Title: Unlinked monotone regression
Authors: Fadoua Balabdaoui - ETH Zurich (Switzerland)
Charles Doss - University of Minnesota (United States)
Cecile Durot - Univ. Paris Nanterre (France) [presenting]
Abstract: The so-called univariate unlinked (sometimes decoupled, or shuffled) regression is considered 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 only observes a vector of realizations from both the response $Y$ and the covariate $X$ where now, it is only known that $Y$ has the same distribution as $m_0(X) + \epsilon$. There is no (observed) pairing of $X$ and $Y$. Despite this, it is actually still possible to derive a consistent non-parametric estimator of $m_0$ under the assumption of monotonicity of $m_0$ and knowledge of the distribution of the noise $\epsilon$. We establish an upper bound on the rate of convergence of such an estimator under minimal assumptions on the distribution of the covariate $X$.