### Abstract

Asymmetric least squares regression is an important method that has wide applications in statistics, econometrics and finance. The existing work on asymmetric least squares only considers the traditional low dimension and large sample setting. In this paper, we systematically study the Sparse Asymmetric LEast Squares (SALES) regression under high dimensions where the penalty functions include the Lasso and nonconvex penalties. We develop a unified efficient algorithm for fitting SALES and establish its theoretical properties. As an important application, SALES is used to detect heteroscedasticity in high-dimensional data. Another method for detecting heteroscedasticity is the sparse quantile regression. However, both SALES and the sparse quantile regression may fail to tell which variables are important for the conditional mean and which variables are important for the conditional scale/variance, especially when there are variables that are important for both the mean and the scale. To that end, we further propose a COupled Sparse Asymmetric LEast Squares (COSALES) regression which can be efficiently solved by an algorithm similar to that for solving SALES. We establish theoretical properties of COSALES. In particular, COSALES using the SCAD penalty or MCP is shown to consistently identify the two important subsets for the mean and scale simultaneously, even when the two subsets overlap. We demonstrate the empirical performance of SALES and COSALES by simulated and real data.

Original language | English (US) |
---|---|

Pages (from-to) | 2661-2694 |

Number of pages | 34 |

Journal | Annals of Statistics |

Volume | 44 |

Issue number | 6 |

DOIs | |

State | Published - Dec 1 2016 |

### Fingerprint

### Keywords

- Asymmetric least squares
- COSALES
- High dimensions
- SALES

### Cite this

**High-dimensional generalizations of asymmetric least squares regression and their applications.** / Gu, Yuwen; Zou, Hui.

Research output: Contribution to journal › Article

*Annals of Statistics*, vol. 44, no. 6, pp. 2661-2694. https://doi.org/10.1214/15-AOS1431

}

TY - JOUR

T1 - High-dimensional generalizations of asymmetric least squares regression and their applications

AU - Gu, Yuwen

AU - Zou, Hui

PY - 2016/12/1

Y1 - 2016/12/1

N2 - Asymmetric least squares regression is an important method that has wide applications in statistics, econometrics and finance. The existing work on asymmetric least squares only considers the traditional low dimension and large sample setting. In this paper, we systematically study the Sparse Asymmetric LEast Squares (SALES) regression under high dimensions where the penalty functions include the Lasso and nonconvex penalties. We develop a unified efficient algorithm for fitting SALES and establish its theoretical properties. As an important application, SALES is used to detect heteroscedasticity in high-dimensional data. Another method for detecting heteroscedasticity is the sparse quantile regression. However, both SALES and the sparse quantile regression may fail to tell which variables are important for the conditional mean and which variables are important for the conditional scale/variance, especially when there are variables that are important for both the mean and the scale. To that end, we further propose a COupled Sparse Asymmetric LEast Squares (COSALES) regression which can be efficiently solved by an algorithm similar to that for solving SALES. We establish theoretical properties of COSALES. In particular, COSALES using the SCAD penalty or MCP is shown to consistently identify the two important subsets for the mean and scale simultaneously, even when the two subsets overlap. We demonstrate the empirical performance of SALES and COSALES by simulated and real data.

AB - Asymmetric least squares regression is an important method that has wide applications in statistics, econometrics and finance. The existing work on asymmetric least squares only considers the traditional low dimension and large sample setting. In this paper, we systematically study the Sparse Asymmetric LEast Squares (SALES) regression under high dimensions where the penalty functions include the Lasso and nonconvex penalties. We develop a unified efficient algorithm for fitting SALES and establish its theoretical properties. As an important application, SALES is used to detect heteroscedasticity in high-dimensional data. Another method for detecting heteroscedasticity is the sparse quantile regression. However, both SALES and the sparse quantile regression may fail to tell which variables are important for the conditional mean and which variables are important for the conditional scale/variance, especially when there are variables that are important for both the mean and the scale. To that end, we further propose a COupled Sparse Asymmetric LEast Squares (COSALES) regression which can be efficiently solved by an algorithm similar to that for solving SALES. We establish theoretical properties of COSALES. In particular, COSALES using the SCAD penalty or MCP is shown to consistently identify the two important subsets for the mean and scale simultaneously, even when the two subsets overlap. We demonstrate the empirical performance of SALES and COSALES by simulated and real data.

KW - Asymmetric least squares

KW - COSALES

KW - High dimensions

KW - SALES

UR - http://www.scopus.com/inward/record.url?scp=84999791351&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84999791351&partnerID=8YFLogxK

U2 - 10.1214/15-AOS1431

DO - 10.1214/15-AOS1431

M3 - Article

AN - SCOPUS:84999791351

VL - 44

SP - 2661

EP - 2694

JO - Annals of Statistics

JF - Annals of Statistics

SN - 0090-5364

IS - 6

ER -