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If I consider universal kriging (or multiple spatial regression) in matrix form as:

${\bf{V = XA + R }}$

where $\bf{R}$ is the residual and $\bf{A}$ are the trend coefficients, then the estimate of ${\bf{\hat A}}$ is:

${\bf{\hat A}}=(\bf{X^{T}C^{-1}X)^{-1}X^{T}C^{-1}V}$

(as I understand it), where $\bf{C}$ is the covariance matrix, if it is known. Then, the variance of the coefficients is:

$\text{VAR}({\bf{\hat A}})=(\bf{X^{T}C^{-1}X)^{-1}}$???

I am getting this from here.

How does one get from the estimate of ${\bf{\hat A}}$, to its variance? i.e. how can I derive that variance?

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    Yes, you are probably right. I'm coming from a geostatistics background where we model a random variable (i.e. rock porosity) often by separating the data into a trend component and a residual component (I have seen it referred to as both error and residual). The trend component is modelled deterministically and the residual is modelled either deterministically or stochastically.2012-06-28

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$\newcommand{\var}{\operatorname{var}}$ First, recall that $ \var(MV) = M\Big(\var(V)\Big)M^T. $ so $ \begin{align} & \var((X^T C^{-1}X)^{-1} X^T C^{-1}V) \\[10pt] & = (X^T C^{-1}X)^{-1} X^T C^{-1}\Big(\var{V}\Big)\Big( (X^T C^{-1}X)^{-1} X^T C^{-1} \Big)^T. \tag{1} \end{align} $

Then, recall that $(AB)^T$ (with $A$ to the left of $B$) is equal to $B^T A^T$ (with $A$ to the right of $B$). With $X^T C^{-1} X$, one cannot invert all three matrices and multiply in the opposite order, since $X$ is not a square matrix. But that matrix is symmetric, i.e. it is its own transpose. And $C$ is also symmetric, and so is $C^{-1}$. So we get: $ \Big( (X^T C^{-1}X)^{-1} X^T C^{-1} \Big)^T = C^{-1}X(X^TC^{-1}X)^{-1}. $ Then $(1)$ becomes $ \begin{align} & (X^T C^{-1}X)^{-1} X^T C^{-1}\Big(\var{V}\Big) C^{-1}X(X^TC^{-1}X)^{-1} \\[10pt] & = (X^T C^{-1}X)^{-1} X^T C^{-1}\Big( C \Big) C^{-1}X(X^TC^{-1}X)^{-1} \\[10pt] & = (X^T C^{-1}X)^{-1} X^T C^{-1} X(X^TC^{-1}X)^{-1} \\[10pt] & = (X^T C^{-1}X)^{-1}. \end{align} $

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    Sadly, I do not have the reputation to up-vote it. I would up-vote it twice if I could.2012-06-28