We consider the partitioned linear model\index{partitioned linear model} y = X 1 β 1 + X 2 β 2 + ε \mathbf {y}= \mathbf {X}_{1} \boldsymbol {\beta }_{1} + \mathbf {X}_{2} \boldsymbol {\beta }_{2} + \boldsymbol {\varepsilon } , for which the standard notation is the triplet M = { y , X β , V } = { y , X 1 β 1 + X 2 β 2 , V } \mathscr {M}= \{ \mathbf {y}, \, \mathbf {X}\boldsymbol {\beta }, \, \mathbf {V}\} = \{ \mathbf {y}, \, \mathbf {X}_{1}\boldsymbol {\beta }_{1} + \mathbf {X}_{2}\boldsymbol {\beta }_{2}, \, \mathbf {V}\} . Vector β ∈ R p \boldsymbol {\beta }\in \mathbb {R}^{p} is a vector of fixed (but unknown) parameters, partitioned as β 1 \boldsymbol {\beta }_{1} and β 2 \boldsymbol {\beta }_{2} . Our aim is to estimate parametric functions like K ′ β \mathbf {K}’ \boldsymbol {\beta } under M \mathscr {M} ; here K \mathbf {K} is a given matrix of p p rows and ′ ’ denotes the transpose. In particular, we are interested in the best linear unbiased estimator\index{best linear unbiased estimator (BLUE)}s, BLUEs, of K ′ β \mathbf {K}’ \boldsymbol {\beta } . Let X ∗ \mathbf {X}_{*} be any matrix such that the column spaces of X \mathbf {X} and X ∗ \mathbf {X}_{*} are equal. Then there exists a matrix A \mathbf {A} such that X ∗ = X A \mathbf {X}_{*} = \mathbf {X}\mathbf {A} and we get the reparametrized model T = { y , X ∗ γ , V } \mathscr {T}= \{ \mathbf {y}, \, \mathbf {X}_{*} \boldsymbol {\gamma }, \, \mathbf {V}\} . Models M \mathscr {M} and T \mathscr {T} are called equivalent in the sense that B L U E ( X β | M ) = B L U E ( X ∗ γ | T ) BLUE( \mathbf {X}\boldsymbol {\beta }\,|\,\mathscr {M}) = BLUE( \mathbf {X}_{*} \boldsymbol {\gamma }\,|\,\mathscr {T}) . In this paper we consider two kinds of choice of A \mathbf {A} : A = ( I p 1 a m p ; − X 1 + X 2 0 a m p ; I p 2 ) , A = ( K : K ⊥ ) , \begin{equation*} \mathbf {A}= \begin {pmatrix} \mathbf {I}_{p_{1}} & - \mathbf {X}_{1}^{+} \mathbf {X}_{2} \\ \mathbf {0}& \mathbf {I}_{p_{2}} \end{pmatrix} , \quad \mathbf {A}= (\mathbf {K}: \mathbf {K}^{\bot } ) \,, \end{equation*} where K ⊥ \mathbf {K}^{\bot } is a matrix spanning the orthocomplement of the column space of K \mathbf {K} and the superscript + {}^{+} refers to the Moore–Penrose inverse.

In 1971, in his seminal paper entitled Unified theory of linear estimation, C.R. Rao considered the properties of best linear unbiased estimators, BLUEs, in the general linear model ℳ(V) = {y, Xβ, V} , where V refers to the covariance matrix of the observable random vector y and X is the model matrix. Both X and the nonnegative definite covariance matrix V are known. Citing Rao, “In Section 5 [of his paper] we raise the question of identification of V given the class of BLUE’s of all estimable functions”. It is precisely Section 5 of Rao’s paper which is in our focus. In particular, we will take a good look at Rao’s Theorems 5.2 and 5.3 which answer the following question: Given the model ℳ(V_0) = {y, Xβ, V_0} , how to characterize the set of all covariance matrices V such that every representation of the BLUE of Xβ under ℳ(V_0) remains BLUE under ℳ(V) . Our attempt is to provide some new insight into this problem area.

In a linear model where data are linearly transformed or compressed, conditions for linear sufficiency provide information about whether BLUEs of Xβ or linear combinations of Xβ (i.e., Kβ in our notation) remain unchanged. When there are changes of error covariance structure to the original model, the conditions that the BLUEs are unchanged are well known. We consider the original linear model, say 𝒜 , and the misspecified model ℬ , which differ only in their error covariance matrices. We explore the connections between the invariance of the linear sufficiency and the invariance of the representations of the BLUEs between 𝒜 and ℬ .

Misspecification of the error covariance in linear models usually leads to incorrect inference and conclusions. We consider two linear models, A$$ \mathcal{A} $$ and B$$ \mathcal{B} $$, with the same design matrix but different error covariance matrices. The conditions under which every representation of the best linear unbiased estimator (BLUE) of any estimable parametric vector under A$$ \mathcal{A} $$ remains BLUE under B$$ \mathcal{B} $$ have been well known since C.R. Rao's paper in 1971: Unified theory of linear estimation, Sankhy & amacr; Ser. A, Vol. 33, pp. 371-394. However, there are no previously published results on retaining the weighted sum of squares of errors (SSE) for non-full-rank design or error covariance matrices, and the question of when the covariance matrix of the BLUEs is also retained has been partially explored only recently. For change in any specified error covariance matrix, we provide necessary and sufficient conditions (nasc) for both BLUEs and their covariance matrix to remain unaltered and to retain this property for all submodels. We also consider nasc for SSE to be unchanged. We decompose SSE under error covariance changes, and derive nasc under which error covariance change leaves hypothesis tests for fixed-effect deletion under normality unaltered. We also show that simultaneous retention of BLUEs and both their covariance and SSE is not possible. We outline the effects of weak and strong error covariance singularity. We provide applications (via data cloning) to maintaining data confidentiality in Official Statistics without using Confidentialised Unit Record Files (CURFs), to certain types of experimental design and to estimation of fixed parameters for linear models for single nucleotide polymorphisms (SNPs) in genetics.

In the world of linear statistical models there is a particular matrix equation, G(X : VX perpendicular to) = (X : 0), which is sufficiently important that it is sometimes called the fundamental BLUE equation. In this equation, X is a model matrix, V is the covariance matrix of y in the linear model y = X beta + epsilon, and we are interested in finding the best linear estimator, BLUE, of X beta. Any solution G for this equation has the property that Gy provides a representation for the BLUE of X beta: this is the message of the the fundamental BLUE equation, whose main developer was the late Professor C. R. Rao in early 1970s. In this article we revisit some interesting features and consequences of this equation. We do not provide essentially new results - the aim is to offer a compact easy-to-follow review including also some recent related results by the authors.

We consider the partitioned linear model M12(V0) = { y, X1β1 + X2 β2, V0 } and the corresponding small model M1(V0) = { y, X1β1 , V0 } . We define the set V1/12 of nonnegative definite matrices V such that every representation of the best linear unbiased estimator, BLUE, of μ1 = X1β1 under M12(V0) remains BLUE under M12(V) . Correspondingly, we can characterize the set V1 of matrices V such that every BLUE of μ1 = X1β1 under M1(V0) remains BLUE under M1(V). In this paper we focus on the mutual relations between the sets V1 and V1/12 .

In this article, we consider the partitioned linear model $$\mathscr {M}_{12}({\textbf {V}}_{0}) = \{ {\textbf {y}}, \, {\textbf {X}}_{1}\boldsymbol{\beta }_{1} + {\textbf {X}}_{2}\boldsymbol{\beta }_{2}, \, {\textbf {V}}_{0} \}$$ and the corresponding small model $$\mathscr {M}_{1}({\textbf {V}}_{0}) = \{ {\textbf {y}}, \, {\textbf {X}}_{1} \boldsymbol{\beta }_{1}, \, {\textbf {V}}_{0} \} .$$ Following Rao [14, Sect. 5], we can characterize the set $$\mathscr {V}_{12}$$ of nonnegative definite matrices $${\textbf {V}}$$ such that every representation of the best linear unbiased estimator, BLUE, of $$\boldsymbol{\mu } = {\textbf {X}}\boldsymbol{\beta }$$ under $$\mathscr {M}_{12}({\textbf {V}}_{0}) $$ remains BLUE under $$\mathscr {M}_{12}({\textbf {V}}) $$ . Correspondingly, we can characterize the set $$\mathscr {V}_{1}$$ of matrices $${\textbf {V}}$$ such that every BLUE of $$\boldsymbol{\mu }_{1} = {\textbf {X}}_{1}\boldsymbol{\beta }_{1}$$ under $$\mathscr {M}_{1}({\textbf {V}}_{0}) $$ remains BLUE under $$\mathscr {M}_{1}({\textbf {V}}) $$ . In the first three sections of this paper, we focus on the mutual relations between the sets $$\mathscr {V}_{1}$$ and $$\mathscr {V}_{12}$$ . In Section 5, we assume that under the small model $$\mathscr {M}_{1}$$ the ordinary least squares estimator, OLSE, of $$\boldsymbol{\mu }_{1} $$ equals the $$\textrm{BLUE}$$ of $$\boldsymbol{\mu }_{1} $$ and give several characterizations for the continuation of the equality of OLSE and BLUE when more X-variables are added.

The necessary and sufficient condition for BLUEs of estimable functions of parameters in a linear fixed effect model being un-altered by a change in error covariance structure is due to Rao [18]. Structural insight into Rao’s condition can be gained by writing the quadratic form that is permitted to be added to the original covariance in block diagonal form. When the original full linear model is made smaller by reducing the number of regressors (which may include interactions of any order), block diagonal or diagonal matrices also provide insight into conditions for the entire set of full, small, and intermediate models each to retain their own BLUEs. The paper outlines the role that such changes in error covariance structure can play in data confidentiality and data encryption, especially when the covariance of the BLUEs is also retained. Extensions to linear mixed models and BLUPs are outlined in principle.

In this article we consider the partitioned linear model M-12 = {y, X-1 beta(1) + X-2 beta(2),V} and the corresponding small model M-1 = {y, X-1 beta(1), V}. We focus on comparing the best linear unbiased estimators, BLUEs, of X-1 beta(1) if , under M-12 and( )M(1. )In other words, we are interested in the effect of adding regressors on the BLUEs. Particular attention is paid on the consistency of the model, that is, whether the realized value of the response vector y belongs to the column space of (X-1 : V) or (X-1 : X-2 : V).

In this paper we introduce extensions of the so-called Frisch-Waugh-Lovell Theorem.This is done by employing the close relationship between the concept of linear sufficiency and the appropriate reduction of linear models.Some specific reduced models which demonstrate alternatives to the Frisch-Waugh-Lovell procedure are discussed.

In this article we consider the partitioned fixed linear model F : y = X1β1 + X2β2 + ε" and the corresponding mixed model M : y =X1β1+X2u+ ε, where ε is a random error vector and u is a random effect vector. In 2006, Isotalo, M¨ols, and Puntanen found conditions under which an arbitrary representation of the best linear unbiased estimator (BLUE) of an estimable parametric function of β1 in the fixed model F remains BLUE in the mixed model M . In this paper we extend the results concerning further equalities arising from models F and M.

We consider the general linear model y = X beta + epsilon, denoted as M = {y, X beta, V}, supplemented with the new unobservable random vector y*, coming from y(*) = X-*beta + e(*), where the covariance matrix of y* is known as well as the cross-covariance matrix between y(*) and y. A linear statistic Fy is called linearly sufficient for X-* beta if there exists a matrix A such that AFy is the best linear unbiased estimator, BLUE, for X-*beta. The concept of linear sufficiency with respect to a predictable random vector is defined in the corresponding way but considering the best linear unbiased predictor, BLUP instead of BLUE. In this paper, we consider the linear sufficiency of Fy with respect to y(*), X-*beta, and epsilon(*). We also apply our results into the linear mixed model. The concept of linear sufficiency was essentially introduced in early 1980s by Baksalary, Kala, and Drygas. Recently, several papers providing further properties of the linear sufficiency have been published by the present authors. Our aim is to provide an easy-to-read review of recent results and while doing that,

We consider the general linear model supplemented with the new (future) unobservable random vector coming from where the expectation of is and the covariance matrix of is known as well as the cross-covariance matrix between and . We denote the supplemented model as The misspecified supplemented model is denoted as and the misspecification concerns the covariance part of the setup. Suppose that is linearly sufficient for estimable parametric function under We give necessary and sufficient conditions that continues to be linearly sufficient for under the model The corresponding properties regarding the linear prediction sufficiency with respect to epsilon(*) and gamma(*) are also studied.

In this article, we go through some crucial developments regarding the equality of the ordinary least squares estimator and the best linear unbiased estimator in the general linear model. C. R. Rao (Proceedings of the Fifth Berkeley Symposium on Mathematical Statistics and Probability. University of California Press, Berkeley, pp. 355–372, 1967) appears to be the first to provide necessary and sufficient conditions for the general case when both the model matrix and the random error term’s covariance matrix are possibly deficient in rank. We describe the background of the problem area and provide some examples. We also consider some personal CRR-related glimpses of our research careers and provide a rather generous list of references.

In this paper we consider the linear sufficiency of \(\mathbf {F}\mathbf {y}\) for \(\mathbf {X}\varvec{\beta }\), for \(\mathbf {Z}\mathbf {u}\) and for \(\mathbf {X}\varvec{\beta }+ \mathbf {Z}\mathbf {u}\), when dealing with the linear mixed model \(\mathbf {y}= \mathbf {X}\varvec{\beta }+ \mathbf {Z}\mathbf {u}+ \mathbf {e}\). In particular, we explore the relations between these sufficiency properties. The usual definition of linear sufficiency means, for example, that the \({{\mathrm{BLUE}}}\) of \(\mathbf {X}\varvec{\beta }\) under the original model can be obtained as \(\mathbf {A}\mathbf {F}\mathbf {y}\) for some matrix \(\mathbf {A}\). Liu et al. (J Multivar Anal 99:1503–1517, 2008) introduced a slightly different definition for the linear sufficiency and we study its relation to the standard definition. We also consider the conditions under which \({{\mathrm{BLUE}}}\)s and/or \({{\mathrm{BLUP}}}\)s under one mixed model continue to be \({{\mathrm{BLUE}}}\)s and/or \({{\mathrm{BLUP}}}\)s under the other mixed model. In particular, we describe the mutual relations of the conditions. These problems were approached differently by Rong and Liu (Stat Pap 51:445–453, 2010) and we will show how their results are related to those obtained by our approach.

In this article we present a short history of the MatTriad Conferences, a series of international conferences on matrix analysis and its applications. The name MatTriad originally comes from the phrase Three Days Full of Matrices. The first MatTriad was held in the Mathematical Research and Conference Center of the Institute of Mathematics of the Polish Academy of Sciences in Będlewo, near Poznań, Poland, 3–5 March 2005, and has since then been organized biennially. The 8th MatTriad was held in Liblice, near Prague, Czech Republic, 8–13 September 2019. The next MatTriad will be held in Curia, near Coimbra, Portugal, September 2021.

A linear statistic Fy, where F is an f × n matrix, is called linearly sufficient for estimable parametric function K β under the model ℳ = {𝐲, 𝐗β, 𝐕} , if there exists a matrix A such that AFy is the BLUE for K β. In this paper we consider some particular aspects of the linear sufficiency in the partitioned linear model where X = (X 1 : X 2) with β being partitioned accordingly. We provide new results and new insightful proofs for some known facts, using the properties of relevant covariance matrices and their expressions via certain orthogonal projectors. Particular attention will be paid to the situation under which adding new regressors (in X 2) does not affect the linear sufficiency of Fy.