Calculating the smoothness of Z $_\mathbf{0}$

The estimate of the covariance matrix for the residual field, $\hat \Lambda_\epsilon$, needs to be transformed to $Z_0$ for calculating the probabilities of clusters found in the $Z$-statistic image. The estimate of the covariance matrix for the $Z_0$ field, $\hat\Lambda_{Z_0}$ can be computed by

$$\hat\Lambda_{Z_0} = \lambda_\nu \cdot \hat\Lambda_\epsilon,$$ (22)

where

$$\lambda_\nu = \int_{-\infty}^{\infty} \frac{(t^2+\nu-1)^2}{(\nu - 1)(\nu - 2)} \frac{T_\nu(t)^3}{p(t)^2} dt,$$ (23)

where $T_\nu$ is the PDF of a $t$-distribution with $\nu$ degrees of freedom and

$$p(t) = \phi(\Phi^{-1}(1 - \Phi_\nu(t))),$$ (24)

where $\phi(z)$ is the PDF of the standard normal distribution.