Confidence Region - Weighted and Generalised Least Squares

Weighted and Generalised Least Squares

Now let us consider the more general case where some distinct elements of have known nonzero covariance (in other words, the errors in the observations are not independently distributed), and/or the standard deviations of the errors are not all equal. Suppose the covariance matrix of is, where V is an n-by-n nonsingular matrix which was equal to in the more specific case handled in the previous section, (where I is the identity matrix,) but here is allowed to have nonzero off-diagonal elements representing the covariance of pairs of individual observations, as well as not necessarily having all the diagonal elements equal.

It is possible to find a nonsingular symmetric matrix P such that

In effect, P is a square root of the covariance matrix V.

The least-squares problem

can then be transformed by left-multiplying each term by the inverse of P, forming the new problem formulation

where

and

A joint confidence region for the parameters, i.e. for the elements of, is then bounded by the ellipsoid given by:

(\mathbf{b} - \boldsymbol{\beta})^\prime \mathbf{Q}^\prime\mathbf{Q}(\mathbf{b} - \boldsymbol{\beta}) = {\frac{p}{n - p}} (\mathbf{Z}^\prime\mathbf{Z} - \mathbf{b}^\prime\mathbf{Q}^\prime\mathbf{Z})F_{1 - \alpha}(p,n-p).

Here F represents the percentage point of the F distribution and the quantities p and n-p are the degrees of freedom which are the parameters of this distribution.

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