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ESTIMATING EQUATIONS AND NON‐LINEAR FUNCTIONAL RELATIONSHIPS
Author(s) -
Morton Richard,
Patefield Mike,
Bowtell Phil
Publication year - 1997
Publication title -
australian journal of statistics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.434
H-Index - 41
eISSN - 1467-842X
pISSN - 0004-9581
DOI - 10.1111/j.1467-842x.1997.tb00532.x
Subject(s) - mathematics , manifold (fluid mechanics) , nonlinear system , normality , vector space , interpretation (philosophy) , function (biology) , space (punctuation) , matrix (chemical analysis) , mathematical analysis , computer science , pure mathematics , statistics , mechanical engineering , physics , materials science , quantum mechanics , evolutionary biology , engineering , composite material , biology , programming language , operating system
summary A nonlinear functional relationship is defined as an R ‐dimensional manifold in P ‐dimensional space. The formulation of the model may be explicitly in terms of R ‐dimensional vectors of incidental parameters or implicitly by a ( P‐R )‐dimensional vector function of constraints. The objective is to estimate and make inference about a K ‐vector of parameters θ which defines the manifold. Each observed P ‐vector has its expectation lying on the manifold, and the error vector has a variance matrix defined in terms of a further vector of parameters The theory of estimating equations in the presence of incidental parameters is extended and applied to the explicit formulation, to give equations suitable for estimating θ given knowledge of only the first two moments. The method has a geometrical interpretation. Estimating equations for are chosen to be those which would be optimal if the normality assumption were true. First order corrections to the biases of these estimates are included. An example where the manifold is a circle centred on the origin is used to illustrate the theory. Further examples incorporate more general features, including the estimation of two variance parameters and estimation in higher dimensions.