I'm trying to tackle the following integral:
$$\int_{\mathbb{S}^{d-1}}{ (\sqrt{\theta^T\Sigma \theta} - \sqrt{\theta^T \hat \Sigma \theta} )^2 d\theta }$$
Where $\mathbb{S}^{d-1} =\{ \theta \in \mathbb{R}^d: \|\theta\|=1\}$, $\Sigma$ and $\hat \Sigma$ are covariance matrices, so they are symmetric and PSD.
Expanding the square, each of the squared terms can be computed as:
$$\int_{\mathbb{S}^{d-1}}{ \theta^T\Sigma \theta} \ d\theta = \frac{| \mathbb{S}^{d-1}|}{d} tr(\Sigma).$$
But I can't make any progress on the product term, i.e.:
$$- 2 \int_{\mathbb{S}^{d-1}}{ \sqrt{\theta^T\Sigma \theta} \sqrt{\theta^T \hat \Sigma \theta} d\theta }.$$
Any help would be appreciated!