is generally positive, indicating a right-skewed distribution with no upper bound. The estimated shape is the largest in Florida. Comparing the posterior means and standard deviations, there is evidence that all three GEV parameters vary spatially. Figure \[fshapescale\] shows that there is strong positive dependence between the shape and scale as one might expect, since for shape in $(0,0.5)$ both the mean and variance of GEV includes the ratio of the scale and shape. For locations with large shapes, there is a negative dependence with the log scale.

![Plot of the posterior mean GEV scale versus posterior mean GEV shape at each site.[]{data-label="fshapescale"}](591f06.eps)

To formally assess the need for spatially-varying GEV parameters, we also refit the model for the historical simulation using the Bayesian variable selection prior of @reich-2010 to test whether the variance $\delta_{j}^2$ equals small constant $\Delta_j^2=0.01^2$. The test is carried out using the mixture prior $\delta_{j} = g_j\delta_j^*
+ (1-g_j)\Delta_0$, where $g_j\sim \operatorname{Bernoulli}(0.5)$ and $\delta_j^{*2}\sim \operatorname{InvGamma}(0.1,0.1$). The intuition behind this prior is that if $g_j=1$, then $\delta_{j}^2\sim \operatorname{InvGamma}(0.1,0.1)$ and the GEV parameter varies spatially; in contrast, if $g_0=0$, then $\delta_{j}^2=\Delta_j^2$, and spatial variation after accounting for spatial covariates $\mathbf{x}$ is negligible. Therefore, the posterior mean of $g_j$ 