### Abstract

We introduce a matrix operator, which we call “vecd” operator. This operator stacks up “diagonals” of a symmetric matrix. This operator is more convenient for some statistical analyses than the commonly used “vech” operator. We show an explicit relationship between the vecd and vech operators. Using this relationship, various properties of the vecd operator are derived. As applications of the vecd operator, we derive concise and explicit expressions of the Wald and score tests for equal variances of a multivariate normal distribution and for the diagonality of variance coefficient matrices in a multivariate generalized autoregressive conditional heteroscedastic (GARCH) model, respectively.

Original language | English |
---|---|

Pages (from-to) | 1-17 |

Number of pages | 17 |

Journal | Communications in Statistics - Theory and Methods |

DOIs | |

Publication status | Accepted/In press - 2017 Aug 30 |

### Fingerprint

### Keywords

- Orthogonal matrix
- Permutation matrix
- Symmetric matrix
- vecd
- vech
- Wald test

### ASJC Scopus subject areas

- Statistics and Probability

### Cite this

**On the relationship between the matrix operators, vech and vecd.** / Nagakura, Daisuke.

Research output: Contribution to journal › Article

}

TY - JOUR

T1 - On the relationship between the matrix operators, vech and vecd

AU - Nagakura, Daisuke

PY - 2017/8/30

Y1 - 2017/8/30

N2 - We introduce a matrix operator, which we call “vecd” operator. This operator stacks up “diagonals” of a symmetric matrix. This operator is more convenient for some statistical analyses than the commonly used “vech” operator. We show an explicit relationship between the vecd and vech operators. Using this relationship, various properties of the vecd operator are derived. As applications of the vecd operator, we derive concise and explicit expressions of the Wald and score tests for equal variances of a multivariate normal distribution and for the diagonality of variance coefficient matrices in a multivariate generalized autoregressive conditional heteroscedastic (GARCH) model, respectively.

AB - We introduce a matrix operator, which we call “vecd” operator. This operator stacks up “diagonals” of a symmetric matrix. This operator is more convenient for some statistical analyses than the commonly used “vech” operator. We show an explicit relationship between the vecd and vech operators. Using this relationship, various properties of the vecd operator are derived. As applications of the vecd operator, we derive concise and explicit expressions of the Wald and score tests for equal variances of a multivariate normal distribution and for the diagonality of variance coefficient matrices in a multivariate generalized autoregressive conditional heteroscedastic (GARCH) model, respectively.

KW - Orthogonal matrix

KW - Permutation matrix

KW - Symmetric matrix

KW - vecd

KW - vech

KW - Wald test

UR - http://www.scopus.com/inward/record.url?scp=85028742742&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85028742742&partnerID=8YFLogxK

U2 - 10.1080/03610926.2017.1353623

DO - 10.1080/03610926.2017.1353623

M3 - Article

AN - SCOPUS:85028742742

SP - 1

EP - 17

JO - Communications in Statistics - Theory and Methods

JF - Communications in Statistics - Theory and Methods

SN - 0361-0926

ER -