Everything about Linear Model totally explained
In
statistics the
linear model is given by
»
where
Y is an
n×1 column vector of random variables,
X is an
n×
p matrix of "known" (for example 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 σ
2.
Much of the theory of linear models is associated with inferring the values of the
parameters β and σ
2. Typically this is done using 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.
Assumptions
Multivariate normal errors
Often one takes the components of the vector of errors to be
independent and
normally distributed, giving
Y a
multivariate normal distribution with mean
Xβ and co-variance matrix σ
2 I, where
I is the
identity matrix. Having observed the values of
X and
Y, the statistician must estimate β and σ
2.
Rank of X
We usually assume that
X is of full
rank p, which allows us to invert the
p ×
p matrix
which is the
best linear unbiased estimator for
. If all of the off-diagonal entries in the matrix
Ω are 0, then one normally estimates β by the method of
weighted least squares, with weights proportional to the reciprocals of the diagonal entries. The GLS estimator is also known as the
Aitken estimator, after
Alexander Aitken, the Professor in the
University of Otago Statistics Department who pioneered it.
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". The variance is also not restricted to being normal.
An example is the
Poisson regression model, which states that
» 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.
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.
Further Information
Get more info on 'Linear Model'.
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