The role of the covariance matrix in the least-squares estimation for a common mean

✍ Scribed by Y.L. Tong

• Publisher: Elsevier Science
• Year: 1997
• Tongue: English
• Weight: 406 KB
• Volume: 264
• Category: Article
• ISSN: 0024-3795
• DOI: 10.1016/s0024-3795(96)00411-9

✦ Synopsis

For n > 1 let X = (X 1 ..... X,)' have a mean vector 01 and covariance matrix ~r2~, where 1 = (1,..., 1)', ~E is a known positive definite matrix, and ~r ~ > 0 is either known or unknown. This model has been found useful when the observations X l ..... X, from a population with mean O are not independent. We show how the variance of 0, the least-squares estimator of 0, depends on the covariance structure of ~. More specifically, we give expressions for Var(#), obtain its lower and upper bounds (which involve only the smallest and the larges,t eigenvalues of ~), and show how the dependence of X l ..... Xn plays a role in Var#. Examples of applications are given for M-matrices, for exchangeable random variables, for a class of covariance matrices with a block-correlation structure, and for twin data.