Unlinked Monotone Regression

Fadoua Balabdaoui; Charles R. Doss; Cécile Durot

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

$Y$ . 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

$\epsilon$ . In this paper, we establish an upper bound on the rate of convergence of such an estimator under minimal assumption on the distribution of the covariate

$X$ . We discuss extensions to the case in which the distribution of the noise is unknown. We develop a second order algorithm for its computation, and we demonstrate its use on synthetic data. Finally, we apply our method (in a fully data driven way, without knowledge of the error distribution) on longitudinal data from the US Consumer Expenditure Survey.

Unlinked Monotone Regression

Abstract