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.

翻译：我们认为所谓的unariate univariet 是不相关的( 有时在未知的回归曲线为单调时“ 脱钩 ”, “ ” 或“ 折叠 ” ) 回归。 在标准的单调回归中, 在标准单调回归中, 一个人看到一对一美元( X, Y) 的响应通过模型( Y= m_ 0( X) +\ epsilon$) 与共变美元挂钩的一对一对一美元( X, 美元) 。 在( 未知的) 单调( 单调) 回归功能和 $\ eepsilon 的未观察到的错误( 假设不受美元驱动 ) 。 在不相悖的回归中, 将一个实现的矢量( $_ 0 ) 和 美元 美元 的递增值连接成一对一对一美元 。 在假设 美元 美元 美元 中, 我们的递增量 数据 的分布方式是 。