TY - JOUR

T1 - Pivotal variable detection of the covariance matrix and its application to high-dimensional factor models

AU - Zhao, Junlong

AU - Zhao, Hongyu

AU - ZHU, Lixing

N1 - Funding Information:
Junlong Zhao was supported by the National Science Foundation of China (Nos. 11471030, 61472475) and KLAS 130026507. Hongyu Zhao was supported in part by NIH Grants GM59507 and CA154295, and NSF Grant DMS 1106738. Lixing Zhu was supported by a GRF Grant from the Research Grants Council of Hong Kong, and the National Science Foundation of China (No. 11671042). The authors thank editor, associate editor and an anonymous referee for the constructive suggestions and comments that led to the improvement of an early manuscript.

PY - 2018/7/1

Y1 - 2018/7/1

N2 - To estimate the high-dimensional covariance matrix, row sparsity is often assumed such that each row has a small number of nonzero elements. However, in some applications, such as factor modeling, there may be many nonzero loadings of the common factors. The corresponding variables are also correlated to one another and the rows are non-sparse or dense. This paper has three main aims. First, a detection method is proposed to identify the rows that may be non-sparse, or at least dense with many nonzero elements. These rows are called dense rows and the corresponding variables are called pivotal variables. Second, to determine the number of rows, a ridge ratio method is suggested, which can be regarded as a sure screening procedure. Third, to handle the estimation of high-dimensional factor models, a two-step procedure is suggested with the above screening as the first step. Simulations are conducted to examine the performance of the new method and a real dataset is analyzed for illustration.

AB - To estimate the high-dimensional covariance matrix, row sparsity is often assumed such that each row has a small number of nonzero elements. However, in some applications, such as factor modeling, there may be many nonzero loadings of the common factors. The corresponding variables are also correlated to one another and the rows are non-sparse or dense. This paper has three main aims. First, a detection method is proposed to identify the rows that may be non-sparse, or at least dense with many nonzero elements. These rows are called dense rows and the corresponding variables are called pivotal variables. Second, to determine the number of rows, a ridge ratio method is suggested, which can be regarded as a sure screening procedure. Third, to handle the estimation of high-dimensional factor models, a two-step procedure is suggested with the above screening as the first step. Simulations are conducted to examine the performance of the new method and a real dataset is analyzed for illustration.

KW - Covariance matrix estimation

KW - Factor model

KW - Pivotal variable detection

KW - Principal component analysis

KW - Row sparsity

KW - Ultra-high dimension

UR - http://www.scopus.com/inward/record.url?scp=85023767064&partnerID=8YFLogxK

U2 - 10.1007/s11222-017-9762-6

DO - 10.1007/s11222-017-9762-6

M3 - Article

AN - SCOPUS:85023767064

SN - 0960-3174

VL - 28

SP - 775

EP - 793

JO - Statistics and Computing

JF - Statistics and Computing

IS - 4

ER -