Bernoulli最新文献_第10页

Spiked eigenvalues of noncentral Fisher matrix with applications 非中心Fisher矩阵的尖峰特征值及其应用

IF 1.5 2区数学

Bernoulli Pub Date : 2021-04-10 DOI: 10.3150/22-bej1579

Xiaozhuo Zhang, Zhiqiang Hou, Z. Bai, Jiang Hu

{"title":"Spiked eigenvalues of noncentral Fisher matrix with applications","authors":"Xiaozhuo Zhang, Zhiqiang Hou, Z. Bai, Jiang Hu","doi":"10.3150/22-bej1579","DOIUrl":"https://doi.org/10.3150/22-bej1579","url":null,"abstract":"In this paper, we investigate the asymptotic behavior of spiked eigenvalues of the noncentral Fisher matrix defined by ${mathbf F}_p={mathbf C}_n(mathbf S_N)^{-1}$, where ${mathbf C}_n$ is a noncentral sample covariance matrix defined by $(mathbf Xi+mathbf X)(mathbf Xi+mathbf X)^*/n$ and $mathbf S_N={mathbf Y}{mathbf Y}^*/N$. The matrices $mathbf X$ and $mathbf Y$ are two independent {Gaussian} arrays, with respective $ptimes n$ and $ptimes N$ and the Gaussian entries of them are textit {independent and identically distributed} (i.i.d.) with mean $0$ and variance $1$. When $p$, $n$, and $N$ grow to infinity proportionally, we establish a phase transition of the spiked eigenvalues of $mathbf F_p$. Furthermore, we derive the textit{central limiting theorem} (CLT) for the spiked eigenvalues of $mathbf F_p$. As an accessory to the proof of the above results, the fluctuations of the spiked eigenvalues of ${mathbf C}_n$ are studied, which should have its own interests. Besides, we develop the limits and CLT for the sample canonical correlation coefficients by the results of the spiked noncentral Fisher matrix and give three consistent estimators, including the population spiked eigenvalues and the population canonical correlation coefficients.","PeriodicalId":55387,"journal":{"name":"Bernoulli","volume":" ","pages":""},"PeriodicalIF":1.5,"publicationDate":"2021-04-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42668925","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: 4

Near-optimal estimation of the unseen under regularly varying tail populations 在有规律变化的尾部种群下对未见情况的近最优估计

IF 1.5 2区数学

Bernoulli Pub Date : 2021-04-07 DOI: 10.3150/23-bej1589

S. Favaro, Zacharie Naulet

{"title":"Near-optimal estimation of the unseen under regularly varying tail populations","authors":"S. Favaro, Zacharie Naulet","doi":"10.3150/23-bej1589","DOIUrl":"https://doi.org/10.3150/23-bej1589","url":null,"abstract":"Given $n$ samples from a population of individuals belonging to different species, what is the number $U$ of hitherto unseen species that would be observed if $lambda n$ new samples were collected? This is an important problem in many scientific endeavors, and it has been the subject of recent works introducing non-parametric estimators of $U$ that are minimax near-optimal and consistent all the way up to $lambda asymplog n$. These works do not rely on any assumption on the underlying unknown distribution $p$ of the population, and therefore, while providing a theory in its greatest generality, worst-case distributions may severely hamper the estimation of $U$ in concrete applications. In this paper, we consider the problem of strengthening the non-parametric framework for estimating $U$. Inspired by the estimation of rare probabilities in extreme value theory, and motivated by the ubiquitous power-law type distributions in many natural and social phenomena, we make use of a semi-parametric assumption regular variation of index $alpha in (0,1)$ for the tail behaviour of $p$. Under this assumption, we introduce an estimator of $U$ that is simple, linear in the sampling information, computationally efficient, and scalable to massive datasets. Then, uniformly over our class of regularly varying tail distributions, we show that the proposed estimator has provable guarantees: i) it is minimax near-optimal, up to a power of $log n$ factor; ii) it is consistent all of the way up to $loglambda asymp n^{alpha/2}/sqrt{log n}$, and this range is the best possible. This work presents the first study on the estimation of the unseen under regularly varying tail distributions. A numerical illustration of our methodology is presented for synthetic data and real data.","PeriodicalId":55387,"journal":{"name":"Bernoulli","volume":" ","pages":""},"PeriodicalIF":1.5,"publicationDate":"2021-04-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44428263","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: 8

Concentration bounds for the empirical angular measure with statistical learning applications 具有统计学习应用的经验角测度的集中界

IF 1.5 2区数学

Bernoulli Pub Date : 2021-04-07 DOI: 10.3150/22-bej1562

St'ephan Cl'emenccon, Hamid Jalalzai, St'ephane Lhaut, Anne Sabourin, J. Segers

{"title":"Concentration bounds for the empirical angular measure with statistical learning applications","authors":"St'ephan Cl'emenccon, Hamid Jalalzai, St'ephane Lhaut, Anne Sabourin, J. Segers","doi":"10.3150/22-bej1562","DOIUrl":"https://doi.org/10.3150/22-bej1562","url":null,"abstract":"The angular measure on the unit sphere characterizes the first-order dependence structure of the components of a random vector in extreme regions and is defined in terms of standardized margins. Its statistical recovery is an important step in learning problems involving observations far away from the center. In the common situation that the components of the vector have different distributions, the rank transformation offers a convenient and robust way of standardizing data in order to build an empirical version of the angular measure based on the most extreme observations. However, the study of the sampling distribution of the resulting empirical angular measure is challenging. It is the purpose of the paper to establish finite-sample bounds for the maximal deviations between the empirical and true angular measures, uniformly over classes of Borel sets of controlled combinatorial complexity. The bounds are valid with high probability and, up to logarithmic factors, scale as the square root of the effective sample size. The bounds are applied to provide performance guarantees for two statistical learning procedures tailored to extreme regions of the input space and built upon the empirical angular measure: binary classification in extreme regions through empirical risk minimization and unsupervised anomaly detection through minimum-volume sets of the sphere.","PeriodicalId":55387,"journal":{"name":"Bernoulli","volume":" ","pages":""},"PeriodicalIF":1.5,"publicationDate":"2021-04-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47877995","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: 11

Inference for partially observed Riemannian Ornstein–Uhlenbeck diffusions of covariance matrices 协方差矩阵部分观测riemanian Ornstein-Uhlenbeck扩散的推论

IF 1.5 2区数学

Bernoulli Pub Date : 2021-04-07 DOI: 10.3150/22-bej1570

Mai Bui, Y. Pokern, P. Dellaportas

引用次数: 6

Multivariate time series models for mixed data 混合数据的多元时间序列模型

IF 1.5 2区数学

Bernoulli Pub Date : 2021-04-02 DOI: 10.3150/22-bej1474

Zinsou Max Debaly, L. Truquet

{"title":"Multivariate time series models for mixed data","authors":"Zinsou Max Debaly, L. Truquet","doi":"10.3150/22-bej1474","DOIUrl":"https://doi.org/10.3150/22-bej1474","url":null,"abstract":"We introduce a general approach for modeling the dynamic of multivariate time series when the data are of mixed type (binary/count/continuous). Our method is quite flexible and conditionally on past values, each coordinate at time $t$ can have a distribution compatible with a standard univariate time series model such as GARCH, ARMA, INGARCH or logistic models whereas past values of the other coordinates play the role of exogenous covariates in the dynamic. The simultaneous dependence in the multivariate time series can be modeled with a copula. Additional exogenous covariates are also allowed in the dynamic. We first study usual stability properties of these models and then show that autoregressive parameters can be consistently estimated equation-by-equation using a pseudo-maximum likelihood method, leading to a fast implementation even when the number of time series is large. Moreover, we prove consistency results when a parametric copula model is fitted to the time series and in the case of Gaussian copulas, we show that the likelihood estimator of the correlation matrix is strongly consistent. We carefully check all our assumptions for two prototypical examples: a GARCH/INGARCH model and logistic/log-linear INGARCH model. Our results are illustrated with numerical experiments as well as two real data sets.","PeriodicalId":55387,"journal":{"name":"Bernoulli","volume":"232 3","pages":""},"PeriodicalIF":1.5,"publicationDate":"2021-04-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41263042","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: 8

Consistency of p-norm based tests in high dimensions: Characterization, monotonicity, domination 高维p-范数检验的一致性：特征性、单调性、支配性

IF 1.5 2区数学

Bernoulli Pub Date : 2021-03-20 DOI: 10.3150/22-bej1552

A. Kock, David Preinerstorfer

{"title":"Consistency of p-norm based tests in high dimensions: Characterization, monotonicity, domination","authors":"A. Kock, David Preinerstorfer","doi":"10.3150/22-bej1552","DOIUrl":"https://doi.org/10.3150/22-bej1552","url":null,"abstract":"Many commonly used test statistics are based on a norm measuring the evidence against the null hypothesis. To understand how the choice of a norm affects power properties of tests in high dimensions, we study the consistency sets of $p$-norm based tests in the prototypical framework of sequence models with unrestricted parameter spaces, the null hypothesis being that all observations have zero mean. The consistency set of a test is here defined as the set of all arrays of alternatives the test is consistent against as the dimension of the parameter space diverges. We characterize the consistency sets of $p$-norm based tests and find, in particular, that the consistency against an array of alternatives cannot be determined solely in terms of the $p$-norm of the alternative. Our characterization also reveals an unexpected monotonicity result: namely that the consistency set is strictly increasing in $p in (0, infty)$, such that tests based on higher $p$ strictly dominate those based on lower $p$ in terms of consistency. This monotonicity allows us to construct novel tests that dominate, with respect to their consistency behavior, all $p$-norm based tests without sacrificing size.","PeriodicalId":55387,"journal":{"name":"Bernoulli","volume":" ","pages":""},"PeriodicalIF":1.5,"publicationDate":"2021-03-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44743516","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: 2

Small sample spaces for Gaussian processes 高斯过程的小样本空间

IF 1.5 2区数学

Bernoulli Pub Date : 2021-03-04 DOI: 10.3150/22-bej1483

T. Karvonen

引用次数: 10

Sufficient dimension reduction and instrument search for data with nonignorable nonresponse 不可忽略无响应数据的充分降维和仪器搜索

IF 1.5 2区数学

Bernoulli Pub Date : 2021-03-01 DOI: 10.3150/20-BEJ1260

Puying Zhao, Lei Wang, Junchao Shao

引用次数: 5

Splitting the sample at the largest uncensored observation 在最大的未经审查的观察中分割样本

IF 1.5 2区数学

Bernoulli Pub Date : 2021-03-01 DOI: 10.3150/21-bej1417

R. Maller, S. Resnick, S. Shemehsavar

{"title":"Splitting the sample at the largest uncensored observation","authors":"R. Maller, S. Resnick, S. Shemehsavar","doi":"10.3150/21-bej1417","DOIUrl":"https://doi.org/10.3150/21-bej1417","url":null,"abstract":"We calculate finite sample and asymptotic distributions for the largest censored and uncensored survival times, and some related statistics, from a sample of survival data generated according to an iid censoring model. These statistics are important for assessing whether there is sufficient follow-up in the sample to be confident of the presence of immune or cured individuals in the population. A key structural result obtained is that, conditional on the value of the largest uncensored survival time, and knowing the number of censored observations exceeding this time, the sample partitions into two independent subsamples, each subsample having the distribution of an iid sample of censored survival times, of reduced size, from truncated random variables. This result provides valuable insight into the construction of censored survival data, and facilitates the calculation of explicit finite sample formulae. We illustrate for distributions of statistics useful for testing for sufficient follow-up in a sample, and apply extreme value methods to derive asymptotic distributions for some of those. MSC 2010 subject classifications: MSC2000 Subject Classifications: Primary 62N01, 62N02, 62N03, 62E10, 62E15, 62E20, G2G05; secondary 62F03, 62F05, 62F12, 62G32.","PeriodicalId":55387,"journal":{"name":"Bernoulli","volume":" ","pages":""},"PeriodicalIF":1.5,"publicationDate":"2021-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46551907","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

Minimum spanning trees of random geometric graphs with location dependent weights 具有位置依赖权的随机几何图的最小生成树

IF 1.5 2区数学

Bernoulli Pub Date : 2021-03-01 DOI: 10.3150/20-BEJ1318

Ghurumuruhan Ganesan

引用次数: 2