Bernoulli最新文献_第7页

Approximation of occupation time functionals 占用时间泛函的逼近

IF 1.5 2区数学

Bernoulli Pub Date : 2021-11-01 DOI: 10.3150/21-BEJ1328

R. Altmeyer

引用次数: 9

Cramér-type moderate deviation for quadratic forms with a fast rate 快速二次型的Cramér型中偏差

IF 1.5 2区数学

Bernoulli Pub Date : 2021-11-01 DOI: 10.3150/22-bej1549

Xiao Fang, Song Liu, Q. Shao

{"title":"Cramér-type moderate deviation for quadratic forms with a fast rate","authors":"Xiao Fang, Song Liu, Q. Shao","doi":"10.3150/22-bej1549","DOIUrl":"https://doi.org/10.3150/22-bej1549","url":null,"abstract":"Let $X_1,dots, X_n$ be independent and identically distributed random vectors in $mathbb{R}^d$. Suppose $mathbb{E} X_1=0$, $mathrm{Cov}(X_1)=I_d$, where $I_d$ is the $dtimes d$ identity matrix. Suppose further that there exist positive constants $t_0$ and $c_0$ such that $mathbb{E} e^{t_0|X_1|}leq c_0<infty$, where $|cdot|$ denotes the Euclidean norm. Let $W=frac{1}{sqrt{n}}sum_{i=1}^n X_i$ and let $Z$ be a $d$-dimensional standard normal random vector. Let $Q$ be a $dtimes d$ symmetric positive definite matrix whose largest eigenvalue is 1. We prove that for $0leq xleq varepsilon n^{1/6}$, begin{equation*} left| frac{mathbb{P}(|Q^{1/2}W|>x)}{mathbb{P}(|Q^{1/2}Z|>x)}-1 right|leq C left( frac{1+x^5}{det{(Q^{1/2})}n}+frac{x^6}{n}right) quad text{for} dgeq 5 end{equation*} and begin{equation*} left| frac{mathbb{P}(|Q^{1/2}W|>x)}{mathbb{P}(|Q^{1/2}Z|>x)}-1 right|leq C left( frac{1+x^3}{det{(Q^{1/2})}n^{frac{d}{d+1}}}+frac{x^6}{n}right) quad text{for} 1leq dleq 4, end{equation*} where $varepsilon$ and $C$ are positive constants depending only on $d, t_0$, and $c_0$. This is a first extension of Cram'er-type moderate deviation to the multivariate setting with a faster convergence rate than $1/sqrt{n}$. The range of $x=o(n^{1/6})$ for the relative error to vanish and the dimension requirement $dgeq 5$ for the $1/n$ rate are both optimal. We prove our result using a new change of measure, a two-term Edgeworth expansion for the changed measure, and cancellation by symmetry for terms of the order $1/sqrt{n}$.","PeriodicalId":55387,"journal":{"name":"Bernoulli","volume":" ","pages":""},"PeriodicalIF":1.5,"publicationDate":"2021-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44915080","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 1

Partial generalized four moment theorem revisited 对部分广义四矩定理的再认识

IF 1.5 2区数学

Bernoulli Pub Date : 2021-11-01 DOI: 10.3150/20-BEJ1310

Dandan Jiang, Z. Bai

引用次数: 3

Bounds for the chi-square approximation of Friedman’s statistic by Stein’s method 用斯坦方法求得弗里德曼统计量的卡方近似的边界

IF 1.5 2区数学

Bernoulli Pub Date : 2021-11-01 DOI: 10.3150/22-bej1530

Robert E. Gaunt, G. Reinert

引用次数: 8

Over-parametrized deep neural networks minimizing the empirical risk do not generalize well 过度参数化的深度神经网络使经验风险最小化，但泛化效果不佳

IF 1.5 2区数学

Bernoulli Pub Date : 2021-11-01 DOI: 10.3150/21-BEJ1323

M. Kohler, A. Krzyżak

引用次数: 11

Minimax semi-supervised set-valued approach to multi-class classification 多类分类的极大极小半监督集值方法

IF 1.5 2区数学

Bernoulli Pub Date : 2021-11-01 DOI: 10.3150/20-BEJ1313

Evgenii Chzhen, Christophe Denis, Mohamed Hebiri

{"title":"Minimax semi-supervised set-valued approach to multi-class classification","authors":"Evgenii Chzhen, Christophe Denis, Mohamed Hebiri","doi":"10.3150/20-BEJ1313","DOIUrl":"https://doi.org/10.3150/20-BEJ1313","url":null,"abstract":"We study supervised and semi-supervised algorithms in the set-valued classification framework with controlled expected size. While the former methods can use only n labeled samples, the latter are able to make use of N additional unlabeled data. We obtain semi-supervised minimax rates of convergence under the α-margin assumption and a β-Hölder condition on the conditional distribution of labels. Our analysis implies that if no further assumption is made, there is no supervised method that outperforms the semi-supervised estimator proposed in this work – the best achievable rate for any supervised method is O(n−1/2), even if the margin assumption is extremely favorable; on the contrary, the developed semi-supervised estimator can achieve faster O((n/ logn)−(1+α)β/(2β+d)) rate of convergence provided that sufficiently many unlabeled samples are available. We also show that under additional smoothness assumption, supervised methods are able to achieve faster rates and the unlabeled sample cannot improve the rate of convergence. Finally, a numerical study supports our theory and emphasizes the relevance of the assumptions we required from an empirical perspective.","PeriodicalId":55387,"journal":{"name":"Bernoulli","volume":" ","pages":""},"PeriodicalIF":1.5,"publicationDate":"2021-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47747471","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 5

Estimating the inter-occurrence time distribution from superposed renewal processes 从叠加更新过程中估计事件间时间分布

IF 1.5 2区数学

Bernoulli Pub Date : 2021-11-01 DOI: 10.3150/21-BEJ1331

Xiaoyu Li, Z. Ye, C. Tang

{"title":"Estimating the inter-occurrence time distribution from superposed renewal processes","authors":"Xiaoyu Li, Z. Ye, C. Tang","doi":"10.3150/21-BEJ1331","DOIUrl":"https://doi.org/10.3150/21-BEJ1331","url":null,"abstract":"Superposition of renewal processes is common in practice, and it is challenging to estimate the distribution of the individual inter-occurrence time associated with the renewal process. This is because with only aggregated event history, the link between the observed recurrence times and the respective renewal processes are completely missing, rendering existing theory and methods inapplicable. In this article, we propose a nonparametric procedure to estimate the inter-occurrence time distribution by properly deconvoluting the renewal equation with the empirical renewal function. By carefully controlling the discretization errors and properly handling challenges due to implicit and non-smooth mapping via the renewal equation, our theoretical analysis establishes the consistency and asymptotic normality of the nonparametric estimators. The proposed nonparametric distribution estimators are then utilized for developing theoretically valid and computationally efficient inferences when a parametric family is assumed for the individual renewal process. Comprehensive simulations show that compared with the existing maximum likelihood method, the proposed parametric estimation procedure is much faster, and the proposed estimators are more robust to round-off errors in the observed data.","PeriodicalId":55387,"journal":{"name":"Bernoulli","volume":"27 1","pages":"2804-2826"},"PeriodicalIF":1.5,"publicationDate":"2021-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44009027","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 3

Testing against uniform stochastic ordering with paired observations 用成对观察对均匀随机排序的检验

IF 1.5 2区数学

Bernoulli Pub Date : 2021-11-01 DOI: 10.3150/21-BEJ1322

Dewei Wang, Chuan-Fa Tang

引用次数: 2

Tail processes for stable-regenerative multiple-stable model 稳定-再生多稳定模型的尾部过程

IF 1.5 2区数学

Bernoulli Pub Date : 2021-10-14 DOI: 10.3150/22-bej1582

Shuyang Bai, Yizao Wang

引用次数: 0

Gibbsianness and non-Gibbsianness for Bernoulli lattice fields under removal of isolated sites 伯努利格场在去除孤立位条件下的Gibbsianness和non-Gibbsianness