Consider shifted step function: \begin{equation} f(x,\theta)=\begin{cases} 0 & x < \theta \\ 1 & x \geq \theta\end{cases} \end{equation}
Here $0 \leq \theta \leq 1$ is a parameter that determines the jump point. The goal is to find the "best" linear over-estimator of $f$. More precisely, for the line $ax+b$, we seek: \begin{equation} (a^*,b^*) = \arg\min_{a,b} \int_0^1 (ax+b-f(x))^2 dx \mbox{ s.t. } \forall x \in [0,1] ; ax+b \geq f(x) \end{equation}
Alternatively, we could use $\sup$ instead of $\ell_2$, \begin{equation} (a^*,b^*) = \arg\min_{a,b} \sup_{x \in [0,1]} |ax+b-f(x)| \mbox{ s.t. } \forall x \in [0,1] ; ax+b \geq f(x) \end{equation}
I am wondering if $(a^*,b^*)$ can be expressed in closed form as a function of $\theta$, for either $\ell_2$ or $\sup$ norms (ideally for both)?
Your help would be appreciated!
Golabi