Least squares estimators and non-negative estimators of variance components

Abstract

Using the least squares method we have a unified procedure for the derivation of estimators for variance components in the linear model of ZZ', where Z is the residual vector from a simple least squares fit for X/3. These least squares estimators are unbiased but not always non-negative. The invariant and unbiased least squares estimators for variance components are the MINQUE estimators. For multivariate normally distributed variables the MINQUE is the same as the MIVQUE.

Non-negative estimators of variance components are permissible estimators. Taking into account the constraints of non-negativity of the variance components, a quadratic programming procedure is suggested.

For multivariate normally distributed variables the ML method or REML method can be used. In practice many computer programs do.

Keywords:

• permissible estimators
• MINQUE
• MINQUE
• ML
• REML
• quadratic or convex programming