We have a random variable, $\mathbf{X} \in \mathbb{R}^+$ such that $\min_X \leq \mathbf{X} \leq \max_X$. Let $\mathbf{x_i}$ be the $i^{th}$ draw of the R.V $\mathbf{X}$. Now, we define a new R.V $\mathbf{Y}$ such that:
$\mathbf{y}_i = \left\{ \begin{array}{ c l } \ \mathbf{x_i} - thresh & \quad \textrm{if } \mathbf{x}_i \geq thresh \\ 0 & \quad \textrm{otherwise} \end{array} \right.$
I need to choose a $thresh$ that maximises the variance across all $\mathbf{y_i}$. Additional condition is that $thresh \in (\min_X, \max_X) $. So my questions are:
Part 1. Can we get an estimate for the parameter $thresh$ without making any assumptions on the distribution of $\mathbf{X}$ apart from the knowledge of $\min_X$ and $\max_X$?
Part 2. If Part 1 is not possible, then can we get an estimate for $thresh$ by assuming $\mathbf{X}$ to have a gaussian distribution with ($\mu$, $\sigma$)
I am not sure if I should post this as two questions.