Let
$ S \sim N(\mu, \sigma^2) $
be a normally distributed random variable with known $\mu$ and $\sigma^2$. Suppose, we observe
$ X = \begin{cases} T & \text{if $S \ge 0$}, \\ -T & \text{if $S<0$},\end{cases} $ where $T \in \mathbb{R}$. The probability distribution of $X$ is given by: $ p(x) = Q\left(\frac{-\mu}{\sigma}\right)\delta(x-T)+Q\left(\frac{\mu}{\sigma}\right)\delta(x+T) $
I want to optimize the value of $T$ such that $X$ conveys as much information about $S$ as possible.
My Attempt:
a. I tried minimizing the Kullback–Leibler divergence between the distribution of $X$ and $S$, but as mentioned here, it is not possible.
b. I tried to calculate the mutual information between the two distributions, it turned out to be independent of $\alpha$.
Is there any other way of formulating this problem? I feel quite confident that there must be such $T$ for which $X$ explains $S$ better, e.g., assume $\mu=10000$ then a value of $T$ near $10000$ will better explain $S$ than say $T=2$? One method in my mind was to match the moments of the two distributions but I am not sure if it is the optimal way in the sense of maximizing the information?