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Sample canonical correlation coefficients of high-dimensional random vectors with finite rank correlations 具有有限秩相关的高维随机向量的样本典型相关系数
IF 1.5 2区 数学
Bernoulli Pub Date : 2021-02-05 DOI: 10.3150/22-bej1525
Zongming Ma, Fan Yang
{"title":"Sample canonical correlation coefficients of high-dimensional random vectors with finite rank correlations","authors":"Zongming Ma, Fan Yang","doi":"10.3150/22-bej1525","DOIUrl":"https://doi.org/10.3150/22-bej1525","url":null,"abstract":"Consider two random vectors $widetilde{mathbf x} in mathbb R^p$ and $widetilde{mathbf y} in mathbb R^q$ of the forms $widetilde{mathbf x}=Amathbf z+mathbf C_1^{1/2}mathbf x$ and $widetilde{mathbf y}=Bmathbf z+mathbf C_2^{1/2}mathbf y$, where $mathbf xin mathbb R^p$, $mathbf yin mathbb R^q$ and $mathbf zin mathbb R^r$ are independent vectors with i.i.d. entries of mean 0 and variance 1, $mathbf C_1$ and $mathbf C_2$ are $p times p$ and $qtimes q$ deterministic covariance matrices, and $A$ and $B$ are $ptimes r$ and $qtimes r$ deterministic matrices. With $n$ independent observations of $(widetilde{mathbf x},widetilde{mathbf y})$, we study the sample canonical correlations between $widetilde{mathbf x} $ and $widetilde{mathbf y}$. We consider the high-dimensional setting with finite rank correlations. Let $t_1ge t_2 ge cdotsge t_r$ be the squares of the nontrivial population canonical correlation coefficients, and let $widetildelambda_1 gewidetildelambda_2gecdotsgewidetildelambda_{pwedge q}$ be the squares of the sample canonical correlation coefficients. If the entries of $mathbf x$, $mathbf y$ and $mathbf z$ are i.i.d. Gaussian, then the following dichotomy has been shown in [7] for a fixed threshold $t_c in(0, 1)$: for $1le i le r$, if $t_it_c$, then $widetildelambda_i$ converges to a deterministic limit $theta_i in (lambda_+,1)$ determined by $t_i$. In this paper, we prove that these results hold universally under the sharp fourth moment conditions on the entries of $mathbf x$ and $mathbf y$. Moreover, we prove the results in full generality, in the sense that they also hold for near-degenerate $t_i$'s and for $t_i$'s that are close to the threshold $t_c$.","PeriodicalId":55387,"journal":{"name":"Bernoulli","volume":" ","pages":""},"PeriodicalIF":1.5,"publicationDate":"2021-02-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41483457","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: 9
The complex behaviour of Galton rank-order statistic 高尔顿秩序统计量的复杂行为
IF 1.5 2区 数学
Bernoulli Pub Date : 2021-02-04 DOI: 10.3150/21-bej1406
E. Barrio, J. A. Cuesta-Albertos, C. Matrán
{"title":"The complex behaviour of Galton rank-order statistic","authors":"E. Barrio, J. A. Cuesta-Albertos, C. Matrán","doi":"10.3150/21-bej1406","DOIUrl":"https://doi.org/10.3150/21-bej1406","url":null,"abstract":"Galton’s rank-order statistic is one of the oldest statistical tools for two-sample comparisons. It is also a very natural index to measure departures from stochastic dominance. Yet, its asymptotic behaviour has been investigated only partially, under restrictive assumptions. This work provides a comprehensive study of this behaviour, based on the analysis of the so-called contact set (a modification of the set in which the quantile functions coincide). We show that a.s. convergence to the population counterpart holds if and only if the contact set has zero Lebesgue measure. When this set is finite we show that the asymptotic behaviour is determined by the local behaviour of a suitable reparameterization of the quantile functions in a neighbourhood of the contact points. Regular crossings result in standard rates and Gaussian limiting distributions, but higher order contacts (in the sense introduced in this work) or contacts at the extremes of the supports may result in different rates and non-Gaussian limits.","PeriodicalId":55387,"journal":{"name":"Bernoulli","volume":" ","pages":""},"PeriodicalIF":1.5,"publicationDate":"2021-02-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47805608","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: 0
A frequency domain bootstrap for general multivariate stationary processes 一般多元平稳过程的频域自举
IF 1.5 2区 数学
Bernoulli Pub Date : 2021-02-03 DOI: 10.3150/22-bej1545
M. Meyer, E. Paparoditis
{"title":"A frequency domain bootstrap for general multivariate stationary processes","authors":"M. Meyer, E. Paparoditis","doi":"10.3150/22-bej1545","DOIUrl":"https://doi.org/10.3150/22-bej1545","url":null,"abstract":"For many relevant statistics of multivariate time series, no valid frequency domain bootstrap procedures exist. This is mainly due to the fact that the distribution of such statistics depends on the fourth-order moment structure of the underlying process in nearly every scenario, except for some special cases like Gaussian time series. In contrast to the univariate case, even additional structural assumptions such as linearity of the multivariate process or a standardization of the statistic of interest do not solve the problem. This paper focuses on integrated periodogram statistics as well as functions thereof and presents a new frequency domain bootstrap procedure for multivariate time series, the multivariate frequency domain hybrid bootstrap (MFHB), to fill this gap. Asymptotic validity of the MFHB procedure is established for general classes of periodogram-based statistics and for stationary multivariate processes satisfying rather weak dependence conditions. A simulation study is carried out which compares the finite sample performance of the MFHB with that of the moving block bootstrap.","PeriodicalId":55387,"journal":{"name":"Bernoulli","volume":" ","pages":""},"PeriodicalIF":1.5,"publicationDate":"2021-02-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46717819","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
Flexible integrated functional depths 灵活的集成功能深度
IF 1.5 2区 数学
Bernoulli Pub Date : 2021-02-01 DOI: 10.3150/20-BEJ1254
Stanislav Nagy, Sami Helander, Germain Van Bever, L. Viitasaari, Pauliina Ilmonen
{"title":"Flexible integrated functional depths","authors":"Stanislav Nagy, Sami Helander, Germain Van Bever, L. Viitasaari, Pauliina Ilmonen","doi":"10.3150/20-BEJ1254","DOIUrl":"https://doi.org/10.3150/20-BEJ1254","url":null,"abstract":"This paper develops a new class of functional depths. A generic member of this class is coined J th order k th moment integrated depth. It is based on the distribution of the cross-sectional halfspace depth of a function in the marginal evaluations (in time) of the random process. Asymptotic properties of the proposed depths are provided: we show that they are uniformly consistent and satisfy an inequality related to the law of the iterated logarithm. Moreover, limiting distributions are derived under mild regularity assumptions. The versatility displayed by the new class of depths makes them particularly amenable for capturing important features of functional distributions. This is illustrated in supervised learning, where we show that the corresponding maximum depth classifiers outperform classical competitors.","PeriodicalId":55387,"journal":{"name":"Bernoulli","volume":" ","pages":""},"PeriodicalIF":1.5,"publicationDate":"2021-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48651137","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
Local scaling limits of Lévy driven fractional random fields lsamvy驱动分数随机场的局部缩放极限
IF 1.5 2区 数学
Bernoulli Pub Date : 2021-02-01 DOI: 10.3150/21-BEJ1439
Vytaut.e Pilipauskait.e, D. Surgailis
{"title":"Local scaling limits of Lévy driven fractional random fields","authors":"Vytaut.e Pilipauskait.e, D. Surgailis","doi":"10.3150/21-BEJ1439","DOIUrl":"https://doi.org/10.3150/21-BEJ1439","url":null,"abstract":"We obtain a complete description of local anisotropic scaling limits for a class of fractional random fields X on R 2 written as stochastic integral with respect to infinitely divisible random measure. The scaling procedure involves increments of X over points the distance between which in the horizontal and vertical directions shrinks as O ( λ ) and O ( λ γ ) respectively as λ ↓ 0 , for some γ > 0 . We consider two types of increments of X : usual increment and rectangular increment, leading to the respective concepts of γ -tangent and γ -rectangent random fields. We prove that for above X both types of local scaling limits exist for any γ > 0 and undergo a transition, being independent of γ > γ 0 and γ < γ 0 , for some γ 0 > 0 ; moreover, the ‘unbalanced’ scaling limits ( γ (cid:54) = γ 0 ) are ( H 1 ,H 2 ) -multi self-similar with one of H i , i = 1 , 2 , equal to 0 or 1 . The paper extends Pilipauskait˙e and Surgailis (2017) and Surgailis (2020) on large-scale anisotropic scaling of random fields on Z 2 and Benassi et al. (2004) on 1 -tangent limits of isotropic fractional Lévy random fields.","PeriodicalId":55387,"journal":{"name":"Bernoulli","volume":" ","pages":""},"PeriodicalIF":1.5,"publicationDate":"2021-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46285121","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
Efron–Petrosian integrals for doubly truncated data with covariates: An asymptotic analysis 具有协变量的双截断数据的Efron–Petrosian积分：一个渐近分析
IF 1.5 2区 数学
Bernoulli Pub Date : 2021-02-01 DOI: 10.3150/20-bej1236
J. Uña-Álvarez, I. Keilegom
{"title":"Efron–Petrosian integrals for doubly truncated data with covariates: An asymptotic analysis","authors":"J. Uña-Álvarez, I. Keilegom","doi":"10.3150/20-bej1236","DOIUrl":"https://doi.org/10.3150/20-bej1236","url":null,"abstract":"In survival analysis, epidemiology and related fields there exists an increasing interest in statistical methods for doubly truncated data. Double truncation appears with interval sampling and other sampling schemes, and refers to situations in which the target variable is subject to two (left and right) random observation limits. Doubly truncated data require specific corrections for the observational bias, and this affects a variety of settings including the estimation of marginal and multivariate distributions, regression problems, and multi-state models. In this work multivariate Efron-Petrosian integrals for doubly truncated data are introduced. These integrals naturally arise when the goal is the estimation of the mean of a general transformation which involves the doubly truncated variable and covariates. An asymptotic representation of the Efron-Petrosian integrals as a sum of iid terms is derived and, from this, consistency and distributional convergence are established. As a by-product, uniform iid representations for the marginal nonparametric maximum likelihood estimator and its corresponding weighting process are provided. Applications to correlation analysis, regression, and competing risks models are presented. A simulation study is reported too.","PeriodicalId":55387,"journal":{"name":"Bernoulli","volume":"27 1","pages":"249-273"},"PeriodicalIF":1.5,"publicationDate":"2021-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45098020","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: 10
A class of models for Bayesian predictive inference 贝叶斯预测推理的一类模型
IF 1.5 2区 数学
Bernoulli Pub Date : 2021-02-01 DOI: 10.3150/20-bej1255
P. Berti, E. Dreassi, L. Pratelli, P. Rigo
{"title":"A class of models for Bayesian predictive inference","authors":"P. Berti, E. Dreassi, L. Pratelli, P. Rigo","doi":"10.3150/20-bej1255","DOIUrl":"https://doi.org/10.3150/20-bej1255","url":null,"abstract":"In a Bayesian framework, to make predictions on a sequence X1, X2, . . . of random observations, the inferrer needs to assign the predictive distributions σn(·) = P ( Xn+1 ∈ · | X1, . . . , Xn ) . In this paper, we propose to assign σn directly, without passing through the usual prior/posterior scheme. One main advantage is that no prior probability has to be assessed. The data sequence (Xn) is assumed to be conditionally identically distributed (c.i.d.) in the sense of [4]. To realize this programme, a class Σ of predictive distributions is introduced and investigated. Such a Σ is rich enough to model various real situations and (Xn) is actually c.i.d. if σn belongs to Σ. Furthermore, when a new observation Xn+1 becomes available, σn+1 can be obtained by a simple recursive update of σn. If μ is the a.s. weak limit of σn, conditions for μ to be a.s. discrete are provided as well.","PeriodicalId":55387,"journal":{"name":"Bernoulli","volume":"27 1","pages":"702-726"},"PeriodicalIF":1.5,"publicationDate":"2021-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46777731","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: 12
Inference problem in generalized fractional Ornstein–Uhlenbeck processes with change-point 具有变点的广义分式Ornstein–Uhlenbeck过程的推理问题
IF 1.5 2区 数学
Bernoulli Pub Date : 2021-02-01 DOI: 10.3150/20-bej1230
S. Nkurunziza
{"title":"Inference problem in generalized fractional Ornstein–Uhlenbeck processes with change-point","authors":"S. Nkurunziza","doi":"10.3150/20-bej1230","DOIUrl":"https://doi.org/10.3150/20-bej1230","url":null,"abstract":"In this paper, we study an inference problem in generalized fractional Ornstein-Uhlenbeck (O-U) processes with an unknown change-point when the drift parameter is suspected to satisfy some constraints. The constraint considered is very general and, the testing problem studied generalizes a very recent inference problem in generalized O-U processes. We derive the unrestricted estimator (UE) and the restricted estimator (RE) and we establish the asymptotic properties of the UE and RE. We also propose some shrinkage-type estimators (SEs) as well as a test for testing the constraint. Finally, we derive the asymptotic power of the proposed test and we study the relative risk dominance of the proposed estimators.","PeriodicalId":55387,"journal":{"name":"Bernoulli","volume":"27 1","pages":"107-134"},"PeriodicalIF":1.5,"publicationDate":"2021-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46139055","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
Measuring association with Wasserstein distances 测量与Wasserstein距离的关联
IF 1.5 2区 数学
Bernoulli Pub Date : 2021-01-31 DOI: 10.3150/21-bej1438
J. Wiesel
{"title":"Measuring association with Wasserstein distances","authors":"J. Wiesel","doi":"10.3150/21-bej1438","DOIUrl":"https://doi.org/10.3150/21-bej1438","url":null,"abstract":"Let $piin Pi(mu,nu)$ be a coupling between two probability measures $mu$ and $nu$ on a Polish space. In this article we propose and study a class of nonparametric measures of association between $mu$ and $nu$, which we call Wasserstein correlation coefficients. These coefficients are based on the Wasserstein distance between $nu$ and the disintegration $pi_{x_1}$ of $pi$ with respect to the first coordinate. We also establish basic statistical properties of this new class of measures: we develop a statistical theory for strongly consistent estimators and determine their convergence rate in the case of compactly supported measures $mu$ and $nu$. Throughout our analysis we make use of the so-called adapted/bicausal Wasserstein distance, in particular we rely on results established in [Backhoff, Bartl, Beiglb\"ock, Wiesel. Estimating processes in adapted Wasserstein distance. 2020]. Our approach applies to probability laws on general Polish spaces.","PeriodicalId":55387,"journal":{"name":"Bernoulli","volume":" ","pages":""},"PeriodicalIF":1.5,"publicationDate":"2021-01-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43545073","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
Rates of convergence for the number of zeros of random trigonometric polynomials 随机三角多项式的零个数的收敛速率
IF 1.5 2区 数学
Bernoulli Pub Date : 2021-01-29 DOI: 10.3150/22-bej1528
L. Coutin, Liliana Peralta
{"title":"Rates of convergence for the number of zeros of random trigonometric polynomials","authors":"L. Coutin, Liliana Peralta","doi":"10.3150/22-bej1528","DOIUrl":"https://doi.org/10.3150/22-bej1528","url":null,"abstract":"In this paper, we quantify the rate of convergence between the distribution of number of zeros of random trigonometric polynomials (RTP) with i.i.d. centered random coefficients and the number of zeros of a stationary centered Gaussian process G, whose covariance function is given by the sinc function. First, we find the convergence of the RTP towards G in the Wasserstein-1 distance, which in turn is a consequence of Donsker Theorem. Then, we use this result to derive the rate of convergence between their respective number of zeros. Since the number of real zeros of the RTP is not a continuous function, we use the Kac-Rice formula to express it as the limit of an integral and, in this way, we approximate it by locally Lipschitz continuous functions.","PeriodicalId":55387,"journal":{"name":"Bernoulli","volume":" ","pages":""},"PeriodicalIF":1.5,"publicationDate":"2021-01-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48867317","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: 0
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