鲁棒估计与拉丁超立方采样：Z估计量的中心极限定理

A recursive procedure for density estimation on the binary hypercube

This paper describes a recursive estimation procedure for multivariate binary densities (probability distributions of vectors of Bernoulli random variables) using orthogonal expansions. For $d$ covariates, there are $2^d$ basis coefficients to estimate, which renders conventional approaches computationally prohibitive when $d$ is large. However, for a wide class of densities that satisfy a certain sparsity condition, our estimator runs in probabilistic polynomial time and adapts to the unknown sparsity of the underlying density in two key ways: (1) it attains near-minimax mean-squared error fo…

A multivariate central limit theorem for randomized orthogonal array sampling designs in computer experiments

Let $f:[0,1)^d \to {\mathbb R}$ be an integrable function. An objective of many computer experiments is to estimate $\int_{[0,1)^d} f(x) dx$ by evaluating f at a finite number of points in [0,1)^d. There is a design issue in the choice of these points and a popular choice is via the use of randomized orthogonal arrays. This article proves a multivariate central limit theorem for a class of randomized orthogonal array sampling designs [Owen (1992a)] as well as for a class of OA-based Latin hypercubes [Tang (1993)].

Discrepancy Bounds for a Class of Negatively Dependent Random Points Including Latin Hypercube Samples

We introduce a class of $γ$-negatively dependent random samples. We prove that this class includes, apart from Monte Carlo samples, in particular Latin hypercube samples and Latin hypercube samples padded by Monte Carlo. For a $γ$-negatively dependent $N$-point sample in dimension $d$ we provide probabilistic upper bounds for its star discrepancy with explicitly stated dependence on $N$, $d$, and $γ$. These bounds generalize the probabilistic bounds for Monte Carlo samples from [Heinrich et al., Acta Arith. 96 (2001), 279--302] and [C.~Aistleitner, J.~Complexity 27 (2011), 531--540], and they …

使用变换的 Fourier-Walsh 对角化在二元超立方体上进行密度估计

Density Estimation on the Binary Hypercube using Transformed Fourier-Walsh Diagonalizations

本文提出一种基于 Fourier-Walsh 对角化与单调变换的通用二元密度估计框架，统一并推广了 Aitchison-Aitken 核，在保持正定性的同时实现速度与灵活性的灵活权衡。

用于计算机实验的序列好格点集

Sequential good lattice point sets for computer experiments

本文提出了一种基于好格点集构造序列拉丁超立方设计的新方法，解决了现有方法在大型设计和空间收缩场景下不够灵活或效率低下的问题，并证明了其具有更优的空间填充性、更低的计算复杂度和更强的适应性。

A central limit theorem for general orthogonal array based space-filling designs

Orthogonal array based space-filling designs (Owen [Statist. Sinica 2 (1992a) 439-452]; Tang [J. Amer. Statist. Assoc. 88 (1993) 1392-1397]) have become popular in computer experiments, numerical integration, stochastic optimization and uncertainty quantification. As improvements of ordinary Latin hypercube designs, these designs achieve stratification in multi-dimensions. If the underlying orthogonal array has strength $t$, such designs achieve uniformity up to $t$ dimensions. Existing central limit theorems are limited to these designs with only two-dimensional stratification based on streng…

Asymptotic properties of Bernstein estimators on the simplex

Bernstein estimators are well-known to avoid the boundary bias problem of traditional kernel estimators. The theoretical properties of these estimators have been studied extensively on compact intervals and hypercubes, but never on the simplex, except for the mean squared error of the density estimator in Tenbusch (1994) when $d = 2$. The simplex is an important case as it is the natural domain of compositional data. In this paper, we make an effort to prove several asymptotic results (bias, variance, mean squared error (MSE), mean integrated squared error (MISE), asymptotic normality, uniform…

从局部测量估计随机热方程的多变量变化

Multivariate change estimation for a stochastic heat equation from local measurements

本文提出了一种基于局部测量的随机热方程多变量变化估计方法，通过联合M估计器实现对扩散率和域的高精度估计，并在不同结构假设下展示了收敛速率和一致性。

Validating Sample Average Approximation Solutions with Negatively Dependent Batches

Sample-average approximations (SAA) are a practical means of finding approximate solutions of stochastic programming problems involving an extremely large (or infinite) number of scenarios. SAA can also be used to find estimates of a lower bound on the optimal objective value of the true problem which, when coupled with an upper bound, provides confidence intervals for the true optimal objective value and valuable information about the quality of the approximate solutions. Specifically, the lower bound can be estimated by solving multiple SAA problems (each obtained using a particular sampling…

通过投影私密学习超立方体上的光滑分布

Privately Learning Smooth Distributions on the Hypercube by Projections

本文针对d维超立方体上Sobolev光滑概率密度的中心私密估计问题，将一维结果推广到非整数光滑度与高维场景，还提出带新增惩罚项的自适应Lepskii方法，可在未知真实光滑度时私密选取最优投影估计器。

随机函数均值的最优推断

Optimal inference for the mean of random functions

针对超立方体上实值随机函数均值的估计与推断问题，提出基于傅里叶级数展开的最优速率估计器，建立L²范数下精确非渐近误差界等理论结果，并通过插入式估计器实现对均值函数Hölder正则性的自适应。

Robust Estimators in High Dimensions without the Computational Intractability

We study high-dimensional distribution learning in an agnostic setting where an adversary is allowed to arbitrarily corrupt an $\varepsilon$-fraction of the samples. Such questions have a rich history spanning statistics, machine learning and theoretical computer science. Even in the most basic settings, the only known approaches are either computationally inefficient or lose dimension-dependent factors in their error guarantees. This raises the following question:Is high-dimensional agnostic distribution learning even possible, algorithmically? In this work, we obtain the first computationall…