A Review of the Linear Sufï¬�ciency and Linear Prediction Sufï¬�ciency in the Linear Model with New Observations

Stephen Haslett; Jarkko Isotalo; RadosÅ‚aw Kala; A Markiewicz; S Puntanen

doi:10.1007/978-3-030-75494-5_11

A Review of the Linear Sufï¬�ciency and Linear Prediction Sufï¬�ciency in the Linear Model with New Observations

Stephen Haslett, Jarkko Isotalo, RadosÅ‚aw Kala, A Markiewicz, S Puntanen

Research output: Chapter in Book/Report/Conference proceeding › Chapter › peer-review

Abstract

We consider the general linear model y = XÎ²Î²Î² + ÎµÎµÎµ, denoted as M = {y, XÎ²Î²Î², V}, supplemented with the new unobservable random vector yâˆ—, coming from yâˆ— = Xâˆ—Î²Î²Î² + ÎµÎµÎµâˆ—, where the covariance matrix of yâˆ— is known as well as the cross-covariance matrix between yâˆ— and y. A linear statistic Fy is called linearly sufï¬cient for Xâˆ—Î²Î²Î² if there exists a matrix A such that AFy is the best linear unbi-ased estimator, BLUE, for Xâˆ—Î²Î²Î². The concept of linear sufï¬ciency with respect to a predictable random vector is deï¬ned in the corresponding way but considering the best linear unbiased predictor, BLUP instead of BLUE. In this paper, we consider the linear sufï¬ciency of Fy with respect to yâˆ—, Xâˆ—Î²Î²Î², and ÎµÎµÎµâˆ—. We also apply our results into the linear mixed model. The concept of linear sufï¬ciency was essentially introduced in early 1980s by Baksalary, Kala, and Drygas. Recently, several papers providing further properties of the linear sufï¬ciency 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 go through some basic concepts related to linear sufï¬ciency. As a review paper, we do not provide many proofs, instead our goal is to explain and clarify the central results.

Original language	English
Title of host publication	Multivariate, Multilinear and Mixed Linear Models
Editors	Katarzyna Filipiak, Augustyn Markiewicz, Dietrich von Rosen
Place of Publication	Switzerland
Publisher	Springer Cham
Pages	265-318
Volume	1
Edition	1
ISBN (Print)	978-3-030-75493-8
DOIs	https://doi.org/10.1007/978-3-030-75494-5_11
Publication status	Published - 2021

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10.1007/978-3-030-75494-5_11

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@inbook{fe6cd66cf7ad437eb0f383ce3c2c599e,

title = "A Review of the Linear Suf{\"i}¬�ciency and Linear Prediction Suf{\"i}¬�ciency in the Linear Model with New Observations",

abstract = "We consider the general linear model y = X{\^I}²{\^I}²{\^I}² + {\^I}µ{\^I}µ{\^I}µ, denoted as M = {y, X{\^I}²{\^I}²{\^I}², V}, supplemented with the new unobservable random vector y{\^a}ˆ—, coming from y{\^a}ˆ— = X{\^a}ˆ—{\^I}²{\^I}²{\^I}² + {\^I}µ{\^I}µ{\^I}µ{\^a}ˆ—, where the covariance matrix of y{\^a}ˆ— is known as well as the cross-covariance matrix between y{\^a}ˆ— and y. A linear statistic Fy is called linearly suf{\"i}¬cient for X{\^a}ˆ—{\^I}²{\^I}²{\^I}² if there exists a matrix A such that AFy is the best linear unbi-ased estimator, BLUE, for X{\^a}ˆ—{\^I}²{\^I}²{\^I}². The concept of linear suf{\"i}¬ciency with respect to a predictable random vector is de{\"i}¬ned in the corresponding way but considering the best linear unbiased predictor, BLUP instead of BLUE. In this paper, we consider the linear suf{\"i}¬ciency of Fy with respect to y{\^a}ˆ—, X{\^a}ˆ—{\^I}²{\^I}²{\^I}², and {\^I}µ{\^I}µ{\^I}µ{\^a}ˆ—. We also apply our results into the linear mixed model. The concept of linear suf{\"i}¬ciency was essentially introduced in early 1980s by Baksalary, Kala, and Drygas. Recently, several papers providing further properties of the linear suf{\"i}¬ciency 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 go through some basic concepts related to linear suf{\"i}¬ciency. As a review paper, we do not provide many proofs, instead our goal is to explain and clarify the central results.",

author = "Stephen Haslett and Jarkko Isotalo and Rados{\AA}‚aw Kala and A Markiewicz and S Puntanen",

year = "2021",

doi = "10.1007/978-3-030-75494-5_11",

language = "English",

isbn = "978-3-030-75493-8",

volume = "1",

pages = "265--318",

editor = "{Katarzyna Filipiak, Augustyn Markiewicz, Dietrich von Rosen}",

booktitle = "Multivariate, Multilinear and Mixed Linear Models",

publisher = "Springer Cham",

edition = "1",

}

A Review of the Linear Sufï¬�ciency and Linear Prediction Sufï¬�ciency in the Linear Model with New Observations. / Haslett, Stephen; Isotalo, Jarkko; Kala, RadosÅ‚aw et al.
Multivariate, Multilinear and Mixed Linear Models. ed. / Katarzyna Filipiak, Augustyn Markiewicz, Dietrich von Rosen. Vol. 1 1. ed. Switzerland: Springer Cham, 2021. p. 265-318.

Research output: Chapter in Book/Report/Conference proceeding › Chapter › peer-review

TY - CHAP

T1 - A Review of the Linear Sufï¬�ciency and Linear Prediction Sufï¬�ciency in the Linear Model with New Observations

AU - Haslett, Stephen

AU - Isotalo, Jarkko

AU - Kala, RadosÅ‚aw

AU - Markiewicz, A

AU - Puntanen, S

PY - 2021

Y1 - 2021

N2 - We consider the general linear model y = XÎ²Î²Î² + ÎµÎµÎµ, denoted as M = {y, XÎ²Î²Î², V}, supplemented with the new unobservable random vector yâˆ—, coming from yâˆ— = Xâˆ—Î²Î²Î² + ÎµÎµÎµâˆ—, where the covariance matrix of yâˆ— is known as well as the cross-covariance matrix between yâˆ— and y. A linear statistic Fy is called linearly sufï¬cient for Xâˆ—Î²Î²Î² if there exists a matrix A such that AFy is the best linear unbi-ased estimator, BLUE, for Xâˆ—Î²Î²Î². The concept of linear sufï¬ciency with respect to a predictable random vector is deï¬ned in the corresponding way but considering the best linear unbiased predictor, BLUP instead of BLUE. In this paper, we consider the linear sufï¬ciency of Fy with respect to yâˆ—, Xâˆ—Î²Î²Î², and ÎµÎµÎµâˆ—. We also apply our results into the linear mixed model. The concept of linear sufï¬ciency was essentially introduced in early 1980s by Baksalary, Kala, and Drygas. Recently, several papers providing further properties of the linear sufï¬ciency 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 go through some basic concepts related to linear sufï¬ciency. As a review paper, we do not provide many proofs, instead our goal is to explain and clarify the central results.

AB - We consider the general linear model y = XÎ²Î²Î² + ÎµÎµÎµ, denoted as M = {y, XÎ²Î²Î², V}, supplemented with the new unobservable random vector yâˆ—, coming from yâˆ— = Xâˆ—Î²Î²Î² + ÎµÎµÎµâˆ—, where the covariance matrix of yâˆ— is known as well as the cross-covariance matrix between yâˆ— and y. A linear statistic Fy is called linearly sufï¬cient for Xâˆ—Î²Î²Î² if there exists a matrix A such that AFy is the best linear unbi-ased estimator, BLUE, for Xâˆ—Î²Î²Î². The concept of linear sufï¬ciency with respect to a predictable random vector is deï¬ned in the corresponding way but considering the best linear unbiased predictor, BLUP instead of BLUE. In this paper, we consider the linear sufï¬ciency of Fy with respect to yâˆ—, Xâˆ—Î²Î²Î², and ÎµÎµÎµâˆ—. We also apply our results into the linear mixed model. The concept of linear sufï¬ciency was essentially introduced in early 1980s by Baksalary, Kala, and Drygas. Recently, several papers providing further properties of the linear sufï¬ciency 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 go through some basic concepts related to linear sufï¬ciency. As a review paper, we do not provide many proofs, instead our goal is to explain and clarify the central results.

U2 - 10.1007/978-3-030-75494-5_11

DO - 10.1007/978-3-030-75494-5_11

M3 - Chapter

SN - 978-3-030-75493-8

VL - 1

SP - 265

EP - 318

BT - Multivariate, Multilinear and Mixed Linear Models

A2 - null, Katarzyna Filipiak, Augustyn Markiewicz, Dietrich von Rosen

PB - Springer Cham

CY - Switzerland

ER -

A Review of the Linear Sufï¬�ciency and Linear Prediction Sufï¬�ciency in the Linear Model with New Observations

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