## Electronic Journal of Statistics

### Data enriched linear regression

#### Abstract

We present a linear regression method for predictions on a small data set making use of a second possibly biased data set that may be much larger. Our method fits linear regressions to the two data sets while penalizing the difference between predictions made by those two models. The resulting algorithm is a shrinkage method similar to those used in small area estimation. We find a Stein-type result for Gaussian responses: when the model has $5$ or more coefficients and $10$ or more error degrees of freedom, it becomes inadmissible to use only the small data set, no matter how large the bias is. We also present both plug-in and AICc-based methods to tune our penalty parameter. Most of our results use an $L_{2}$ penalty, but we obtain formulas for $L_{1}$ penalized estimates when the model is specialized to the location setting. Ordinary Stein shrinkage provides an inadmissibility result for only $3$ or more coefficients, but we find that our shrinkage method typically produces much lower squared errors in as few as $5$ or $10$ dimensions when the bias is small and essentially equivalent squared errors when the bias is large.

#### Article information

Source
Electron. J. Statist., Volume 9, Number 1 (2015), 1078-1112.

Dates
First available in Project Euclid: 27 May 2015

https://projecteuclid.org/euclid.ejs/1432732305

Digital Object Identifier
doi:10.1214/15-EJS1027

Mathematical Reviews number (MathSciNet)
MR3352068

Zentralblatt MATH identifier
1328.62457

#### Citation

Chen, Aiyou; Owen, Art B.; Shi, Minghui. Data enriched linear regression. Electron. J. Statist. 9 (2015), no. 1, 1078--1112. doi:10.1214/15-EJS1027. https://projecteuclid.org/euclid.ejs/1432732305

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