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**Some properties of projectors associated with the WLSE under a general linear model.**
*(English)*
Zbl 1141.62043

Summary: Projectors associated with a particular estimator in a general linear model play an important role in characterizing statistical properties of the estimator. A variety of new properties were derived on projectors associated with the weighted least-squares estimator (WLSE). These properties include maximal and minimal possible ranks, rank invariance, uniqueness, idempotency, and other equalities involving the projectors. Applications of these properties were also suggested. The proofs of the main theorems demonstrate how to use the matrix rank method for deriving various equalities involving the projectors under the general linear model.

### MSC:

62H12 | Estimation in multivariate analysis |

62J05 | Linear regression; mixed models |

15A09 | Theory of matrix inversion and generalized inverses |

### Keywords:

weighted least-squares estimator; projectors; generalized inverses of matrices; rank formulas for partitioned matrices; elementary block matrix operations (EBMOs)
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\textit{Y. Tian} and \textit{Y. Takane}, J. Multivariate Anal. 99, No. 6, 1070--1082 (2008; Zbl 1141.62043)

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### References:

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