A New Class of IMSE-Based Criteria for Optimal Designs in Multi-response Random Coefficient Regression Models

Abstract

A new class of criteria for optimal designs in random coefficient regression (RCR) models with r responses is presented, which is based on the integrated mean squared error (IMSE) for the prediction of random effects. This class, referred to as

{\text{IMSE}}_{r,L}

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\textrm{IMSE}_{r,L}$$\end{document}

-class of criteria, is invariant with respect to different parameterizations of the model and contains

\text{IMSE}

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\textrm{IMSE}$$\end{document}

- and G-optimality as special cases for the prediction in univariate response situations. General equivalence theorems for

{\text{IMSE}}_{r,L}

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-criteria are established for

L\in [1,\infty )

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and

L=\infty

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$L=\infty $$\end{document}

, respectively, which are used to check

{\text{IMSE}}_{r,L}

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\textrm{IMSE}_{r,L}$$\end{document}

-optimality of designs.

{\text{IMSE}}_{r,L}

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\textrm{IMSE}_{r,L}$$\end{document}

-optimal designs for linear and quadratic bi-response RCR models are given for illustration.

Keywords

Optimal designs / Integrated mean squared error matrix / IMSE-optimality / Prediction / Mixed effects model

Funding

National Natural Science Foundation of China(11971318)

Natural Science Foundation of Anhui Province (CN)(2008085QA15)

Shanghai Rising-Star Program(20QA1407500)

RIGHTS & PERMISSIONS

School of Mathematical Sciences, University of Science and Technology of China and Springer-Verlag GmbH Germany, part of Springer Nature