We present a posterior analysis of kernel mixtures of thinned completely random measures 6 (CRMs) for multivariate intensities, in the context of competing risks models. The construction 7 of the thinned CRMs is derived from a common Poisson random measure that includes the thin8 ning probabilities in its intensity and is transferable to existing Poisson partition calculus results 9 for the posterior analysis (James 2002, 2005). We derive the posterior thinned CRMs, provide 10 generalizations of both the Blackwell and MacQueen Pólya urn formula and the (weighted) Chi11 nese restaurant process for the variates and partitions generated from the thinned CRMs, and 12 we outline strategies for the further development of Monte Carlo simulation for estimation. 13
{"title":"Thinned completely random measures with applications in competing risks models","authors":"J. Lau, E. Cripps","doi":"10.3150/21-bej1361","DOIUrl":"https://doi.org/10.3150/21-bej1361","url":null,"abstract":"We present a posterior analysis of kernel mixtures of thinned completely random measures 6 (CRMs) for multivariate intensities, in the context of competing risks models. The construction 7 of the thinned CRMs is derived from a common Poisson random measure that includes the thin8 ning probabilities in its intensity and is transferable to existing Poisson partition calculus results 9 for the posterior analysis (James 2002, 2005). We derive the posterior thinned CRMs, provide 10 generalizations of both the Blackwell and MacQueen Pólya urn formula and the (weighted) Chi11 nese restaurant process for the variates and partitions generated from the thinned CRMs, and 12 we outline strategies for the further development of Monte Carlo simulation for estimation. 13","PeriodicalId":55387,"journal":{"name":"Bernoulli","volume":" ","pages":""},"PeriodicalIF":1.5,"publicationDate":"2022-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44065770","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}
Multiscale models involving “slow” and “fast” components appear naturally in various fields, such as nonlinear oscillations, chemical kinetics, biology, climate dynamics, etc, see, e.g., [3,12,22,33] and the references therein. The averaging principle of multiscale models describes the asymptotic behavior of the slow components as the scale parameter → 0. In [23], Khasminskii considered a class of multiscale stochastic differential equations (SDEs for short) driven by Wiener noise, i.e., dX t = A(X t , Y t )dt+ dWt, X 0 = x ∈ R, dY t = 1 B(X t , Y t )dt+ 1 √ dWt, Y 0 = y ∈ R,
{"title":"Strong and weak convergence rates for slow–fast stochastic differential equations driven by α-stable process","authors":"Xiaobin Sun, Longjie Xie, Yingchao Xie","doi":"10.3150/21-bej1345","DOIUrl":"https://doi.org/10.3150/21-bej1345","url":null,"abstract":"Multiscale models involving “slow” and “fast” components appear naturally in various fields, such as nonlinear oscillations, chemical kinetics, biology, climate dynamics, etc, see, e.g., [3,12,22,33] and the references therein. The averaging principle of multiscale models describes the asymptotic behavior of the slow components as the scale parameter → 0. In [23], Khasminskii considered a class of multiscale stochastic differential equations (SDEs for short) driven by Wiener noise, i.e., dX t = A(X t , Y t )dt+ dWt, X 0 = x ∈ R, dY t = 1 B(X t , Y t )dt+ 1 √ dWt, Y 0 = y ∈ R,","PeriodicalId":55387,"journal":{"name":"Bernoulli","volume":" ","pages":""},"PeriodicalIF":1.5,"publicationDate":"2022-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45836944","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}
We consider a stationary random field indexed by an increasing sequence of subsets of $mathbb{Z}^d$ obeying a very broad geometrical assumption on how the sequence expands. Under certain mixing and local conditions, we show how the tail distribution of the individual variables relates to the tail behavior of the maximum of the field over the index sets in the limit as the index sets expand. Furthermore, in a framework where we let the increasing index sets be scalar multiplications of a fixed set $C$, potentially with different scalars in different directions, we use two cluster definitions to define associated cluster counting point processes on the rescaled index set $C$; one cluster definition divides the index set into more and more boxes and counts a box as a cluster if it contains an extremal observation. The other cluster definition that is more intuitive considers extremal points to be in the same cluster, if they are close in distance. We show that both cluster point processes converge to a Poisson point process on $C$. Additionally, we find a limit of the mean cluster size. Finally, we pay special attention to the case without clusters.
{"title":"Extremal clustering and cluster counting for spatial random fields","authors":"Anders Rønn-Nielsen, Mads Stehr","doi":"10.3150/22-bej1561","DOIUrl":"https://doi.org/10.3150/22-bej1561","url":null,"abstract":"We consider a stationary random field indexed by an increasing sequence of subsets of $mathbb{Z}^d$ obeying a very broad geometrical assumption on how the sequence expands. Under certain mixing and local conditions, we show how the tail distribution of the individual variables relates to the tail behavior of the maximum of the field over the index sets in the limit as the index sets expand. Furthermore, in a framework where we let the increasing index sets be scalar multiplications of a fixed set $C$, potentially with different scalars in different directions, we use two cluster definitions to define associated cluster counting point processes on the rescaled index set $C$; one cluster definition divides the index set into more and more boxes and counts a box as a cluster if it contains an extremal observation. The other cluster definition that is more intuitive considers extremal points to be in the same cluster, if they are close in distance. We show that both cluster point processes converge to a Poisson point process on $C$. Additionally, we find a limit of the mean cluster size. Finally, we pay special attention to the case without clusters.","PeriodicalId":55387,"journal":{"name":"Bernoulli","volume":" ","pages":""},"PeriodicalIF":1.5,"publicationDate":"2022-01-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45742852","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}
This paper proves that, under a monotonicity condition, the invariant probability measure of a McKean--Vlasov process can be approximated by weighted empirical measures of some processes including itself. These processes are described by distribution dependent or empirical measure dependent stochastic differential equations constructed from the equation for the McKean--Vlasov process. Convergence of empirical measures is characterized by upper bound estimates for their Wasserstein distance to the invariant measure. The theoretical results are demonstrated via a mean-field Ornstein--Uhlenbeck process.
{"title":"Empirical approximation to invariant measures for McKean–Vlasov processes: Mean-field interaction vs self-interaction","authors":"Kai Du, Yifan Jiang, Jinfeng Li","doi":"10.3150/22-bej1550","DOIUrl":"https://doi.org/10.3150/22-bej1550","url":null,"abstract":"This paper proves that, under a monotonicity condition, the invariant probability measure of a McKean--Vlasov process can be approximated by weighted empirical measures of some processes including itself. These processes are described by distribution dependent or empirical measure dependent stochastic differential equations constructed from the equation for the McKean--Vlasov process. Convergence of empirical measures is characterized by upper bound estimates for their Wasserstein distance to the invariant measure. The theoretical results are demonstrated via a mean-field Ornstein--Uhlenbeck process.","PeriodicalId":55387,"journal":{"name":"Bernoulli","volume":" ","pages":""},"PeriodicalIF":1.5,"publicationDate":"2021-12-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46848921","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}
We establish precise right-tail small deviation estimates for the largest eigenvalue of real symmetric and complex Hermitian matrices whose entries are independent random variables with uniformly bounded moments. The proof relies on a Green function comparison along a continuous interpolating matrix flow for a long time. Less precise estimates are also obtained in the left tail.
{"title":"Small deviation estimates for the largest eigenvalue of Wigner matrices","authors":"L'aszl'o ErdHos, Yuanyuan Xu","doi":"10.3150/22-bej1490","DOIUrl":"https://doi.org/10.3150/22-bej1490","url":null,"abstract":"We establish precise right-tail small deviation estimates for the largest eigenvalue of real symmetric and complex Hermitian matrices whose entries are independent random variables with uniformly bounded moments. The proof relies on a Green function comparison along a continuous interpolating matrix flow for a long time. Less precise estimates are also obtained in the left tail.","PeriodicalId":55387,"journal":{"name":"Bernoulli","volume":" ","pages":""},"PeriodicalIF":1.5,"publicationDate":"2021-12-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45539686","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}
We study random normal matrix models whose eigenvalues tend to be distributed within a narrow"band"around the unit circle of width proportional to $frac1n$, where $n$ is the size of matrices. For general radially symmetric potentials with various boundary conditions, we derive the scaling limits of the correlation functions, some of which appear in the previous literature notably in the context of almost-Hermitian random matrices. We also obtain that fluctuations of the maximal and minimal modulus of the ensembles follow the Gumbel or exponential law depending on the boundary conditions.
{"title":"Random normal matrices in the almost-circular regime","authors":"Sunggyu Byun, Seong-Mi Seo","doi":"10.3150/22-bej1514","DOIUrl":"https://doi.org/10.3150/22-bej1514","url":null,"abstract":"We study random normal matrix models whose eigenvalues tend to be distributed within a narrow\"band\"around the unit circle of width proportional to $frac1n$, where $n$ is the size of matrices. For general radially symmetric potentials with various boundary conditions, we derive the scaling limits of the correlation functions, some of which appear in the previous literature notably in the context of almost-Hermitian random matrices. We also obtain that fluctuations of the maximal and minimal modulus of the ensembles follow the Gumbel or exponential law depending on the boundary conditions.","PeriodicalId":55387,"journal":{"name":"Bernoulli","volume":" ","pages":""},"PeriodicalIF":1.5,"publicationDate":"2021-12-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43843904","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}
For a generalization of Johnstone's spiked model, a covariance matrix with eigenvalues all one but $M$ of them, the number of features $N$ comparable to the number of samples $n: N=N(n), M=M(n), gamma^{-1} leq frac{N}{n} leq gamma$ where $gamma in (0,infty),$ we obtain consistency rates in the form of CLTs for separated spikes tending to infinity fast enough whenever $M$ grows slightly slower than $n: lim_{n to infty}{frac{sqrt{log{n}}}{log{frac{n}{M(n)}}}}=0.$ Our results fill a gap in the existing literature in which the largest range covered for the number of spikes has been $o(n^{1/6})$ and reveal a certain degree of flexibility for the centering in these CLTs inasmuch as it can be empirical, deterministic, or a sum of both. Furthermore, we derive consistency rates of their corresponding empirical eigenvectors to their true counterparts, which turn out to depend on the relative growth of these eigenvalues.
对于Johnstone的尖刺模型的推广，协方差矩阵的特征值除了$M$之外都是，特征数量$N$与样本数量$n: N=N(n), M=M(n), gamma^{-1} leq frac{N}{n} leq gamma$相当，其中$gamma in (0,infty),$我们以clt的形式获得一致性率，当$M$的增长速度略慢于$n: lim_{n to infty}{frac{sqrt{log{n}}}{log{frac{n}{M(n)}}}}=0.$时，分离的峰值趋于无穷大。我们的结果填补了现有文献中的空白，其中峰值数量覆盖的最大范围是$o(n^{1/6})$，并揭示了这些clt中定心的一定程度的灵活性因为它可以是经验的，确定的，或两者的总和。此外，我们推导出它们对应的经验特征向量与它们的真对应物的一致性率，这取决于这些特征值的相对增长。
{"title":"On the eigenstructure of covariance matrices with divergent spikes","authors":"Simona Diaconu","doi":"10.3150/22-bej1498","DOIUrl":"https://doi.org/10.3150/22-bej1498","url":null,"abstract":"For a generalization of Johnstone's spiked model, a covariance matrix with eigenvalues all one but $M$ of them, the number of features $N$ comparable to the number of samples $n: N=N(n), M=M(n), gamma^{-1} leq frac{N}{n} leq gamma$ where $gamma in (0,infty),$ we obtain consistency rates in the form of CLTs for separated spikes tending to infinity fast enough whenever $M$ grows slightly slower than $n: lim_{n to infty}{frac{sqrt{log{n}}}{log{frac{n}{M(n)}}}}=0.$ Our results fill a gap in the existing literature in which the largest range covered for the number of spikes has been $o(n^{1/6})$ and reveal a certain degree of flexibility for the centering in these CLTs inasmuch as it can be empirical, deterministic, or a sum of both. Furthermore, we derive consistency rates of their corresponding empirical eigenvectors to their true counterparts, which turn out to depend on the relative growth of these eigenvalues.","PeriodicalId":55387,"journal":{"name":"Bernoulli","volume":" ","pages":""},"PeriodicalIF":1.5,"publicationDate":"2021-12-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48763865","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}
In this work, we generalise the stochastic local time space integration introduced in cite{Ei00} to the case of Brownian sheet. %We develop a stochastic local time-space calculus with respect to the Brownian sheet. This allows us to prove a generalised two-parameter It^o formula and derive Davie type inequalities for the Brownian sheet. Such estimates are useful to obtain regularity bounds for some averaging type operators along Brownian sheet curves.
在这项工作中，我们推广了cite{Ei00}中引入的随机局部时间空间积分到布朗页的情况。 %We develop a stochastic local time-space calculus with respect to the Brownian sheet. This allows us to prove a generalised two-parameter Itô formula and derive Davie type inequalities for the Brownian sheet. Such estimates are useful to obtain regularity bounds for some averaging type operators along Brownian sheet curves.
{"title":"Stochastic integration with respect to local time of the Brownian sheet and regularising properties of Brownian sheet paths","authors":"Antoine-Marie Bogso, M. Dieye, O. M. Pamen","doi":"10.3150/22-BEJ1555","DOIUrl":"https://doi.org/10.3150/22-BEJ1555","url":null,"abstract":"In this work, we generalise the stochastic local time space integration introduced in cite{Ei00} to the case of Brownian sheet. %We develop a stochastic local time-space calculus with respect to the Brownian sheet. This allows us to prove a generalised two-parameter It^o formula and derive Davie type inequalities for the Brownian sheet. Such estimates are useful to obtain regularity bounds for some averaging type operators along Brownian sheet curves.","PeriodicalId":55387,"journal":{"name":"Bernoulli","volume":" ","pages":""},"PeriodicalIF":1.5,"publicationDate":"2021-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44896763","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}
In the density estimation model, we investigate the problem of constructing adaptive honest confidence sets with radius measured in Wasserstein distance $W_p$, $pgeq1$, and for densities with unknown regularity measured on a Besov scale. As sampling domains, we focus on the $d-$dimensional torus $mathbb{T}^d$, in which case $1leq pleq 2$, and $mathbb{R}^d$, for which $p=1$. We identify necessary and sufficient conditions for the existence of adaptive confidence sets with diameters of the order of the regularity-dependent $W_p$-minimax estimation rate. Interestingly, it appears that the possibility of such adaptation of the diameter depends on the dimension of the underlying space. In low dimensions, $dleq 4$, adaptation to any regularity is possible. In higher dimensions, adaptation is possible if and only if the underlying regularities belong to some interval of width at least $d/(d-4)$. This contrasts with the usual $L_p-$theory where, independently of the dimension, adaptation requires regularities to lie in a small fixed-width window. For configurations allowing these adaptive sets to exist, we explicitly construct confidence regions via the method of risk estimation, centred at adaptive estimators. Those are the first results in a statistical approach to adaptive uncertainty quantification with Wasserstein distances. Our analysis and methods extend more globally to weak losses such as Sobolev norm distances with negative smoothness indices.
{"title":"On adaptive confidence sets for the Wasserstein distances","authors":"N. Deo, Thibault Randrianarisoa","doi":"10.3150/22-bej1535","DOIUrl":"https://doi.org/10.3150/22-bej1535","url":null,"abstract":"In the density estimation model, we investigate the problem of constructing adaptive honest confidence sets with radius measured in Wasserstein distance $W_p$, $pgeq1$, and for densities with unknown regularity measured on a Besov scale. As sampling domains, we focus on the $d-$dimensional torus $mathbb{T}^d$, in which case $1leq pleq 2$, and $mathbb{R}^d$, for which $p=1$. We identify necessary and sufficient conditions for the existence of adaptive confidence sets with diameters of the order of the regularity-dependent $W_p$-minimax estimation rate. Interestingly, it appears that the possibility of such adaptation of the diameter depends on the dimension of the underlying space. In low dimensions, $dleq 4$, adaptation to any regularity is possible. In higher dimensions, adaptation is possible if and only if the underlying regularities belong to some interval of width at least $d/(d-4)$. This contrasts with the usual $L_p-$theory where, independently of the dimension, adaptation requires regularities to lie in a small fixed-width window. For configurations allowing these adaptive sets to exist, we explicitly construct confidence regions via the method of risk estimation, centred at adaptive estimators. Those are the first results in a statistical approach to adaptive uncertainty quantification with Wasserstein distances. Our analysis and methods extend more globally to weak losses such as Sobolev norm distances with negative smoothness indices.","PeriodicalId":55387,"journal":{"name":"Bernoulli","volume":" ","pages":""},"PeriodicalIF":1.5,"publicationDate":"2021-11-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44374358","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}
This work focuses on the nonparametric estimation of a drift function from N discrete repeated independent observations of a diffusion process over a fixed time interval [0, T ]. We study a ridge estimator obtained by the minimization of a constrained least squares contrast. The resulting projection estimator is based on the B-spline basis. Under mild assumptions, this estimator is universally consistent with respect to an integrate norm. We establish that, up to a logarithmic factor and when the estimation is performed on a compact interval, our estimation procedure reaches the best possible rate of convergence. Furthermore, we build an adaptive estimator that achieves this rate. Finally, we illustrate our procedure through an intensive simulation study which highlights the good performance of the proposed estimator in various models.
{"title":"A ridge estimator of the drift from discrete repeated observations of the solution of a stochastic differential equation","authors":"Christophe Denis, C. Dion-Blanc, Miguel Martinez","doi":"10.3150/21-BEJ1327","DOIUrl":"https://doi.org/10.3150/21-BEJ1327","url":null,"abstract":"This work focuses on the nonparametric estimation of a drift function from N discrete repeated independent observations of a diffusion process over a fixed time interval [0, T ]. We study a ridge estimator obtained by the minimization of a constrained least squares contrast. The resulting projection estimator is based on the B-spline basis. Under mild assumptions, this estimator is universally consistent with respect to an integrate norm. We establish that, up to a logarithmic factor and when the estimation is performed on a compact interval, our estimation procedure reaches the best possible rate of convergence. Furthermore, we build an adaptive estimator that achieves this rate. Finally, we illustrate our procedure through an intensive simulation study which highlights the good performance of the proposed estimator in various models.","PeriodicalId":55387,"journal":{"name":"Bernoulli","volume":"27 1","pages":"2675-2713"},"PeriodicalIF":1.5,"publicationDate":"2021-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47383768","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}
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