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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.
Generalizations[edit | edit source]
Generalized linear models[edit | edit source]
Generalized linear models, for which rather than
- E(Y) = Xβ,
- g(E(Y)) = Xβ,
where g is the "link function". An example is the Poisson regression model, which states that
- Yi has a Poisson distribution with expected value eγ+δxi.
General linear model[edit | edit source]
See also[edit | edit source]
- 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.
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