Some recent results in linear estimation

Abstract

Some recent work of the author on 'Unified Theory of Linear Estimation' is described. The general Gauss-Markoff (GGM) model (Y, Xβ, σ²V), is considered, where V=E[(Y-Xβ) (Y-Xβ)'] is possibly singular and X possibly deficient in rank. Aitken's procedure of least squares is not applicable when V is singular. The object of the paper is to lay down procedures which are valid in all situations and which do not require prior examination of the ranks of V and X. Two unified methods are suggested. One is a numerical approach called the Inverse Partitioned Matrix (IPM) method. Another is an analogue of the least squares theory, called the Unified Least Squares (ULS) method. It has been pointed out that singularity of V imposes some restriction on the parameter β, which have to be taken into account in constructing unbiased estimators.