model, we see the process is very smooth for $\alpha=0.1$ and has little discernable spatial pattern with $\alpha= 0.9$.

![Random draws on a $50\times50$ grid from the random effect model for various $\alpha$. Each sample has a knot at each data point, GEV parameters $\mu({\mathbf{s}})=\log
[\sigma({\mathbf{s}})]=0$ and $\xi({\mathbf{s}})=-0.1$, and bandwidth $\tau=2$.[]{data-label="fsamples"}](591f01.eps)

The parameter $\alpha$ clearly plays an important role in this model. It determines the magnitude of the nugget effect, the form of spatial dependence function in (\[ECgeneral\]), and the shape and scale of the conditional distributions in (\[auxmodel\]). To illustrate the links between the contribution of $\alpha$ to these aspects of the model, we consider the extreme cases with $\alpha=0$ and $\alpha=1$. With $\alpha= 1$, $p(A|\alpha)$ concentrates its mass on $A=1$, and thus $\theta({\mathbf{s}})= \sum_{l=1}^Lw_{l}({\mathbf{s}})=1$. In this case, the conditional and marginal GEV parameters are the same, for example, $\mu^*({\mathbf{s}})= \mu({\mathbf{s}})$, there is no residual dependence with $\theta
({\mathbf{s}}_i,{\mathbf{s}}_j)=2$, and thus $Y({\mathbf{s}}){\stackrel{\mathrm{indep}}{\sim}}\operatorname{GEV}[\mu({\mathbf{s}}),\sigma({\mathbf{s}}),\xi({\mathbf{s}})]$. On the other hand, if $\alpha\approx0$, then the conditional scale $\sigma^*({\mathbf{s}})\approx0$ and $Y({\mathbf{s}}) \approx\mu^*({\mathbf{s}})$, a continuous spatial process with strong small-scale spatial dependence $\theta({\mathbf{s}},{\mathbf{s}})\approx1$.

Spatial prediction (analogous to Kriging) at a new location ${\mathbf{s}}^*$ is straight-forward under this hierarchical model. Predictions are made by simply computing $\theta({\mathbf{s}}^*)=[\sum_{l=1}^LA_{l}w_{l}({\mathbf{s}}^*)^{1/\alpha
