ABSTRACT

Non-linear aggregation strategies have recently been proposed in response to the problem of how to combine, in a non-linear way, estimators of the regression function (see for instance [Biau, G., Fischer, A., Guedj, B., and Malley, J. (2016), ‘COBRA: A Combined Regression Strategy’, Journal of Multivariate Analysis, 146, 18–28.]), classification rules (see [Cholaquidis, A., Fraiman, R., Kalemkerian, J., and Llop, P. (2016), ‘A Nonlinear Aggregation Type Classifier’, Journal of Multivariate Analysis, 146, 269–281.]), among others. Although there are several linear strategies to aggregate density estimators, most of them are hard to compute (even in moderate dimensions). Our approach aims to overcome this problem by estimating the density at a point x using not just sample points close to x but in a neighbourhood of the (estimated) level set $f\left(x\right)$. We show that the mean squared error of our proposal is at most equal to the mean squared error of the best density estimator used for the aggregation plus a second term that tends to zero. This fact is illustrated through a simulation study. A Central Limit Theorem is also proven.

摘要

响应于如何以非线性方式组合回归函数的估计量的问题，最近提出了非线性聚合策略（例如，参见[Biau，G.，Fischer，A.，Guedj，B. ，and Malley，J.（2016），“ COBRA：一种组合回归策略”，《多变量分析杂志》，第146页，第18-28页。），分类规则（请参阅[Cholaquidis，A.，Fraiman，R.，Kalemkerian， J.和Llop，P.（2016），“非线性聚合类型分类器”，《多元分析杂志》，第146页，第269-281页。]）等。尽管有几种线性策略可以汇总密度估计量，但大多数方法都难以计算（即使是中等尺寸）。我们的方法旨在通过估计点x处的密度来克服此问题。不仅使用接近x的采样点，而且使用（估计的）水平集的邻域$F（X）$。我们表明，我们的建议的均方误差最多等于用于聚合的最佳密度估计量的均方误差加上趋于零的第二项。通过仿真研究可以说明这一事实。还证明了中心极限定理。