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Statistical Modeling of Implicit Functional Relations

Author

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  • Stan Lipovetsky

    (Independent Researcher, Minneapolis, MN 55305, USA)

Abstract

This study considers the statistical estimation of relations presented by implicit functions. Such structures define mutual interconnections of variables rather than outcome variable dependence by predictor variables considered in regular regression analysis. For a simple case of two variables, pairwise regression modeling produces two different lines of each variable dependence using another variable, but building an implicit relation yields one invertible model composed of two simple regressions. Modeling an implicit linear relation for multiple variables can be expressed as a generalized eigenproblem of the covariance matrix of the variables in the metric of the covariance matrix of their errors. For unknown errors, this work describes their estimation by the residual errors of each variable in its regression by the other predictors. Then, the generalized eigenproblem can be reduced to the diagonalization of a special matrix built from the variables’ covariance matrix and its inversion. Numerical examples demonstrate the eigenvector solution’s good properties for building a unique equation of the relations between all variables. The proposed approach can be useful in practical regression modeling with all variables containing unobserved errors, which is a common situation for the applied problems.

Suggested Citation

  • Stan Lipovetsky, 2023. "Statistical Modeling of Implicit Functional Relations," Stats, MDPI, vol. 6(3), pages 1-18, August.
  • Handle: RePEc:gam:jstats:v:6:y:2023:i:3:p:55-906:d:1224842
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    References listed on IDEAS

    as
    1. Draper, Norman R. & Yang, Yonghong (Fred), 1997. "Generalization of the geometric mean functional relationship," Computational Statistics & Data Analysis, Elsevier, vol. 23(3), pages 355-372, January.
    2. Ronald Christensen, 2022. "Comment on “On Optimal Correlation-Based Prediction,” by Bottai et al. (2022)," The American Statistician, Taylor & Francis Journals, vol. 76(4), pages 322-322, October.
    3. Francq, Bernard G. & Govaerts, Bernadette, 2014. "Measurement methods comparison with errors-in-variables regressions. From horizontal to vertical OLS regression, review and new perspectives," LIDAM Reprints ISBA 2014015, Université catholique de Louvain, Institute of Statistics, Biostatistics and Actuarial Sciences (ISBA).
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