://projecteuclid.org/euclid.aos/1564797856

#### Supplemental materials

- Supplement to “Semi-supervised inference: General theory and estimation of means”. The supplement contains additional proofs for the main results of the paper.Digital Object Identifier: doi:10.1214/18-AOS1756SUPPSupplemental files are immediately available to subscribers. Non-subscribers gain access to supplemental files with the purchase of the article.