Unlinked monotone regression

Understand

We consider so-called univariate unlinked (sometimes ``decoupled,'' or ``shuffled'') regression when the unknown regression curve is monotone.

• In standard monotone regression, one observes a pair $(X,Y)$ where a response $Y$ is linked to a covariate $X$ through the model $Y= m_0(X) + \epsilon$, with $m_0$ the (unknown) monotone regression function and $\epsilon$ the unobserved error (assumed to be independent of $X$).
• In the unlinked regression setting one gets only to observe a vector of realizations from both the response $Y$ and from the covariate $X$ where now $Y \stackrel{d}{=} m_0(X) + \epsilon$.
• There is no (observed) pairing of $X$ and $Y$.