\begin{align}
    q_r &= \frac{\sqrt{a^2 \alpha ^2+2 a \alpha  (\lambda -a) \nu _r+(a+\lambda )^2 \nu _r^2}-a \alpha +(a-\lambda )
   \nu _r}{2 \nu _r} \label{q_equicorr}\\
    \hat{q}_r &=\frac{\sqrt{a^2 \alpha ^2+2 a \alpha  (\lambda -a) \nu _r+(a+\lambda )^2 \nu _r^2}+a \alpha -(a+\lambda )
   \nu _r}{2 a \lambda } \label{qhat_equicorr}
\end{align}
