Characterizing Relationships Between Estimations Under a General Linear Model with Explicit and Implicit Restrictions by Rank of Matrix

Abstract

In the investigation of the restricted linear model ℳ _r  = {y, X β | A β = b, σ² Σ}, the parameter constraints A β = b are often handled by transforming the model into certain implicitly restricted model. Any estimation derived from the explicitly and implicitly restricted models on the vector β and its functions should be equivalent, although the expressions of the estimation under the two models may be different. However, people more likely want to directly compare different expressions of estimations and yield a conclusion on their equivalence by using some algebraic operations on expressions of estimations. In this article, we give some results on equivalence of the well-known OLSEs and BLUEs under the explicitly and implicitly restricted linear models by using some expansion formulas for ranks of matrices.

Keywords:

• BLUE
• Equality of estimations
• Implicitly restricted model
• Matrix rank method
• Moore–Penrose inverse
• OLSE
• Restricted linear model

Mathematics Subject Classification:

• Primary 62J05, 62H12
• Secondary 15A09

Acknowledgment

The author thanks two anonymous referees for their helpful comments and suggestions on this article.