Ismaila Ba, Jean‐François Coeurjolly, F. Cuevas-Pacheco
The class of Gibbs point processes (GPP) is a large class of spatial point processes able to model both clustered and repulsive point patterns. They are specified by their conditional intensity, which for a point pattern $mathbf{x}$ and a location $u$, is roughly speaking the probability that an event occurs in an infinitesimal ball around $u$ given the rest of the configuration is $mathbf{x}$. The most simple and natural class of models is the class of pairwise interaction point processes where the conditional intensity depends on the number of points and pairwise distances between them. This paper is concerned with the problem of estimating the pairwise interaction function non parametrically. We propose to estimate it using an orthogonal series expansion of its logarithm. Such an approach has numerous advantages compared to existing ones. The estimation procedure is simple, fast and completely data-driven. We provide asymptotic properties such as consistency and asymptotic normality and show the efficiency of the procedure through simulation experiments and illustrate it with several datasets.
吉布斯点过程(Gibbs point processes, GPP)是一类能够模拟聚类和排斥点模式的空间点过程。它们由它们的条件强度指定，对于点模式$mathbf{x}$和位置$u$，粗略地说，事件发生在$u$周围的无限小球中，给定其余配置为$mathbf{x}$的概率。最简单和最自然的一类模型是成对相互作用点过程，其中条件强度取决于点的数量和它们之间的成对距离。本文研究了非参数估计两两相互作用函数的问题。我们建议用它的对数的正交级数展开来估计它。与现有的方法相比，这种方法有许多优点。估算过程简单、快速且完全由数据驱动。我们提供了渐近性质，如一致性和渐近正态性，并通过仿真实验证明了该过程的有效性，并用几个数据集说明了它。
{"title":"Pairwise interaction function estimation of stationary Gibbs point processes using basis expansion","authors":"Ismaila Ba, Jean‐François Coeurjolly, F. Cuevas-Pacheco","doi":"10.1214/23-aos2284","DOIUrl":"https://doi.org/10.1214/23-aos2284","url":null,"abstract":"The class of Gibbs point processes (GPP) is a large class of spatial point processes able to model both clustered and repulsive point patterns. They are specified by their conditional intensity, which for a point pattern $mathbf{x}$ and a location $u$, is roughly speaking the probability that an event occurs in an infinitesimal ball around $u$ given the rest of the configuration is $mathbf{x}$. The most simple and natural class of models is the class of pairwise interaction point processes where the conditional intensity depends on the number of points and pairwise distances between them. This paper is concerned with the problem of estimating the pairwise interaction function non parametrically. We propose to estimate it using an orthogonal series expansion of its logarithm. Such an approach has numerous advantages compared to existing ones. The estimation procedure is simple, fast and completely data-driven. We provide asymptotic properties such as consistency and asymptotic normality and show the efficiency of the procedure through simulation experiments and illustrate it with several datasets.","PeriodicalId":22375,"journal":{"name":"The Annals of Statistics","volume":"72 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2021-10-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"77327526","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Two-level parallel flats designs","authors":"Chunyan Wang, Robert W. Mee","doi":"10.1214/21-aos2071","DOIUrl":"https://doi.org/10.1214/21-aos2071","url":null,"abstract":"","PeriodicalId":22375,"journal":{"name":"The Annals of Statistics","volume":"7 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2021-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86996243","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Peter Bickel, Marta Fiocco, Mathisca de Gunst, Friedrich Götze
{"title":"Willem van Zwet’s research","authors":"Peter Bickel, Marta Fiocco, Mathisca de Gunst, Friedrich Götze","doi":"10.1214/21-aos2060","DOIUrl":"https://doi.org/10.1214/21-aos2060","url":null,"abstract":"","PeriodicalId":22375,"journal":{"name":"The Annals of Statistics","volume":"26 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2021-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"90449198","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Willem van Zwet, teacher and thesis advisor","authors":"Sara van de Geer, C. Klaassen","doi":"10.1214/21-aos2069","DOIUrl":"https://doi.org/10.1214/21-aos2069","url":null,"abstract":"","PeriodicalId":22375,"journal":{"name":"The Annals of Statistics","volume":"106 2","pages":""},"PeriodicalIF":0.0,"publicationDate":"2021-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"91434200","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The proposal and study of dependent Bayesian nonparametric models has been one of the most active research lines in the last two decades, with random vectors of measures representing a natural and popular tool to define them. Nonetheless a principled approach to understand and quantify the associated dependence structure is still missing. In this work we devise a general, and non model-specific, framework to achieve this task for random measure based models, which consists in: (a) quantify dependence of a random vector of probabilities in terms of closeness to exchangeability, which corresponds to the maximally dependent coupling with the same marginal distributions, i.e. the comonotonic vector; (b) recast the problem in terms of the underlying random measures (in the same Fréchet class) and quantify the closeness to comonotonicity; (c) define a distance based on the Wasserstein metric, which is ideally suited for spaces of measures, to measure the dependence in a principled way. Several results, which represent the very first in the area, are obtained. In particular, useful bounds in terms of the underlying Lévy intensities are derived relying on compound Poisson approximations. These are then specialized to popular models in the Bayesian literature leading to interesting insights.
{"title":"Measuring dependence in the Wasserstein distance for Bayesian nonparametric models","authors":"Marta Catalano, A. Lijoi, Igor Prünster","doi":"10.1214/21-aos2065","DOIUrl":"https://doi.org/10.1214/21-aos2065","url":null,"abstract":"The proposal and study of dependent Bayesian nonparametric models has been one of the most active research lines in the last two decades, with random vectors of measures representing a natural and popular tool to define them. Nonetheless a principled approach to understand and quantify the associated dependence structure is still missing. In this work we devise a general, and non model-specific, framework to achieve this task for random measure based models, which consists in: (a) quantify dependence of a random vector of probabilities in terms of closeness to exchangeability, which corresponds to the maximally dependent coupling with the same marginal distributions, i.e. the comonotonic vector; (b) recast the problem in terms of the underlying random measures (in the same Fréchet class) and quantify the closeness to comonotonicity; (c) define a distance based on the Wasserstein metric, which is ideally suited for spaces of measures, to measure the dependence in a principled way. Several results, which represent the very first in the area, are obtained. In particular, useful bounds in terms of the underlying Lévy intensities are derived relying on compound Poisson approximations. These are then specialized to popular models in the Bayesian literature leading to interesting insights.","PeriodicalId":22375,"journal":{"name":"The Annals of Statistics","volume":"33 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2021-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"84138473","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Set structured global empirical risk minimizers are rate optimal in general dimensions","authors":"Q. Han","doi":"10.1214/21-aos2049","DOIUrl":"https://doi.org/10.1214/21-aos2049","url":null,"abstract":"","PeriodicalId":22375,"journal":{"name":"The Annals of Statistics","volume":"1 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2021-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75555767","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
A considerable portion of the work on mixed orthogonal arrays applies specifically to arrays of strength 2. Although strength t = 2 is arguably the most important case for statistical applications, there is an urgent need for better methods for t ≥ 3. However, the knowledge on the existence of arrays for t ≥ 3 is rather limited. In this paper, new construction methods for symmetric and asymmetric orthogonal arrays (OAs) with high strength are proposed by using lower strength orthogonal partitions of spaces and OAs. A positive answer is provided to the open problem in Hedayat, Sloane and Stufken ( Orthogonal Arrays: Theory and Applications (1999) Springer) on developing better methods and tools for the construction of mixed orthogonal arrays with strength t ≥ 3. Not only are the methods straightforward, but also they are useful for constructing symmetric or asymmetric OAs of arbitrary strengths, numbers of levels and various sizes. The constructed OAs can be utilized to generate more OAs. The resulting OAs have a high degree of flexibility and many other desirable properties. Some selective OAs are tabulated for practical uses.
{"title":"Construction of mixed orthogonal arrays with high strength","authors":"S. Pang, Jing Wang, D. Lin, Min-Qian Liu","doi":"10.1214/21-aos2063","DOIUrl":"https://doi.org/10.1214/21-aos2063","url":null,"abstract":"A considerable portion of the work on mixed orthogonal arrays applies specifically to arrays of strength 2. Although strength t = 2 is arguably the most important case for statistical applications, there is an urgent need for better methods for t ≥ 3. However, the knowledge on the existence of arrays for t ≥ 3 is rather limited. In this paper, new construction methods for symmetric and asymmetric orthogonal arrays (OAs) with high strength are proposed by using lower strength orthogonal partitions of spaces and OAs. A positive answer is provided to the open problem in Hedayat, Sloane and Stufken ( Orthogonal Arrays: Theory and Applications (1999) Springer) on developing better methods and tools for the construction of mixed orthogonal arrays with strength t ≥ 3. Not only are the methods straightforward, but also they are useful for constructing symmetric or asymmetric OAs of arbitrary strengths, numbers of levels and various sizes. The constructed OAs can be utilized to generate more OAs. The resulting OAs have a high degree of flexibility and many other desirable properties. Some selective OAs are tabulated for practical uses.","PeriodicalId":22375,"journal":{"name":"The Annals of Statistics","volume":"25 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2021-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"84393177","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We study the rank of the instantaneous or spot covariance matrix $Sigma_X(t)$ of a multidimensional continuous semi-martingale $X(t)$. Given high-frequency observations $X(i/n)$, $i=0,ldots,n$, we test the null hypothesis $rank(Sigma_X(t))le r$ for all $t$ against local alternatives where the average $(r+1)$st eigenvalue is larger than some signal detection rate $v_n$. A major problem is that the inherent averaging in local covariance statistics produces a bias that distorts the rank statistics. We show that the bias depends on the regularity and a spectral gap of $Sigma_X(t)$. We establish explicit matrix perturbation and concentration results that provide non-asymptotic uniform critical values and optimal signal detection rates $v_n$. This leads to a rank estimation method via sequential testing. For a class of stochastic volatility models, we determine data-driven critical values via normed p-variations of estimated local covariance matrices. The methods are illustrated by simulations and an application to high-frequency data of U.S. government bonds.
{"title":"Inference on the maximal rank of time-varying covariance matrices using high-frequency data","authors":"M. Reiß, Lars Winkelmann","doi":"10.17169/REFUBIUM-32210","DOIUrl":"https://doi.org/10.17169/REFUBIUM-32210","url":null,"abstract":"We study the rank of the instantaneous or spot covariance matrix $Sigma_X(t)$ of a multidimensional continuous semi-martingale $X(t)$. Given high-frequency observations $X(i/n)$, $i=0,ldots,n$, we test the null hypothesis $rank(Sigma_X(t))le r$ for all $t$ against local alternatives where the average $(r+1)$st eigenvalue is larger than some signal detection rate $v_n$. \u0000A major problem is that the inherent averaging in local covariance statistics produces a bias that distorts the rank statistics. We show that the bias depends on the regularity and a spectral gap of $Sigma_X(t)$. We establish explicit matrix perturbation and concentration results that provide non-asymptotic uniform critical values and optimal signal detection rates $v_n$. This leads to a rank estimation method via sequential testing. For a class of stochastic volatility models, we determine data-driven critical values via normed p-variations of estimated local covariance matrices. The methods are illustrated by simulations and an application to high-frequency data of U.S. government bonds.","PeriodicalId":22375,"journal":{"name":"The Annals of Statistics","volume":"40 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2021-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"89116949","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Additive regression is studied in a very general setting where both the response and predictors are allowed to be non-Euclidean. The response takes values in a general separable Hilbert space, whereas the predictors take values in general semimetric spaces, which covers a very wide range of nonstandard response variables and predictors. A general framework of estimating additive models is presented for semimetric space-valued predictors. In particular, full details of implementation and the corresponding theory are given for predictors taking values in Hilbert spaces and/or Riemannian manifolds. The existence of the estimators, convergence of a backfitting algorithm, rates of convergence and asymptotic distributions of the estimators are discussed. The finite sample performance of the estimators is investigated by means of two simulation studies. Finally, three data sets covering several types of nonEuclidean data are analyzed to illustrate the usefulness of the proposed general approach.
{"title":"Additive regression for non-Euclidean responses and predictors","authors":"Jeong Min Jeon, B. Park, I. Van Keilegom","doi":"10.1214/21-aos2048","DOIUrl":"https://doi.org/10.1214/21-aos2048","url":null,"abstract":"Additive regression is studied in a very general setting where both the response and predictors are allowed to be non-Euclidean. The response takes values in a general separable Hilbert space, whereas the predictors take values in general semimetric spaces, which covers a very wide range of nonstandard response variables and predictors. A general framework of estimating additive models is presented for semimetric space-valued predictors. In particular, full details of implementation and the corresponding theory are given for predictors taking values in Hilbert spaces and/or Riemannian manifolds. The existence of the estimators, convergence of a backfitting algorithm, rates of convergence and asymptotic distributions of the estimators are discussed. The finite sample performance of the estimators is investigated by means of two simulation studies. Finally, three data sets covering several types of nonEuclidean data are analyzed to illustrate the usefulness of the proposed general approach.","PeriodicalId":22375,"journal":{"name":"The Annals of Statistics","volume":"77 4","pages":""},"PeriodicalIF":0.0,"publicationDate":"2021-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"91403899","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Preface: Section of memorial articles for Willem van Zwet","authors":"R. Samworth, Ming Yuan","doi":"10.1214/21-aos2112","DOIUrl":"https://doi.org/10.1214/21-aos2112","url":null,"abstract":"","PeriodicalId":22375,"journal":{"name":"The Annals of Statistics","volume":"438 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2021-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76545850","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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