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In statistics the linear model is a model given by

where Y is an n×1 column vector of random variables, X is an n×p matrix of "known" (i.e., observable and non-random) quantities, whose rows correspond to statistical units, β is a p×1 vector of (unobservable) parameters, and ε is an n×1 vector of "errors", which are uncorrelated random variables each with expected value 0 and variance σ². Often one takes the components of the vector of errors to be independent and normally distributed. Having observed the values of X and Y, the statistician must estimate β and σ². Typically the parameters β are estimated by the method of maximum likelihood, which in the case of normal errors is equivalent (by the Gauss-Markov theorem) to the method of least squares.

If, rather than taking the variance of ε to be σ²I, where I is the n×n identity matrix, one assumes the variance is σ²M, where M is a known matrix other than the identity matrix, then one estimates β by the method of "generalized least squares", in which, instead of minimizing the sum of squares of the residuals, one minimizes a different quadratic form in the residuals — the quadratic form being the one given by the matrix M^-1. This leads to the estimator

which is the Best Linear Unbiased Estimator for . If all of the off-diagonal entries in the matrix M are 0, then one normally estimates β by the method of "weighted least squares", with weights proportional to the reciprocals of the diagonal entries.

Ordinary linear regression is a very closely related topic.

Contents 1 Generalizations 1.1 Generalized linear models 1.2 General linear model 1.3 See also

[edit] Generalizations

[edit] Generalized linear models

Generalized linear models, for which rather than

E(Y) = Xβ,

one has

g(E(Y)) = Xβ,

where g is the "link function". An example is the Poisson regression model, which states that

Y_i has a Poisson distribution with expected value e^γ+δx_i.

The link function is the natural logarithm function. Having observed x_i and Y_i for i = 1, ..., n, one can estimate γ and δ by the method of maximum likelihood.

[edit] General linear model

The general linear model (or multivariate regression model) is a linear model with multiple measurements per object. Each object may be represented in a vector.

[edit] See also

• ANOVA, or analysis of variance, is historically a precursor to the development of linear models. Here the model parameters themselves are not computed, but X column contributions and their significance are identified using the ratios of within-group variances to the error variance and applying the F test.de:Lineares Modell

This page uses content from the English-language version of Wikipedia. The original article was at Linear model. The list of authors can be seen in the page history. As with Psychology Wiki, the text of Wikipedia is available under the GNU Free Documentation License.