On the de la Garza phenomenon

By Min Yang. 2009

85 citations recorded by the source corpus.

Deriving optimal designs for nonlinear models is, in general, challenging. One crucial step is to determine the number of support points needed. Current tools handle this on a case-by-case basis. Each combination of model, optimality criterion and objective requires its own proof. The celebrated de la Garza Phenomenon states that under a (p — 1)th-degree polynomial regression model, any optimal design can be based on at most p design points, the minimum number of support points such that all parameters are estimable. Does this conclusion also hold for nonlinear models? If the answer is yes, it would be relatively easy to derive any optimal design, analytically or numerically. In this paper, a novel approach is developed to address this question. Using this new approach, it can be easily shown that the de la Garza phenomenon exists for many commonly studied nonlinear models, such as the Emax model, exponential model, three- and four-parameter log-linear models, Emax-PK1 model, as well as many classical polynomial regression models. The proposed approach unifies and extends many well-known results in the optimal design literature. It has four advantages over current tools: (i) it can be applied to many forms of nonlinear models; to continuous or discrete data; to data with homogeneous or nonhomogeneous errors; (ii) it can be applied to any design region; (iii) it can be applied to multiple-stage optimal design and (iv) it can be easily implemented.

Subfields: Computational Theory and Mathematics, Geometry and Topology, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/271b483913c09fad30aba7c0a168ec1ecf07340c

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Citations in the corpus

Design of Experiments in Nonlinear Models: Asymptotic Normality, Optimality Criteria and Small-Sample Properties (2013, methodology): “The generalization to models with more than two parameters is presented in (Yang, 2010) and (Dette and Melas, 2011), extending de la Garza (1954) result on polynomial regression to general nonlinear models.”
IDENTIFYING LOCALLY OPTIMAL DESIGNS FOR NONLINEAR MODELS: A SIMPLE EXTENSION WITH PROFOUND CONSEQUENCES (2012, methodology): “With a proper choice of Ψ functions, it can be shown that the result in Yang (2010) yields a complete class of designs with at most five points, including U and V .”
Optimal subsampling for quantile regression in big data (2020, background): “In the asymptotic variance-covariance matrix n−1D−1N VπD −1 N , only Vπ depends on πi’s while DN does not involve πi’s, and D −1 N VπD −1 N ≤ D−1N Vπ′D−1N if and only if Vπ ≤ Vπ′ in the Loewner ordering (Yang, 2010).”
A note on the de la Garza phenomenon for locally optimal designs (2011, methodology): “Note that—in contrast to Theorem 1 and 2 of Yang (2010)—our Theorem 3.1 does not require the differentiability of the functions j .”