L. Douge
In this paper, the uniformly asymptotic normality for sample quantiles of associated random variables is investigated under some conditions on the decay of the covariances. We obtain the rate of normal approximation of order $O(n^{-1/2}\log^2 n)$ if the covariances decrease exponentially to $0$. The best rate is shown as $O(n^{-1/3})$ under a polynomial decay of the covariances.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
继续探索
New nonuniform Berry--Esseen-type bounds for sums of independent random variables are obtained, motivated by recent studies concerning such bounds for nonlinear statistics. The proofs are based on the Chen--Shao concentration techniques within the framework of the Stein method.
As an extension of a central limit theorem established by Svante Janson, we prove a Berry-Esseen inequality for a sum of independent and identically distributed random variables conditioned by a sum of independent and identically distributed integer-valued random variables.
Stratified Permutational Berry--Esseen Bounds and Their Applications to Statistics
本文通过Stein方法推导出适用于异质层的分层排列Berry-Esseen界，并应用于统计问题，提升理论量化和统计推断能力。
Berry-Esseen Theorem for Sample Quantiles with Locally Dependent Data
本文针对局部相依数据推导了样本分位数的 Gauss 中心极限定理，通过将问题转化为示性随机变量之和并应用 Stein 方法给出了明确的收敛速度，并将其推广至联合收敛。
A Chernoff-type distribution is a nonnormal distribution defined by the slope at zero of the greatest convex minorant of a two-sided Brownian motion with a polynomial drift. While a Chernoff-type distribution is known to appear as the distributional limit in many non-regular statistical estimation problems, the accuracy of Chernoff-type approximations has remained largely unknown. In the present paper, we tackle this problem and derive Berry-Esseen bounds for Chernoff-type limit distributions in the canonical non-regular statistical estimation problem of isotonic (or monotone) regression. The …
Let $H$ be a real separable Hilbert space and $(a_k)_{k\in\mathbb{Z}}$ a sequence of bounded linear operators from $H$ to $H$. We consider the linear process $X$ defined for any $k$ in $\mathbb{Z}$ by $X_k=\sum_{j\in\mathbb{Z}}a_j(\varepsilon_{k-j})$ where $(\varepsilon_k)_{k\in\mathbb{Z}}$ is a sequence of i.i.d. centered $H$-valued random variables. We investigate the rate of convergence in the CLT for $X$ and in particular we obtain the usual Berry-Esseen's bound provided that $\sum_{j\in\mathbb{Z}}\vert j\vert\|a_j\|_{\mathcal{L}(H)}<+\infty$ and $\varepsilon_0$ belongs to $L_H^{\infty}$.
Let $X_n=\sum_{i=1}^{\infty}a_iε_{n-i}$, where the $ε_i$ are i.i.d. with mean 0 and at least finite second moment, and the $a_i$ are assumed to satisfy $|a_i|=O(i^{-β})$ with $β>1/2$. When $1/2<β<1$, $X_n$ is usually called a long-range dependent or long-memory process. For a certain class of Borel functions $K(x_1,...,x_{d+1})$, $d\ge0$, from ${\mathcal{R}}^{d+1}$ to $\mathcal{R}$, which includes indicator functions and polynomials, the stationary sequence $K(X_n,X_{n+1},...,X_{n+d})$ is considered. By developing a finite orthogonal expansion of $K(X_n,...,X_{n+d})$, the Berry--Esseen type bo…
Berry-Esseen theorems for the asymptotic normality of incomplete U-statistics with Bernoulli sampling
本文研究了带有伯努利抽样的不完整U统计量在不同样本与计算预算规模下的渐近正态性，并建立了伯恩-艾森界以评估其正态近似精度。
The Berry-Esseen Bound for High-dimensional Self-normalized Sums
该研究针对高维自归一化统计量逐坐标最大值的高斯近似问题，在弱矩假设下推导出显式Berry-Esseen界，填补了高维CLT领域的相关研究空白。
Berry--Esseen theorems, LLT, Edgeworth expansions, and large deviations for Locally Stationary Markov Chains
本文针对局部平稳马尔可夫链的部分和，证明了Berry--Esseen定理、局部中心极限定理、Edgeworth展开及大偏差原理，并给出渐近均值和方差的显式公式及最优收敛速率。
Wasserstein-$1$ distance and nonuniform Berry-Esseen bound for a supercritical branching process in a random environment
本文研究了超临界随机环境中分支过程在Wasserstein-1距离下的收敛速率，并给出了非均匀Berry-Esseen界，同时探讨了其在临界性参数和种群规模估计中的应用。
Let $T>0,α>\frac12$. In the present paper we consider the $α$-Brownian bridge defined as $dX_t=-α\frac{X_t}{T-t}dt+dW_t,~ 0\leq t< T$, where $W$ is a standard Brownian motion. We investigate the optimal rate of convergence to normality of the maximum likelihood estimator (MLE) for the parameter $ α$ based on the continuous observation $\{X_s,0\leq s\leq t\}$ as $t\uparrow T$. We prove that an optimal rate of Kolmogorov distance for central limit theorem on the MLE is given by $\frac{1}{\sqrt{|\log(T-t)|}}$, as $t\uparrow T$. First we compute an upper bound and then find a lower bound with the …
Berry--Esseen bounds and bootstrap approximations for the Hilbert-space norm of $U$-statistics
本文针对非退化U统计量的希尔伯特空间范数，建立了非渐近Berry–Esseen界与三类自助近似，得到的检验具渐近正确性且分离率达极小极大最优。
Parameter estimation for fractional stochastic heat equations : Berry-Esséen bounds in CLTs
本文研究加性时空噪声驱动的分数阶随机热方程漂移系数的 MLE，分别在连续观测与均匀离散观测下建立中心极限定理的 Wasserstein 距离 Berry-Esséen 界及一致性、渐近正态性。
Incomplete U-Statistics of Equireplicate Designs: Berry-Esseen Bound and Efficient Construction
本文提出基于超图理论和组合设计的U统计量新方法，推导Berry-Esseen界并高效构造不完全U统计量，解决退化情况下的高斯极限分布问题。
Central Limit Theorem and Near classical Berry-Esseen rate for self normalized sums in high dimensions
本文研究高维自归一化和的中心极限定理及Berry-Esseen速率，证明多项式矩条件下的最优截断速率和最优Berry-Esseen速率。
Leave-One-Out Neighborhood Smoothing for Graphons: Berry--Esseen Bounds, Confidence Intervals, and Edge-Wise No-Leakage Tuning
本文提出留一法邻域平滑方法，通过解耦邻域选择与边平均，恢复条件独立性，从而实现对图ons中单个边概率的置信区间估计和诚实调参。
Asymptotic and finite-sample distributions of one- and two-sample empirical relative entropy
研究两个概率分布散度统计检验中经验相对熵的分布，推导有限样本集中不等式、渐近分布及贝里 - 埃森界，引入新方法获和统计量非线性函数的贝里 - 埃森不等式，涵盖单双样本经验相对熵。
A sum of observations derived by a simple random sampling design from a population of independent random variables is studied. A procedure finding a general term of Edgeworth asymptotic expansion is presented. The Lindeberg condition of asymptotic normality, Berry-Esseen bound, Edgeworth asymptotic expansions under weakened conditions and Cramer type large deviation results are derived.
We establish a Berry--Esseen bound for general multivariate nonlinear statistics by developing a new multivariate-type randomized concentration inequality. The bound is the best possible for many known statistics. As applications, Berry--Esseen bounds for M-estimators and averaged stochastic gradient descent algorithms are obtained.
In this paper, a new technique is introduced to obtain non-uniform Berry-Esseen bounds of normal and nonnormal approximation for unbounded exchangeable pairs. This technique does not rely on the concentration inequalities developed by Chen and Shao \cite{cls1, cls2} and can be applied to the quadratic forms, general Curie-Weiss model and an independence test. In particular, our non-uniform result about the independence test is under 6th moment condition, while the uniform bound in Chen and Shao \cite{cs2} requires 24th moment condition.
New Berry--Esseen-type bounds, with explicit constant factors, for the distribution of the Student statistic and, equivalently, for that of the self-normalized sum of independent zero-mean random variables are obtained. These bounds are compared with the corresponding existing results.
In this work, we provide a $(n/m)^{-1/2}$-rate finite sample Berry-Esseen bound for $m$-dependent high-dimensional random vectors over the class of hyper-rectangles. This bound imposes minimal assumptions on the random vectors such as nondegenerate covariances and finite third moments. The proof uses inductive relationships between anti-concentration inequalities and Berry--Esseen bounds, which are inspired by the telescoping method of Chen and Shao (2004) and the recursion method of Kuchibhotla and Rinaldo (2020). Performing a dual induction based on the relationships, we obtain tight Berry-E…
In this paper, we derive a necessary and sufficient condition on the parameters of the Hypergeometric distribution for weak convergence to a Normal limit. We establish a Berry-Esseen theorem for the Hypergeometric distribution solely under this necessary and sufficient condition. We further derive a nonuniform Berry-Esseen bound where the tails of the difference between the Hypergeometric and the Normal distribution functions are shown to decay at a sub-Gaussian rate.
We study accuracy of bootstrap procedures for estimation of quantiles of a smooth function of a sum of independent sub-Gaussian random vectors. We establish higher-order approximation bounds with error terms depending on a sample size and a dimension explicitly. These results lead to improvements of accuracy of a weighted bootstrap procedure for general log-likelihood ratio statistics. The key element of our proofs of the bootstrap accuracy is a multivariate higher-order Berry-Esseen inequality. We consider a problem of approximation of distributions of two sums of zero mean independent random…
Berry-Esseen Bounds and Moderate Deviations for Catoni-Type Robust Estimation
本文研究了Catoni型稳健估计器在重尾数据均值估计和线性回归中的应用，建立了伯恩-埃森界和中等偏差原理，解决了方差已知和未知情况下的问题。
Efficient Concentration with Gaussian Approximation
针对有界随机变量，提出具有渐近最优宽度、有限样本有效性和次高斯衰减的可计算集中不等式，通过零偏耦合和Stein交换对方法紧界非均匀Kolmogorov和Wasserstein距离，构建高效置信区间和经验Berry-Esseen界。
Refined Berry-Esseen bounds under local dependence
本文在局部依赖下通过Stein方法和新集中不等式，改进了自归一化和非自归一化和的Berry-Esseen界，并应用于图依赖等场景。
Berry-Esseen Bounds for Statistics of Non-Stationary, $ϕ$-Mixing Random Variables
通过改进的斯蒂因方法，将Bentkus等人结果推广到非平稳ϕ-混合随机变量序列的广泛统计量，推导伯恩-艾森界，并应用于非平稳ϕ-混合随机变量的伯恩-艾森速率分析。
Let $\{X_n\}_{n\ge0}$ be a $V$-geometrically ergodic Markov chain. Given some real-valued functional $F$, define $M_n(α):=n^{-1}\sum_{k=1}^nF(α,X_{k-1},X_k)$, $α\in\mathcal{A}\subset \mathbb {R}$. Consider an $M$ estimator $\hatα_n$, that is, a measurable function of the observations satisfying $M_n(\hatα_n)\leq \min_{α\in\mathcal{A}}M_n(α)+c_n$ with $\{c_n\}_{n\geq1}$ some sequence of real numbers going to zero. Under some standard regularity and moment assumptions, close to those of the i.i.d. case, the estimator $\hatα_n$ satisfies a Berry--Esseen theorem uniformly with respect to the under…
New Berry-Esseen bounds for parameter estimation of Gaussian processes observed at high frequency
本文提出了一种在高频观测下估计高斯过程参数的新方法，通过改进的Berry-Esseen界提升了估计精度。
Berry-Esseen bounds for multivariate martingale difference sequences in the Kolmogorov distance
该研究推导了Kolmogorov距离下多元鞅差序列的Berry-Esseen界，其界与序列长度呈$n^{-1/4}$阶、与维度呈$\text{polylog}(d)$阶相关，并将其应用于马尔可夫链生成的鞅序列的高维超矩形Berry-Esseen界推导。
Let $Z:=\{Z_t,t\geq0\}$ be a stationary Gaussian process. We study two estimators of $\mathbb{E}[Z_0^2]$, namely $\widehat{f}_T(Z):= \frac{1}{T} \int_{0}^{T} Z_{t}^{2}dt$, and $\widetilde{f}_n(Z) :=\frac{1}{n} \sum_{i =1}^{n} Z_{t_{i}}^{2}$, where $ t_{i} = i Δ_{n}$, $ i=0,1,\ldots, n $, $Δ_{n}\rightarrow 0$ and $ T_{n} := n Δ_{n}\rightarrow \infty$. We prove that the two estimators are strongly consistent and establish Berry-Esseen bounds for a central limit theorem involving $\widehat{f}_T(Z)$ and $\widetilde{f}_n(Z)$. We apply these results to asymptotically stationary Gaussian processes an…
In this note, we provide a Berry--Esseen bounds for rectangles in high-dimensions when the random vectors have non-singular covariance matrices. Under this assumption of non-singularity, we prove an $n^{-1/2}$ scaling for the Berry--Esseen bound for sums of mean independent random vectors with a finite third moment. The proof is essentially the method of compositions proof of multivariate Berry--Esseen bound from Senatov (2011). Similar to other existing works (Kuchibhotla et al. 2018, Fang and Koike 2020a), this note considers the applicability and effectiveness of classical CLT proof techniq…
The inductive size bias coupling technique and Stein's method yield a Berry-Esseen theorem for the number of urns having occupancy $d \ge 2$ when $n$ balls are uniformly distributed over $m$ urns. In particular, there exists a constant $C$ depending only on $d$ such that $$ \sup_{z \in \mathbb{R}}|P(W_{n,m} \le z) -P(Z \le z)| \le C \left( \frac{1+(\frac{n}{m})^3}{σ_{n,m}} \right) \quad \mbox{for all $n \ge d$ and $m \ge 2$,} $$ where $W_{n,m}$ and $σ_{n,m}^2$ are the standardized count and variance, respectively, of the number of urns with $d$ balls, and $Z$ is a standard normal random variab…
Moderate Deviation and Berry-Esseen Bounds in the $p$-Spin Curie-Weiss Model
本文针对 $p$-spin Curie-Weiss 模型，证明了磁化强度的 Cramér 型中偏差定理与 Berry-Esseen 界，给出了高斯与非高斯波动下的收敛速度，并推导了逆温度最大伪似然估计的 Berry-Esseen 界。
Nonuniform Berry-Esseen bounds for Studentized U-statistics
本文针对学生化 U-统计量，在三阶矩条件下证明速率为 $1/\sqrt{n}$ 的非一致 Berry-Esseen 界，并通过加入随 $n$ 指数衰减的修正项解决既有界失效的问题。
We provide finite sample bounds on the Normal approximation to the law of the least squares estimator of the projection parameters normalized by the sandwich-based standard errors. Our results hold in the increasing dimension setting and under minimal assumptions on the data generating distribution. In particular, we do not assume a linear regression function and only require the existence of finitely many moments for the response and the covariates. Furthermore, we construct confidence sets for the projection parameters in the form of hyper-rectangles and establish finite sample bounds on the…
An optimal bound on the quantiles of a certain kind of distributions is given. Such a bound is used in applications to Berry--Esseen-type bounds for nonlinear statistics.
Improved Berry-Esseen bounds for multivariate nonlinear statistics in convex distance
本文针对多变量非线性统计量在凸距离下建立两个非渐近Berry-Esseen界，改进维数增长条件，并应用于随机逼近、时间差分学习和U-统计量，给出显式高斯近似误差及收敛充分条件。
We study rates of convergence in central limit theorems for the partial sum of squares of general Gaussian sequences, using tools from analysis on Wiener space. No assumption of stationarity, asymptotically or otherwise, is made. The main theoretical tool is the so-called Optimal Fourth Moment Theorem \cite{NP2015}, which provides a sharp quantitative estimate of the total variation distance on Wiener chaos to the normal law. The only assumptions made on the sequence are the existence of an asymptotic variance, that a least-squares-type estimator for this variance parameter has a bias and a va…
Another look at Stein's method for Studentized nonlinear statistics with an application to U-statistics
该研究重新探讨用Stein方法建立学生化非线性统计量的一致Berry-Esseen界，明确变量删失与删失变量和的指数型随机集中不等式为核心工具，并给出学生化U-统计量的一致Berry-Esseen界，体现其对核阶数的依赖。
Sharp Berry-Esseen Bounds for the Log Determinant of a Gaussian Sample Correlation Matrix
该研究针对高斯样本相关矩阵对数行列式，推导了零相关下的埃奇沃思展开式与统一的Kolmogorov等价式，还证明了正定总体相关矩阵下的有限样本Berry-Esseen界，且相关结果有Lean 4等价公式并经内核验证。
Linear wavelet density estimators are wavelet projections of the empirical measure based on independent, identically distributed observations. We study here the law of the iterated logarithm (LIL) and a Berry-Esseen type theorem. These results are proved under different assumptions on the density $f$ that are different from those needed for similar results in the case of convolution kernels (KDE): whereas the smoothness requirements are much less stringent than for the KDE, Riemann integrability assumptions are needed in order to compute the asymptotic variance, which gives the scaling constan…
In the so called lightbulb process, on days $r=1,..., n$, out of $n$ lightbulbs, all initially off, exactly $r$ bulbs, selected uniformly and independent of the past, have their status changed from off to on, or vice versa. With $X$ the number of bulbs on at the terminal time $n$, an even integer, and $μ=n/2, σ^2=Var(X)$, we have $$ \sup_{z \in \mathbb{R}} |P(\frac{X-μ}σ \le z)-P(Z \le z)| \le \frac{n}{2σ^2} \barΔ_0 + 1.64 \frac{n}{σ^3}+ \frac{2}σ $$ where $Z$ is a standard normal random variable, and $$ \barΔ_0 = 1/2\sqrt{n}} + \frac{1}{2n} + 1/3 e^{-n/2} \qmq {for $n \ge 6$,} $$ yielding a b…
扫码添加微信好友，提出您的宝贵建议 👇
💡 备注请填写：网站反馈
发表机构由 AI 从论文文本抽取，如发现错误，欢迎反馈修正建议。