论文标题
多元单调回归函数的近似，表征和连续性
Approximation, characterization, and continuity of multivariate monotonic regression functions
论文作者
论文摘要
我们处理多元函数的单调回归$ f：q \ to \ to \ mathbb {r} $上的紧凑型矩形域$ q $ q $ q $ q $ in $ \ mathbb {r}^d $，其中单调性是从某些协调方向和某些其他辅助方向上以同性恋的广泛性来理解的。像往常一样，给定功能$ f $的单调回归是单调函数$ f^*$，其均方根最小（加权）均方体距离$ f $。 We establish a simple general approach to compute monotonic regression functions: namely, we show that the monotonic regression $f^*$ of a given function $f$ can be approximated arbitrarily well -- with simple bounds on the approximation error in both the $2$-norm and the $\infty$-norm -- by the monotonic regression $f_n^*$ of grid-constant functions $f_n$.我们还建立了连续函数$ f $的单调回归$ f^*$的连续性，以及$ f^*$的显式平均公式。最后，我们处理了广义单调回归，在该回归中，与标准单调回归的均方距离被更复杂的距离测量所取代，例如，在最大的平滑似然估计中产生的距离。我们将看到，这种广义单调回归问题的解决方案仅由标准单调回归$ f^*$给出。
We deal with monotonic regression of multivariate functions $f: Q \to \mathbb{R}$ on a compact rectangular domain $Q$ in $\mathbb{R}^d$, where monotonicity is understood in a generalized sense: as isotonicity in some coordinate directions and antitonicity in some other coordinate directions. As usual, the monotonic regression of a given function $f$ is the monotonic function $f^*$ that has the smallest (weighted) mean-squared distance from $f$. We establish a simple general approach to compute monotonic regression functions: namely, we show that the monotonic regression $f^*$ of a given function $f$ can be approximated arbitrarily well -- with simple bounds on the approximation error in both the $2$-norm and the $\infty$-norm -- by the monotonic regression $f_n^*$ of grid-constant functions $f_n$. We also establish the continuity of the monotonic regression $f^*$ of a continuous function $f$ along with an explicit averaging formula for $f^*$. And finally, we deal with generalized monotonic regression where the mean-squared distance from standard monotonic regression is replaced by more complex distance measures which arise, for instance, in maximum smoothed likelihood estimation. We will see that the solution of such generalized monotonic regression problems is simply given by the standard monotonic regression $f^*$.
