Abstract
The model equation for a linear model, i.e., y = Xβ + e, has two distinct terms on the right-hand side: Xβ and e. Because Xβ = E(y), we refer to any additional assumptions made about Xβ (i.e., additional to X being nonrandom and known and β being nonrandom and unknown) as the model’s mean structure, and to any additional assumptions made about e [i.e., additional to E(e) = 0] as the model’s error structure . In this chapter, we give a brief overview of various linear models that are distinguishable from one another on the basis of these two types of structure, and we describe some terminology associated with each type. We also describe prediction-extended versions of these models, i.e., models for the mean structure and error structure of the joint distribution of y and a related, though unobservable, random vector u. Throughout the remainder of the book, many aspects of linear model theory will be exemplified by the models introduced here.
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References
Aitken, A. C. (1935). On least squares and linear combination of observations. Proceedings of the Royal Society of Edinburgh, 55, 42–48.
Christensen, R. (2011). Plane answers to complex questions (4th ed.) New York: Springer.
Gauss, C. F. (1855). Methode des moindres Carres. Paris: Mallet-Bachelier.
Graybill, F. A. (1976). Theory and application of the linear model. Belmont, CA: Wadsworth.
Harville, D. A. (2018). Linear models and the relevant distributions and matrix algebra. Boca Raton, FL: CRC.
Hinkelmann, K., & Kempthorne, O. (1994). Design and analysis of experiments. New York: Wiley.
Khuri, A. I. (2010). Linear model methodology. Boca Raton, FL: Chapman & Hall/CRC Press.
Myers, R. H. & Milton, J. S. (1991). A first course in the theory of linear statistical models. Boston: PWS-KENT.
Plackett, R. L. (1949). A historical note on the method of least squares. Biometrika, 36, 458–460.
Ravishanker, N. & Dey, D. K. (2002). A first course in linear model theory. Boca Raton, FL: Chapman & Hall/CRC Press.
Rencher, A. C. (2000). Linear models in statistics. New York: Wiley.
Searle, S. R. (1971). Linear models. New York: Wiley.
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Zimmerman, D.L. (2020). Types of Linear Models. In: Linear Model Theory. Springer, Cham. https://doi.org/10.1007/978-3-030-52063-2_5
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DOI: https://doi.org/10.1007/978-3-030-52063-2_5
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