## The Annals of Statistics

- Ann. Statist.
- Volume 47, Number 2 (2019), 884-908.

### Partial least squares prediction in high-dimensional regression

R. Dennis Cook and Liliana Forzani

#### Abstract

We study the asymptotic behavior of predictions from partial least squares (PLS) regression as the sample size and number of predictors diverge in various alignments. We show that there is a range of regression scenarios where PLS predictions have the usual root-$n$ convergence rate, even when the sample size is substantially smaller than the number of predictors, and an even wider range where the rate is slower but may still produce practically useful results. We show also that PLS predictions achieve their best asymptotic behavior in abundant regressions where many predictors contribute information about the response. Their asymptotic behavior tends to be undesirable in sparse regressions where few predictors contribute information about the response.

#### Article information

**Source**

Ann. Statist., Volume 47, Number 2 (2019), 884-908.

**Dates**

Received: January 2017

Revised: December 2017

First available in Project Euclid: 11 January 2019

**Permanent link to this document**

https://projecteuclid.org/euclid.aos/1547197242

**Digital Object Identifier**

doi:10.1214/18-AOS1681

**Mathematical Reviews number (MathSciNet)**

MR3909954

**Zentralblatt MATH identifier**

07033155

**Subjects**

Primary: 62J05: Linear regression

Secondary: 62F12: Asymptotic properties of estimators

**Keywords**

Abundant regressions dimension reduction sparse regressions

#### Citation

Cook, R. Dennis; Forzani, Liliana. Partial least squares prediction in high-dimensional regression. Ann. Statist. 47 (2019), no. 2, 884--908. doi:10.1214/18-AOS1681. https://projecteuclid.org/euclid.aos/1547197242

#### Supplemental materials

- Supplement to “Partial least squares prediction in high-dimensional regression”. Proofs for all lemmas, propositions and theorems are provided in the online supplement to this article.Digital Object Identifier: doi:10.1214/18-AOS1681SUPPSupplemental files are immediately available to subscribers. Non-subscribers gain access to supplemental files with the purchase of the article.