Some Further Remarks on the Linear Sufficiency in the Linear Model

Abstract

In this article we consider the linear sufficiency of statistic $\mathbf {F}\mathbf {y}$ when estimating the estimable parametric function of $\varvec{\beta }$ under the linear model $\mathscr {A}= \{\mathbf {y}, \mathbf {X}\varvec{\beta }, \mathbf {V}\}$. We review some properties that have not been received much attention in the literature and provide some new results and insight into the meaning of the linear sufficiency . In particular, we consider the best linear unbiased estimation ($\mathrm {BLUE})$ under the transformed model $\mathscr {A}_{t}= \{ \mathbf {F}\mathbf {y}, \mathbf {F}\mathbf {X}\varvec{\beta }, \mathbf {F}\mathbf {V}\mathbf {F}' \}$ and study the possibilities to measure the relative linear sufficiency of $\mathbf {F}\mathbf {y}$ by comparing the $\mathrm {BLUE}$s under $\mathscr {A}$ and $\mathscr {A}_{t}$. We also consider some new properties of the Euclidean norm of the distance of the BLUEs under $\mathscr {A}$ and $\mathscr {A}_{t}$. The concept of linear sufficiency was essentially introduced in early 1980s by Baksalary, Kala and Drygas, but to our knowledge the concept of relative linear sufficiency nor the Euclidean norm of the difference between the $\mathrm {BLUE}$s under $\mathscr {A}$ and $\mathscr {A}_{t}$ have not appeared in the literature. To make the article more self-readable we go through some basic concepts related to linear sufficiency . We also provide a rather extensive list of relevant references.