Some Properties of Linear Prediction Sufficiency in the Linear Model

In this paper we consider the linear sufficiency of Fy for Xβ, for Zu and for Xβ + Zu, when dealing with the linear mixed model y = Xβ + Zu + e. In particular, we explore the relations between these sufficiency properties. The usual definition of linear sufficiency means, for example, that the BLUE of Xβ under the original model can be obtained as AFy for some matrix A. Liu et al. (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 BLUEs and/or BLUPs under one mixed model continue to be BLUEs and/or BLUPs under the other mixed model. In particular, we describe the mutual relations of the conditions. These problems were approached differently by Rong & Liu (2010) and we will show how their results are related to those obtained by our approach.

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“…The following theorem can now be proved along the same lines as Theorem 3.4 in Isotalo et al (2017). We omit the proof.…”

Section: Linear Sufficiency In a Linear Mixed Modelmentioning

confidence: 84%

Some properties of linear sufficiency and the BLUPs in the linear mixed model

et al. 2017

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“…For the following Lemma, see, e.g., Baksalary & Kala (1981, Drygas (1983), Tian & Puntanen (2009, Th. 2.8), Kala, Puntanen & Tian (2017, Th. 2), Isotalo & Puntanen (2006b), and Isotalo et al (2018).…”

Section: Conditions For Linear Sufficiencymentioning

confidence: 99%

“…Thus the results concerning the linear sufficiency in the misspecified mixed model can be directly obtained from the corresponding properties of the models with new observations. For the linear sufficiency in the mixed model, see also Isotalo et al (2018) and Markiewicz & Puntanen (2018c).…”

mentioning

confidence: 99%

Linear prediction sufficiency in the misspecified linear model

Markiewicz

Puntanen

2019

Communications in Statistics - Theory and Methods

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

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