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Linear model

In statistics the linear model can be expressed by saying
  • Y=Xβ+ε
where Y is an nx1 column vector of random variables, X is an nxp matrix of "known" (i.e., observable and non-random) quanitities, whose rows correspond to statistical units, β is a px1 vector of (unobservable) parameters, and ε is an nx1 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 least squares.

If, rather than taking the variance of ε to be σ2I, where I is the nxn 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. 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.

"Generalized linear models", rather than saying

  • E(Y)=Xβ,
say
  • f(E(Y))=Xβ,
where f is the "link function". An example is the "Poisson regression model", which says
  • Yi has a Poisson distribution with expected value eγ+δxi.
The link function is the natural logarithm function. Having observed xi and Yi for i=1,...,n, one can estimate γ and δ by the method of maximum likelihood.



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