Unlinked monotone regression

TL;DR

This paper introduces a method for estimating a monotone regression function from unlinked data where responses and covariates are not paired, providing theoretical guarantees, an efficient algorithm, and real data application.

Contribution

It develops the first consistent non-parametric estimator for unlinked monotone regression and analyzes its convergence rate under minimal assumptions.

Findings

Estimator achieves a provable convergence rate.

Algorithm effectively computes the estimator on synthetic data.

Method successfully applied to real longitudinal survey data.

Abstract

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.