Let $L$ be a linear differential operator acting on functions defined over an open set $mathcal{D}subset mathbb{R}^d$. In this article, we characterize the measurable second order random fields $U = (U(x))_{xinmathcal{D}}$ whose sample paths all verify the partial differential equation (PDE) $L(u) = 0$, solely in terms of their first two moments. When compared to previous similar results, the novelty lies in that the equality $L(u) = 0$ is understood in the sense of distributions, which is a powerful functional analysis framework mostly designed to study linear PDEs. This framework enables to reduce to the minimum the required differentiability assumptions over the first two moments of $(U(x))_{xinmathcal{D}}$ as well as over its sample paths in order to make sense of the PDE $L(U_{omega})=0$. In view of Gaussian process regression (GPR) applications, we show that when $(U(x))_{xinmathcal{D}}$ is a Gaussian process (GP), the sample paths of $(U(x))_{xinmathcal{D}}$ conditioned on pointwise observations still verify the constraint $L(u)=0$ in the distributional sense. We finish by deriving a simple but instructive example, a GP model for the 3D linear wave equation, for which our theorem is applicable and where the previous results from the literature do not apply in general.
设$L$是作用于开集$mathcal{D}子集mathbb{R}^ D $上定义的函数的线性微分算子。在本文中，我们描述了可测量的二阶随机场$U = (U(x))_{x In mathcal{D}}$，其样本路径都验证了偏微分方程(PDE) $L(U) = 0$，仅根据它们的前两个矩。与以往的类似结果相比，新颖之处在于，等式$L(u) = 0$被理解为分布的意义，这是一个功能强大的泛函分析框架，主要用于研究线性偏微分方程。这个框架能够将$(U(x))_{xinmathcal{D}}$的前两个矩以及它的样本路径上所需的可微性假设减少到最小，以便使PDE $L(U_{omega})=0$有意义。针对高斯过程回归(GPR)的应用，我们证明了当$(U(x))_{x In mathcal{D}}$是高斯过程(GP)时，$(U(x))_{x In mathcal{D}}$的样本路径在点向观测条件下仍然在分布意义上验证了约束$L(U)=0$。最后，我们推导了一个简单但具有指导意义的例子，即三维线性波动方程的GP模型，对于这个模型，我们的定理是适用的，而以前的文献结果一般不适用。
{"title":"Characterization of the second order random fields subject to linear distributional PDE constraints","authors":"Iain Henderson, P. Noble, O. Roustant","doi":"10.3150/23-bej1588","DOIUrl":"https://doi.org/10.3150/23-bej1588","url":null,"abstract":"Let $L$ be a linear differential operator acting on functions defined over an open set $mathcal{D}subset mathbb{R}^d$. In this article, we characterize the measurable second order random fields $U = (U(x))_{xinmathcal{D}}$ whose sample paths all verify the partial differential equation (PDE) $L(u) = 0$, solely in terms of their first two moments. When compared to previous similar results, the novelty lies in that the equality $L(u) = 0$ is understood in the sense of distributions, which is a powerful functional analysis framework mostly designed to study linear PDEs. This framework enables to reduce to the minimum the required differentiability assumptions over the first two moments of $(U(x))_{xinmathcal{D}}$ as well as over its sample paths in order to make sense of the PDE $L(U_{omega})=0$. In view of Gaussian process regression (GPR) applications, we show that when $(U(x))_{xinmathcal{D}}$ is a Gaussian process (GP), the sample paths of $(U(x))_{xinmathcal{D}}$ conditioned on pointwise observations still verify the constraint $L(u)=0$ in the distributional sense. We finish by deriving a simple but instructive example, a GP model for the 3D linear wave equation, for which our theorem is applicable and where the previous results from the literature do not apply in general.","PeriodicalId":55387,"journal":{"name":"Bernoulli","volume":" ","pages":""},"PeriodicalIF":1.5,"publicationDate":"2023-01-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48871886","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 Euclidean spaces, the empirical mean vector as an estimator of the population mean is known to have polynomial concentration unless a strong tail assumption is imposed on the underlying probability measure. The idea of median-of-means tournament has been considered as a way of overcoming the sub-optimality of the empirical mean vector. In this paper, to address the sub-optimal performance of the empirical mean in a more general setting, we consider general Polish spaces with a general metric, which are allowed to be non-compact and of infinite-dimension. We discuss the estimation of the associated population Frechet mean, and for this we extend the existing notion of median-of-means to this general setting. We devise several new notions and inequalities associated with the geometry of the underlying metric, and using them we study the concentration properties of the extended notions of median-of-means as the estimators of the population Frechet mean. We show that the new estimators achieve exponential concentration under only a second moment condition on the underlying distribution, while the empirical Frechet mean has polynomial concentration. We focus our study on spaces with non-positive Alexandrov curvature since they afford slower rates of convergence than spaces with positive curvature. We note that this is the first work that derives non-asymptotic concentration inequalities for extended notions of the median-of-means in non-vector spaces with a general metric.
{"title":"Exponential concentration for geometric-median-of-means in non-positive curvature spaces","authors":"H. Yun, B. Park","doi":"10.3150/22-BEJ1569","DOIUrl":"https://doi.org/10.3150/22-BEJ1569","url":null,"abstract":"In Euclidean spaces, the empirical mean vector as an estimator of the population mean is known to have polynomial concentration unless a strong tail assumption is imposed on the underlying probability measure. The idea of median-of-means tournament has been considered as a way of overcoming the sub-optimality of the empirical mean vector. In this paper, to address the sub-optimal performance of the empirical mean in a more general setting, we consider general Polish spaces with a general metric, which are allowed to be non-compact and of infinite-dimension. We discuss the estimation of the associated population Frechet mean, and for this we extend the existing notion of median-of-means to this general setting. We devise several new notions and inequalities associated with the geometry of the underlying metric, and using them we study the concentration properties of the extended notions of median-of-means as the estimators of the population Frechet mean. We show that the new estimators achieve exponential concentration under only a second moment condition on the underlying distribution, while the empirical Frechet mean has polynomial concentration. We focus our study on spaces with non-positive Alexandrov curvature since they afford slower rates of convergence than spaces with positive curvature. We note that this is the first work that derives non-asymptotic concentration inequalities for extended notions of the median-of-means in non-vector spaces with a general metric.","PeriodicalId":55387,"journal":{"name":"Bernoulli","volume":"1 1","pages":""},"PeriodicalIF":1.5,"publicationDate":"2022-11-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41551240","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 paper, we focus on the characterization of a Brownian bridge measure on the pinned path space over a compact Riemannian manifold. In the case when the Riemannian manifold is simply connected, we prove that the integration by parts formula can characterize the Brownian bridge measure. Otherwise, we show that it is not always true by constructing an illustrating example.
{"title":"On the characterization of Brownian bridge measure on the pinned path space over a compact Riemannian manifold","authors":"Fuzhou Gong, Xiaoxia Sun","doi":"10.3150/21-bej1420","DOIUrl":"https://doi.org/10.3150/21-bej1420","url":null,"abstract":"In this paper, we focus on the characterization of a Brownian bridge measure on the pinned path space over a compact Riemannian manifold. In the case when the Riemannian manifold is simply connected, we prove that the integration by parts formula can characterize the Brownian bridge measure. Otherwise, we show that it is not always true by constructing an illustrating example.","PeriodicalId":55387,"journal":{"name":"Bernoulli","volume":" ","pages":""},"PeriodicalIF":1.5,"publicationDate":"2022-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43712086","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}
ZHIDONG BAI1,*, KWOK PUI CHOI2, YASUNORI FUJIKOSHI3 and JIANG HU 1,† 1KLASMOE and School of Mathematics & Statistics, Northeast Normal University, China. E-mail: *baizd@nenu.edu.cn; †huj156@nenu.edu.cn 2Department of Statistics and Applied Probability, National University of Singapore, Singapore. E-mail: stackp@nus.edu.sg 3Department of Mathematics, Graduate School of Science, Hiroshima University, Japan. E-mail: fujikoshi y@yahoo.co.jp
{"title":"Asymptotics of AIC, BIC and Cp model selection rules in high-dimensional regression","authors":"Z. Bai, K. P. Choi, Y. Fujikoshi, Jiang Hu","doi":"10.3150/21-bej1422","DOIUrl":"https://doi.org/10.3150/21-bej1422","url":null,"abstract":"ZHIDONG BAI1,*, KWOK PUI CHOI2, YASUNORI FUJIKOSHI3 and JIANG HU 1,† 1KLASMOE and School of Mathematics & Statistics, Northeast Normal University, China. E-mail: *baizd@nenu.edu.cn; †huj156@nenu.edu.cn 2Department of Statistics and Applied Probability, National University of Singapore, Singapore. E-mail: stackp@nus.edu.sg 3Department of Mathematics, Graduate School of Science, Hiroshima University, Japan. E-mail: fujikoshi y@yahoo.co.jp","PeriodicalId":55387,"journal":{"name":"Bernoulli","volume":" ","pages":""},"PeriodicalIF":1.5,"publicationDate":"2022-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41794760","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 online computation of expectations of additive state functionals under general path probability measures proportional to products of unnormalised transition densities. These transition densities are assumed to be intractable but possible to estimate, with or without bias. Using pseudo-marginalisation techniques we are able to extend the particle- based, rapid incremental smoother (PaRIS) algorithm proposed in [J. Olsson and J. Westerborn. Efficient particle-based online smoothing in general hidden Markov models: The PaRIS algorithm. Bernoulli , 23(3):1951–1996, 2017] to this setting. The resulting algorithm, which has a linear complex- ity in the number of particles and constant memory requirements, applies to a wide range of challenging path-space Monte Carlo problems, includ- ing smoothing in partially observed diffusion processes and models with intractable likelihood. The algorithm is furnished with several theoretical results, including a central limit theorem, establishing its convergence and numerical stability. Moreover, under strong mixing assumptions we estab- lish a novel O ( nε ) bound on the asymptotic bias of the algorithm, where n is the path length and ε controls the bias of the density estimators.
：我们考虑在与未规范化跃迁密度的乘积成比例的一般路径概率测度下，可加状态泛函的期望的在线计算。这些跃迁密度被认为是难以处理的，但可以估计，无论有没有偏差。使用伪边缘化技术，我们能够将[J.Olsson和J.Westerborn.Eefficient particle based online smoothing in general hidden Markov models:the PaRIS algorithm.Bernoulli，23（3）：1951–19962017]中提出的基于粒子的快速增量平滑（PaRIS）算法扩展到该设置。由此产生的算法在粒子数量和恒定内存需求方面具有线性复杂性，适用于一系列具有挑战性的路径空间蒙特卡罗问题，包括在部分观察到的扩散过程和具有棘手可能性的模型中进行平滑。该算法得到了一些理论结果，包括中心极限定理、收敛性和数值稳定性。此外，在强混合假设下，我们在算法的渐近偏差上建立了一个新的O（nε）界，其中n是路径长度，ε控制密度估计器的偏差。
{"title":"A pseudo-marginal sequential Monte Carlo online smoothing algorithm","authors":"P. Gloaguen, Sylvain Le Corff, J. Olsson","doi":"10.3150/21-bej1431","DOIUrl":"https://doi.org/10.3150/21-bej1431","url":null,"abstract":": We consider online computation of expectations of additive state functionals under general path probability measures proportional to products of unnormalised transition densities. These transition densities are assumed to be intractable but possible to estimate, with or without bias. Using pseudo-marginalisation techniques we are able to extend the particle- based, rapid incremental smoother (PaRIS) algorithm proposed in [J. Olsson and J. Westerborn. Efficient particle-based online smoothing in general hidden Markov models: The PaRIS algorithm. Bernoulli , 23(3):1951–1996, 2017] to this setting. The resulting algorithm, which has a linear complex- ity in the number of particles and constant memory requirements, applies to a wide range of challenging path-space Monte Carlo problems, includ- ing smoothing in partially observed diffusion processes and models with intractable likelihood. The algorithm is furnished with several theoretical results, including a central limit theorem, establishing its convergence and numerical stability. Moreover, under strong mixing assumptions we estab- lish a novel O ( nε ) bound on the asymptotic bias of the algorithm, where n is the path length and ε controls the bias of the density estimators.","PeriodicalId":55387,"journal":{"name":"Bernoulli","volume":" ","pages":""},"PeriodicalIF":1.5,"publicationDate":"2022-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48369375","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 concerns the asymptotic properties of the quadratic functionals and associated ordinary least squares estimator in the explosive first-order Gaussian autoregressive process. By the deviation inequalities for multiple Wiener-Itô integrals and asymptotic analysis techniques, Cramér-type moderate deviations are achieved under the explosive and mildly explosive frameworks. As applications, the global and local powers for the unit root test are shown to approach one at exponential rates. Simulation experiments are conducted to confirm the theoretical results.
{"title":"Deviation inequalities and Cramér-type moderate deviations for the explosive autoregressive process","authors":"Hui Jiang, Yilong Wan, Guangyu Yang","doi":"10.3150/21-bej1432","DOIUrl":"https://doi.org/10.3150/21-bej1432","url":null,"abstract":"This paper concerns the asymptotic properties of the quadratic functionals and associated ordinary least squares estimator in the explosive first-order Gaussian autoregressive process. By the deviation inequalities for multiple Wiener-Itô integrals and asymptotic analysis techniques, Cramér-type moderate deviations are achieved under the explosive and mildly explosive frameworks. As applications, the global and local powers for the unit root test are shown to approach one at exponential rates. Simulation experiments are conducted to confirm the theoretical results.","PeriodicalId":55387,"journal":{"name":"Bernoulli","volume":" ","pages":""},"PeriodicalIF":1.5,"publicationDate":"2022-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45150218","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}
LI LIU1 WEN SU2 GUOSHENG YIN2 YING ZHANG3 and XINGQIU ZHAO4 1School of Mathematics and Statistics, Wuhan University, Wuhan, Hubei, 430072, China E-mail: lliu.math@whu.edu.cn 2Department of Statistics and Actuarial Science, The University of Hong Kong, Hong Kong E-mail: jenna.wen.su@connect.hku.hk E-mail: gyin@hku.hk; Corresponding author 3Department of Biostatistics, University of Nebraska Medical Center, Omaha, NE, USA E-mail: ying.zhang@unmc.edu 4Department of Applied Mathematics, The Hong Kong Polytechnic University, Hong Kong E-mail: xingqiu.zhao@polyu.edu.hk
{"title":"Nonparametric inference for reversed mean models with panel count data","authors":"Li Liu, Wen Su, G. Yin, Xingqiu Zhao, Ying Zhang","doi":"10.3150/21-bej1444","DOIUrl":"https://doi.org/10.3150/21-bej1444","url":null,"abstract":"LI LIU1 WEN SU2 GUOSHENG YIN2 YING ZHANG3 and XINGQIU ZHAO4 1School of Mathematics and Statistics, Wuhan University, Wuhan, Hubei, 430072, China E-mail: lliu.math@whu.edu.cn 2Department of Statistics and Actuarial Science, The University of Hong Kong, Hong Kong E-mail: jenna.wen.su@connect.hku.hk E-mail: gyin@hku.hk; Corresponding author 3Department of Biostatistics, University of Nebraska Medical Center, Omaha, NE, USA E-mail: ying.zhang@unmc.edu 4Department of Applied Mathematics, The Hong Kong Polytechnic University, Hong Kong E-mail: xingqiu.zhao@polyu.edu.hk","PeriodicalId":55387,"journal":{"name":"Bernoulli","volume":" ","pages":""},"PeriodicalIF":1.5,"publicationDate":"2022-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48488992","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}
Importance sampling is a widely used technique to estimate properties of a distribution. The resulting estimator is unbiased but may have huge, potentially infinite, variance. This paper proposes trading-off some bias for variance by adaptively winsorizing the importance sampling estimator. The novel procedure is based on the Balancing Principle (or Lepskii’s Method). As a consequence, it offers a principled way to perform winsorization with finitesample guarantees in the form of an oracle inequality. In various examples, the proposed estimator is shown to have smaller mean squared error and mean absolute deviation than leading alternatives such as the traditional importance sampling estimator or winsorizing it via cross-validation.
{"title":"Robust importance sampling with adaptive winsorization","authors":"Paulo Orenstein","doi":"10.3150/21-bej1440","DOIUrl":"https://doi.org/10.3150/21-bej1440","url":null,"abstract":"Importance sampling is a widely used technique to estimate properties of a distribution. The resulting estimator is unbiased but may have huge, potentially infinite, variance. This paper proposes trading-off some bias for variance by adaptively winsorizing the importance sampling estimator. The novel procedure is based on the Balancing Principle (or Lepskii’s Method). As a consequence, it offers a principled way to perform winsorization with finitesample guarantees in the form of an oracle inequality. In various examples, the proposed estimator is shown to have smaller mean squared error and mean absolute deviation than leading alternatives such as the traditional importance sampling estimator or winsorizing it via cross-validation.","PeriodicalId":55387,"journal":{"name":"Bernoulli","volume":" ","pages":""},"PeriodicalIF":1.5,"publicationDate":"2022-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42728548","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}
{"title":"Tracy-Widom limit for the largest eigenvalue of high-dimensional covariance matrices in elliptical distributions","authors":"Wen Jun, Xie Jiahui, Yu Long, Zhou Wang","doi":"10.3150/21-bej1443","DOIUrl":"https://doi.org/10.3150/21-bej1443","url":null,"abstract":"","PeriodicalId":55387,"journal":{"name":"Bernoulli","volume":" ","pages":""},"PeriodicalIF":1.5,"publicationDate":"2022-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46157652","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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