Let's say I have a list of values that I got sampling the normally distributed random variables $[X_i]_{i=1}^{i=N}$ once (as in 1 value from each RV), and I know that $X_i\sim N(b\cdot k_i,\sigma^2)$ where $k_i\in \mathbb{N}, b\in\mathbb{R}_{\gt0}$ are unknown. (sigma may be known or not, it doesn't really matter in my case, just assume it's known)
I want to find $b$ and $k_i$ that minimize the value $max_i(E((X_i-b\cdot k_i)^2))$ (as in the maximum error across all i's)
for example for the following values: [$0.14791560173034668$, $0.3048287630081177$, $0.14991354942321777$, $0.7385870218276978$] we would get something along the lines of $b = 0.15$ and $k = \left[1,2,1,5\right]$
I know this problem isn't really well defined since it's possible to pick a really small $b$ and big $K$'s, but let's assume there is a plausible limit of the sort.
The reason I want the max instead of the sum is that for example the list [$0.1, 0.1, 0.1, 0.1, ....,0.15$] would give me $b\approx0.1$ for a large N. Maybe there is a better way to go about this problem instead of getting the max, if there is I'd be happy if anyone told me.
also if there is a formal name for a problem like this I'd like to know, since it's a bit similar to things I've previously seen.