## The Annals of Statistics

- Ann. Statist.
- Volume 24, Number 6 (1996), 2297-2778

### Heuristics of instability and stabilization in model selection

#### Abstract

In model selection, usually a "best" predictor is chosen from a collection ${\hat{\mu}(\cdot, s)}$ of predictors where $\hat{\mu}(\cdot, s)$ is the minimum least-squares predictor in a collection $\mathsf{U}_s$ of
predictors. Here *s* is a complexity parameter; that is, the smaller *s*, the lower dimensional/smoother the models in $\mathsf{U}_s$.

If $\mathsf{L}$ is the data used to derive the sequence ${\hat{\mu}(\cdot, s)}$, the procedure is called unstable if a small change in $\mathsf{L}$ can cause large changes in ${\hat{\mu}(\cdot, s)}$. With a crystal ball, one could pick the predictor in ${\hat{\mu}(\cdot, s)}$ having minimum prediction error. Without prescience, one uses test sets, cross-validation and so forth. The difference in prediction error between the crystal ball selection and the statistician's choice we call predictive loss. For an unstable procedure the predictive loss is large. This is shown by some analytics in a simple case and by simulation results in a more complex comparison of four different linear regression methods. Unstable procedures can be stabilized by perturbing the data, getting a new predictor sequence ${\hat{\mu'}(\cdot, s)}$ and then averaging over many such predictor sequences.

#### Article information

**Source**

Ann. Statist. Volume 24, Number 6 (1996), 2350-2383.

**Dates**

First available: 16 September 2002

**Permanent link to this document**

http://projecteuclid.org/euclid.aos/1032181158

**Mathematical Reviews number (MathSciNet)**

MR1425957

**Digital Object Identifier**

doi:10.1214/aos/1032181158

**Zentralblatt MATH identifier**

0867.62055

**Subjects**

Primary: 62H99: None of the above, but in this section

**Keywords**

Regression subset selection predictive loss cross-validation prediction error

#### Citation

Breiman, Leo. Heuristics of instability and stabilization in model selection. The Annals of Statistics 24 (1996), no. 6, 2350--2383. doi:10.1214/aos/1032181158. http://projecteuclid.org/euclid.aos/1032181158.