An extension of the General Equivalence Theorem to nonlinear models

Biometrika ◽  
1973 ◽  
Vol 60 (2) ◽  
pp. 345-348 ◽  
Author(s):  
LYNDA V. WHITE
Keyword(s):  
Nonlinear Models ◽  
Equivalence Theorem ◽  
General Equivalence Theorem

scholarly journals [[EQUATION]]-optimal Designs for Hierarchical Linear Models: an Equivalence Theorem and a Nature-inspired Meta-heuristic Algorithm

2021 ◽  
Author(s):  
Xin Liu ◽  
RongXian Yue ◽  
Zizhao Zhang ◽  
Weng Kee Wong
Keyword(s):  
Heuristic Algorithm ◽  
Linear Models ◽  
Optimization Problems ◽  
Nonlinear Models ◽  
Equivalence Theorem ◽  
Hierarchical Linear Models ◽  
Optimal Designs ◽  
Approximate Design ◽  
Random Intercept ◽  
Mathematical Question

Abstract Hierarchical linear models are widely used in many research disciplines and estimation issues for such models are generally well addressed. Design issues are relatively much less discussed for hierarchical linear models but there is an increasing interest as these models grow in popularity. This paper discusses [[EQUATION]] -optimality for predicting individual parameters in such models and establishes an equivalence theorem for confirming the [[EQUATION]] -optimality of an approximate design. Because the criterion is non-differentiable and requires solving multiple nested optimization problems, it is much harder to find and study [[EQUATION]] -optimal designs analytically. We propose a nature-inspired meta-heuristic algorithm called competitive swarm optimizer (CSO) to generate [[EQUATION]] -optimal designs for linear mixed models with different means and covariance structures. We further demonstrate that CSO is flexible and generally effective for finding the widely used locally [[EQUATION]] -optimal designs for nonlinear models with multiple interacting factors and some of the random effects are correlated. Our numerical results for a few examples suggest that [[EQUATION]] and [[EQUATION]] -optimal designs may be equivalent and we establish that [[EQUATION]] and [[EQUATION]] -optimal designs for hierarchical linear models are equivalent when the models have only a random intercept only. The challenging mathematical question whether their equivalence applies more generally to other hierarchical models remains elusive.

scholarly journals Optimality criteria for design in nonlinear models with constraints

2012 ◽  
Vol 51 (1) ◽  
pp. 141-149
Author(s):  
Andrej Pázman
Keyword(s):  
Nonlinear Regression ◽  
Regression Models ◽  
Nonlinear Models ◽  
Equivalence Theorem ◽  
Optimality Criteria ◽  
Nonlinear Regression Models

ABSTRACT We shall present different expressions for optimality criteria in nonlinear regression models, and compare them with corresponding expressions in models without constraints. We also present how to formulate the equivalence theorem in models with constraints.

Computable and Continuous Partial Homomorphisms on Metric Partial Algebras

2003 ◽  
Vol 9 (3) ◽  
pp. 299-334 ◽  
Author(s):  
Viggo Stoltenberg-Hansen ◽  
John V. Tucker
Keyword(s):  
Banach Spaces ◽  
Effective Properties ◽  
Equivalence Theorem ◽  
Linear Operators ◽  
General Situation ◽  
Universal Algebras ◽  
Effective Metric ◽  
Partial Algebras ◽  
General Equivalence Theorem ◽  
Closed Linear Operators

AbstractWe analyse the connection between the computability and continuity of functions in the case of homomorphisms between topological algebraic structures. Inspired by the Pour-El and Richards equivalence theorem between computability and boundedness for closed linear operators on Banach spaces, we study the rather general situation of partial homomorphisms between metric partial universal algebras. First, we develop a set of basic notions and results that reveal some of the delicate algebraic, topological and effective properties of partial algebras. Our main computability concepts are based on numerations and include those of effective metric partial algebras and effective partial homomorphisms. We prove a general equivalence theorem that includes a version of the Pour-El and Richards Theorem, and has other applications. Finally, the Pour-El and Richards axioms for computable sequence structures on Banach spaces are generalised to computable partial sequence structures on metric algebras, and we prove their equivalence with our computability model based on numerations.

scholarly journals R-optimal designs for second-order Scheffé model with qualitative factors

2021 ◽  
Vol 7 (3) ◽  
pp. 4540-4551
Author(s):  
Ling Ling ◽  
◽  
Guanghui Li ◽  
Xiaoyuan Zhu ◽  
Chongqi Zhang ◽  
...  
Keyword(s):  
Optimal Design ◽  
Relative Efficiency ◽  
Design Problem ◽  
Equivalence Theorem ◽  
Decision Function ◽  
Optimal Designs ◽  
Optimal Design Problem ◽  
Qualitative Factors ◽  
General Equivalence Theorem ◽  
Analytical Expressions

<abstract><p>Considering a mixture model with qualitative factors, the $ R $-optimal design problem is investigated when the levels of the qualitative factor interact with the mixture factors. In this paper, the conditions for $ R $-optimality of designs with mixture and qualitative factors are presented. General analytical expressions are also derived for the decision function under the $ R $-optimal designs, in order to verify that the resulting designs satisfy the general equivalence theorem. In addition, the relative efficiency of the $ R $-optimal design is discussed.</p></abstract>

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