论文标题
未链接单调回归
Unlinked monotone regression
论文作者
论文摘要
We consider so-called univariate unlinked (sometimes ``decoupled,'' or ``shuffled'') regression when the unknown regression curve is monotone.在标准单调回归中，人们观察到一对$（x，y）$，其中响应$ y $通过模型$ y = m_0（x） +ε$链接到协变量$ x $，带有$ m_0 $ $ m_0 $（未知）单调回归函数，$ε$ the nocked obsed误差（假定独立于$ x $）。在未链接回归设置中，只能从响应$ y $和协变量$ x $中观察一个实现向量，其中$ y \ y \ stackrel {d} {=} {=} m_0（x） +ε$。 $ x $和$ y $没有（观察到的）配对。尽管如此，实际上仍然可以在单调性$ m_0 $的假设下得出一致的非参数估计器$ M_0 $，并了解噪声$ε$的分布。在本文中，我们建立了对这种估计量的收敛速率，该估计量在对协变量$ x $的分布的最小假设下。我们讨论了噪声分布未知的情况的扩展。我们为其计算开发了二阶算法，并证明了其在合成数据上的使用。最后，我们在美国消费者支出调查的纵向数据上应用我们的方法（以完全数据驱动的方式，不了解错误分布）。
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 $Y= m_0(X) + ε$, with $m_0$ the (unknown) monotone regression function and $ε$ 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 $Y \stackrel{d}{=} m_0(X) + ε$. 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 $ε$. 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.
