论文标题
多种等渗回归和其他单调模型的置信区间
Confidence intervals for multiple isotonic regression and other monotone models
论文作者
论文摘要
我们考虑在多种同位素回归模型中构建点置置信区间的问题。最近，[HZ19]在该模型中获得了所谓的块最大块和最小估计器[FLN17]的点上极限分布理论，但是由于极限分布的滋扰参数涉及真实回归函数的多个未知的部分衍生剂，因此推断仍然是一个困难的问题。 在本文中，我们表明，可以通过利用超出块最大值和最小最大估计器中的点估计的信息来有效消除这种困难的滋扰参数。形式上，让$ \ hat {u}（x_0）$（分别为$ \ hat {v}（x_0）$）是最大化的下左端（分别最小化右上线）在块中最大值（max min-max）估计器和$ \ hat {f} _n _n $是块最大值的估计器和minmax minmin和minmin和minmin和minmin和minmin and Max Max的平均值。如果$ f_0 $的所有（一阶）部分衍生物在$ x_0 $上都不逐渐呈现，则以下关键限制分布理论：$$ \ sqrt {n _ {\ hat {u}，\ hat {v}}（x_0）} \ big（\ hat {f} _n（x_0）-f_0（x_0）-f_0（x_0）\ big）\ rientsquigarrow =quigigarrowσ $$这里$ n _ {\ hat {u}，\ hat {v}}（x_0）$是块中的设计点数量，$ [\ hat {u}（x_0），\ hat {v}（x_0）$，$σ$是错误的标准差，是$ rors and is a n nimently and in l} c {滋扰参数。这立即获得$ f_0（x_0）$的置信区间，并具有渐近的置信度和甲骨文长度。值得注意的是，置信区间的构建，即使在单变量环境中的新事物，除了一次使用块最大值和最小最大最大估计器进行等值频率回归外，不需要努力，并且可以轻松地适应其他常见的单调模型。进行广泛的模拟以支持我们的理论。
We consider the problem of constructing pointwise confidence intervals in the multiple isotonic regression model. Recently, [HZ19] obtained a pointwise limit distribution theory for the so-called block max-min and min-max estimators [FLN17] in this model, but inference remains a difficult problem due to the nuisance parameter in the limit distribution that involves multiple unknown partial derivatives of the true regression function. In this paper, we show that this difficult nuisance parameter can be effectively eliminated by taking advantage of information beyond point estimates in the block max-min and min-max estimators. Formally, let $\hat{u}(x_0)$ (resp. $\hat{v}(x_0)$) be the maximizing lower-left (resp. minimizing upper-right) vertex in the block max-min (resp. min-max) estimator, and $\hat{f}_n$ be the average of the block max-min and min-max estimators. If all (first-order) partial derivatives of $f_0$ are non-vanishing at $x_0$, then the following pivotal limit distribution theory holds: $$ \sqrt{n_{\hat{u},\hat{v}}(x_0)}\big(\hat{f}_n(x_0)-f_0(x_0)\big)\rightsquigarrow σ\cdot \mathbb{L}_{1_d}. $$ Here $n_{\hat{u},\hat{v}}(x_0)$ is the number of design points in the block $[\hat{u}(x_0),\hat{v}(x_0)]$, $σ$ is the standard deviation of the errors, and $\mathbb{L}_{1_d}$ is a universal limit distribution free of nuisance parameters. This immediately yields confidence intervals for $f_0(x_0)$ with asymptotically exact confidence level and oracle length. Notably, the construction of the confidence intervals, even new in the univariate setting, requires no more efforts than performing an isotonic regression for once using the block max-min and min-max estimators, and can be easily adapted to other common monotone models. Extensive simulations are carried out to support our theory.
