How to find $\phi$, that denotes the correlation of signals among informed traders? Since I do not have an answer on Quantitative Finance in my question I cross-post here the problem to tag some other categories
The following assumptions are part of the paper of Back, Chao and Willard and I can not solve for the statistic that is denoted as $\phi$ in the sequel. I would be glad if anyone could help me. Below i set the assumptions and the equations of interest
Suppose that in the market, there are $N\geq 1$ informed agents, who trade a risky asset continuously in the time interval $[0,1)$. Each agent $i$ receives a mean-zero signal $\tilde{s}^i$ at time 0. We assume the signals and the liquidation value of the asset have a nondegenerate joint normal distribution that is symmetric in the signals. Symmetry means that the joint distribution of the asset value and the signals $\tilde{s}^1,...,\tilde{s}^N$ is invariant to a permutation of the indices $1,...,N$. Let $\tilde{v}$ denote the expectation of the liquidation value conditional on the combined information of the informed traders. By normality, $\tilde{v}$ is an affine function of the $\tilde{s}^i$. By rescaling the $\tilde{s}^i$  if necessary, we can assume without loss of generality that
\begin{equation}\tilde{v}=\bar{v}+\Sigma^{N}_{i=1} s^i\end{equation}
for a constant $\bar{v}$. For simplicity, we assume $\bar{v}=0$. Let
\begin{align}\phi=\frac{var(\tilde{v})}{var(N\tilde{s}^i)}\end{align}
The statistic $\phi$ is a measure of the quality of each agent’s information. Specifically, it is the $R^2$ in the linear regression of $\tilde{v}$ on $\tilde{s}^i$, that is, it is the percentage of the variance in $\tilde{v}$ that is explained by the trader’s information.
Letting $\rho$ denote the correlation coefficient of $\tilde{s}^i$ with $\tilde{s}^j$ for $i\neq j$, one can compute $\phi$ for $N&gt;1$ as
\begin{equation}\phi=\frac{1}{N}+\frac{N-1}{N}\rho\end{equation}
If $\phi=1$, then either $N=1$ or the $\tilde{s}^i$ are perfectly corellated. In either case each informed trader has perfect information about $\tilde{v}$.
My questions are the following

what does it mean intuitively &quot;a nondegenerate joint normal distribution&quot; and in particular I would like to understand the term nondegenerate.
What does it mean &quot;invariant to indices&quot; ?
the liquidation value is equal to the sume of the signals, does this come from the assumption that it it an affine function of the $\tilde{s}^i$?
How do we find that measure $\phi$? is it from the linear regression of $\tilde{v}$ on $\tilde{s}^i$?
How $\phi$ is tranformed to
\begin{equation}\phi=\frac{1}{N}+\frac{N-1}{N}\rho\end{equation}

Here it is a link from the paper
A degenerate joint normal is distribution is one in which you cannot find a PDF for the distribution. They assume you can. (The covariance matrix is invertible).

Let $f(s_1,s_2\dots,s_n,v)$ be the distribution.
If I was to exchange $s_1$ for $s_2$, $f(s_2,s_1,\dots,s_n,v) = f(s_1,s_2\dots,s_n,v)$, the distribution does not change. And you can exchange as many indices (signals) as you'd like.

The assumption that the distribution is jointly normal implies $\tilde{v}$ is an affine function of $\tilde{s}_i$, which then implies the sum of signals.

Regress $\tilde{v}$ on $\tilde{s}_i$ and calculate $R^2$ (the Coefficient of determination).

This is something I hope someone else can answer.