A note on kernel density estimation at a parametric rate
J. E. Chacón, J. Montanero, A. G. Nogales
发表信息：
Chacón, J.E., Montanero, J. and Nogales, A.G. (2007) A note on kernel density estimation at a parametric rate. Journal of Nonparametric Statistics, 19, 13-21
In the context of kernel density estimation, we give a characterization of the kernels for which the parametric mean integrated squared error rate $n^{-1}$ may be obtained, where $n$ is the sample size. Also, for the cases where this rate is attainable, we give an asymptotic bandwidth choice that makes the kernel estimator consistent in mean integrated squared error at that rate and a numerical example showing the superior performance of the superkernel estimator when the bandwidth is properly chosen.
We derive the optimal signed variable in general case kernels for the classical statistic density estimation, which are some generalization of the famous Epanechnikov's ones.
A note on extreme values and kernel estimators of sample boundaries
在前一篇文章中，我们研究了二维有界集上边缘的核估计，它是基于Poisson点过程的极值。最初的文件“Geffroy J.（1964）Sur un problème d 'estimation géométrique.Publications de l' Institut de Staubique de l 'Université de Paris，XIII，191-200”关于这个主题的处理边界作为边界的支撑集的密度和点作为一个随机样本。Girard，S.和Jacob，P.（2004）极值和点过程边界的核估计.ESAIM：概率和统计，8，150-168”，我们能够从点过程情况推导出随机样本情况。本说明为此提供了一些基本说明，包括一种可能引起普遍兴趣的方法。
We propose a new fully non-parametric two-step adaptive bandwidth selection method for kernel estimators of spatial point process intensity functions based on the Campbell-Mecke formula and Abramson's square root law. We present a simulation study to assess its performance relative to the Cronie-Van Lieshout global bandwidth selector and apply the technique to data on induced earthquakes in theGroningen gas field.
Multi-dimensional Hawkes process (MHP) is a class of self and mutually exciting point processes that find wide range of applications -- from prediction of earthquakes to modelling of order books in high frequency trading. This paper makes two major contributions, we first find an unbiased estimator for the log-likelihood estimator of the Hawkes process to enable efficient use of the stochastic gradient descent method for maximum likelihood estimation. The second contribution is, we propose a specific single hidden layered neural network for the non-parametric estimation of the underlying kerne…
University College London（伦敦大学学院）；University of Technology Sydney（悉尼科技大学）；Toyota Technological Institute at Chicago（芝加哥丰田技术研究院）
We propose a class of kernel-based two-sample tests, which aim to determine whether two sets of samples are drawn from the same distribution. Our tests are constructed from kernels parameterized by deep neural nets, trained to maximize test power. These tests adapt to variations in distribution smoothness and shape over space, and are especially suited to high dimensions and complex data. By contrast, the simpler kernels used in prior kernel testing work are spatially homogeneous, and adaptive only in lengthscale. We explain how this scheme includes popular classifier-based two-sample tests as…
Non-parametric goodness-of-fit testing procedures based on kernel Stein discrepancies (KSD) are promising approaches to validate general unnormalised distributions in various scenarios. Existing works focused on studying kernel choices to boost test performances. However, the choices of (non-unique) Stein operators also have considerable effect on the test performances. Inspired by the standardisation technique that was originally developed to better derive approximation properties for normal distributions, we present a unifying framework, called standardisation-function kernel Stein discrepan…
Estimating with kernel smoothers the mean of functional data in a finite population setting. A note on variance estimation in presence of partially observed trajectories
Hasso Plattner Institute for Digital Engineering（哈索·普拉特纳数字工程学院）；University of Kaiserslautern-Landau（凯泽斯劳滕-兰道大学）；Hasso Plattner Institute for Digital Health at the Icahn School of Medicine at Mount Sinai（西奈山伊坎医学院哈索·普拉特纳数字健康研究所）
We study optimal procedures for estimating a linear functional based on observational data. In many problems of this kind, a widely used assumption is strict overlap, i.e., uniform boundedness of the importance ratio, which measures how well the observational data covers the directions of interest. When it is violated, the classical semi-parametric efficiency bound can easily become infinite, so that the instance-optimal risk depends on the function class used to model the regression function. For any convex and symmetric function class $\mathcal{F}$, we derive a non-asymptotic local minimax b…
