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General Linear Models

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International Encyclopedia of Statistical Science

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Lety be a random variable such that E(y) = μ, or \(y = \mu + \epsilon \), where ε is a random error with E(ε) = 0. Suppose that \(\mu = {x}_{1}{\beta }_{1} + \cdots + {x}_{p}{\beta }_{p}\), where x 1, …, x p are p variables and β1, …, β p are p unknown parameters. The model

$$ y = {x}_{1}{\beta }_{1} + \cdots + {x}_{p}{\beta }_{p} + \epsilon , $$
(1)

is the well-known multiple linear regression model (see Linear Regression Models). Here y is called dependent (or response) variable, x 1, …, x p are called independent (or explanatory) variables or regressors, and β1, …, β p are called regression coefficients. Letting

$$({y}_{1},{x}_{11},\ldots ,{x}_{1p}),\ldots ,({y}_{n},{x}_{n1},\ldots ,{x}_{np})$$

be a sequence of the observations of Y and x 1, …, x p , we have

$$ {y}_{i} = {x}_{i1}{\beta }_{1} + \cdots + {x}_{ip}{\beta }_{p} + {\epsilon }_{i},\quad i = 1,\ldots ,n, $$
(2)

where ε1, …, ε n are the corresponding random errors. Denote y n = (y 1, …, y n )T, X n = (x 1, …, x ...

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References and Further Reading

  • Draper NR, Smith H (1998) Applied regression analysis, 3rd edn. Wiley, New York

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  • Fang KT, Zhang YT (1990) Generalized multivariate analysis. Springer/Science Press, Berlin/Beijing

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  • Gauss CF (1809) Least squares. Werke 4:1–93, Göttingen

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  • Mardia KV, Kent JT, Bibby JM (1979) Multivariate analysis. Academic, London

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  • Markov AA (1900) Ischislenie veroyatnostej, SPb

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  • Muirhead RJ (1982) Aspects Of multivariate statistical theory. Wiley, New York

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  • Rao CR (1973) Linear statistical inference and applications. Wiley, New York

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  • Rao CR, Toutenburg H, Shalabh C, Heumann C (2008) Linear models and generalizations. Least squares and alternatives. 3rd edn. Springer, Berlin/Heidelberg

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  • Seber GAF, Lee AJ (2003) Linear regression analysis. 2nd edn. Wiley, New York

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Author information

Authors and Affiliations

  1. York University, Toronto, ON, Canada

    Yuehua Wu (Professor)

Authors
  1. Yuehua Wu
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Editor information

Editors and Affiliations

  1. Department of Statistics and Informatics, Faculty of Economics, University of Kragujevac, City of Kragujevac, Serbia

    Miodrag Lovric

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Wu, Y. (2011). General Linear Models. In: Lovric, M. (eds) International Encyclopedia of Statistical Science. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-04898-2_277

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