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I'm taking a spatial stats class, and I'm on the road so I can't ask the prof for help. Would appreciate help understanding what is going on here.

The problem is set up with $Y = X_s'\beta + e$ (with $s$ indexing coordinates). $e$ is assumed to be second-order stationary, with the variogram

$$ \gamma(h) = \left\{\begin{array}{l r} 0 & h=0\\ \tau^2 + \sigma^2\left(\frac{3h}{2\rho}-\frac{h^3}{2\rho^3}\right) & 0 < h \leq \rho\\ \tau^2 + \sigma^2 & h > \rho\\ \end{array}\right. $$ where $h$ is distance between two points. For second-order stationary processes, apparently it is true that $C(h) = lim_{h' \rightarrow \infty} \gamma(h') - \gamma(h)$, where $C(h)$ is a covariance function, which is only a function of distance, because the processes is second-order stationary. This means that the covariance function is

$$ C(h) = \left\{\begin{array}{l r} \tau^2 + \sigma^2 & h=0\\ \sigma^2\left(1-\left(\frac{3h}{2\rho}-\frac{h^3}{2\rho^3}\right)\right) & 0 < h \leq \rho\\ 0 & h > \rho\\ \end{array}\right. $$ which makes intuitive sense.

Now for the trickier part (for me anyway): we are told to assume that $e$ is a gaussian process. We are asked if we might calculate a profile likelihood as a function only of $\rho$.

The likelihood function is that of a standard multivariate normal.

My question: how do I go about the algebra to (either) find a profile likelihood that is only a function of $\rho$, or perhaps a function of $\rho$ and $\tau$? If the latter, how do I deal with the fact that the parameter $\tau$ will show up on the diagonals of the covariance matrix, but not on the off diagonals? It seems like it makes the algebra intractable.

Hints appreciated.

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So, the answer to my own question occurred to me after sleeping on it, as it often does. The covariance function up there in the question is equivalent to $$C(h) = \tau^2I + \sigma^2\left(1-\left(\frac{3h}{2\rho}-\frac{h^3}{2\rho^3}\right)\right)$$

which can be tractably profiled into a bivariate likelihood function.

For posterity.

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