Space-filling designs based on orthogonal arrays are attractive for computer experiments for they can be easily generated with desirable low-dimensional stratification properties. Nonetheless, it is not very clear how they behave and how to construct good such designs under other space-filling criteria. In this paper, we justify orthogonal array-based designs under a broad class of space-filling criteria, which include commonly used distance-, orthogonality- and discrepancy-based measures. To identify designs with even better space-filling properties, we partition orthogonal array-based designs into classes by allowable level permutations and show that the average performance of each class of designs is determined by two types of stratifications, with one of them being achieved by strong orthogonal arrays of strength 2+. Based on these results, we investigate various new and exist-ing constructions of space-filling orthogonal array-based designs, including some strong orthogonal arrays of strength 2+ and mappable nearly orthogonal arrays.
{"title":"A study of orthogonal array-based designs under a broad class of space-filling criteria","authors":"Guanzhou Chen, Boxin Tang","doi":"10.1214/22-aos2215","DOIUrl":"https://doi.org/10.1214/22-aos2215","url":null,"abstract":"Space-filling designs based on orthogonal arrays are attractive for computer experiments for they can be easily generated with desirable low-dimensional stratification properties. Nonetheless, it is not very clear how they behave and how to construct good such designs under other space-filling criteria. In this paper, we justify orthogonal array-based designs under a broad class of space-filling criteria, which include commonly used distance-, orthogonality- and discrepancy-based measures. To identify designs with even better space-filling properties, we partition orthogonal array-based designs into classes by allowable level permutations and show that the average performance of each class of designs is determined by two types of stratifications, with one of them being achieved by strong orthogonal arrays of strength 2+. Based on these results, we investigate various new and exist-ing constructions of space-filling orthogonal array-based designs, including some strong orthogonal arrays of strength 2+ and mappable nearly orthogonal arrays.","PeriodicalId":22375,"journal":{"name":"The Annals of Statistics","volume":"339 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2022-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80741450","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 herein establish an asymptotic representation theorem for locally asymptotically normal quantum statistical models. This theorem enables us to study the asymptotic efficiency of quantum estimators such as quantum regular estimators and quantum minimax estimators, leading to a universal tight lower bound beyond the i.i.d. assumption. This formulation complements the theory of quantum contiguity developed in the previous paper [Fujiwara and Yamagata, Bernoulli 26 (2020) 2105-2141], providing a solid foundation of the theory of weak quantum local asymptotic normality.
本文建立了局部渐近正态量子统计模型的渐近表示定理。这个定理使我们能够研究量子估计量的渐近效率，如量子正则估计量和量子极大极小估计量，从而得到一个超越i.i.d假设的普遍紧下界。该公式补充了先前论文[Fujiwara and Yamagata, Bernoulli 26(2020) 2105-2141]中发展的量子连续理论，为弱量子局部渐近正态性理论提供了坚实的基础。
{"title":"Efficiency of estimators for locally asymptotically normal quantum statistical models","authors":"A. Fujiwara, Koichi Yamagata","doi":"10.1214/23-aos2285","DOIUrl":"https://doi.org/10.1214/23-aos2285","url":null,"abstract":"We herein establish an asymptotic representation theorem for locally asymptotically normal quantum statistical models. This theorem enables us to study the asymptotic efficiency of quantum estimators such as quantum regular estimators and quantum minimax estimators, leading to a universal tight lower bound beyond the i.i.d. assumption. This formulation complements the theory of quantum contiguity developed in the previous paper [Fujiwara and Yamagata, Bernoulli 26 (2020) 2105-2141], providing a solid foundation of the theory of weak quantum local asymptotic normality.","PeriodicalId":22375,"journal":{"name":"The Annals of Statistics","volume":"51 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2022-09-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"84640042","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}
Block-based resampling estimators have been intensively investigated for weakly dependent time processes, which has helped to inform implementation (e.g., best block sizes). However, little is known about resampling performance and block sizes under strong or long-range dependence. To establish guideposts in block selection, we consider a broad class of strongly dependent time processes, formed by a transformation of a stationary long-memory Gaussian series, and examine block-based resampling estimators for the variance of the prototypical sample mean; extensions to general statistical functionals are also considered. Unlike weak dependence, the properties of resampling estimators under strong dependence are shown to depend intricately on the nature of non-linearity in the time series (beyond Hermite ranks) in addition the long-memory coefficient and block size. Additionally, the intuition has often been that optimal block sizes should be larger under strong dependence (say $O(n^{1/2})$ for a sample size $n$) than the optimal order $O(n^{1/3})$ known under weak dependence. This intuition turns out to be largely incorrect, though a block order $O(n^{1/2})$ may be reasonable (and even optimal) in many cases, owing to non-linearity in a long-memory time series. While optimal block sizes are more complex under long-range dependence compared to short-range, we provide a consistent data-driven rule for block selection, and numerical studies illustrate that the guides for block selection perform well in other block-based problems with long-memory time series, such as distribution estimation and strategies for testing Hermite rank.
{"title":"On optimal block resampling for Gaussian-subordinated long-range dependent processes","authors":"Qihao Zhang, S. Lahiri, D. Nordman","doi":"10.1214/22-aos2242","DOIUrl":"https://doi.org/10.1214/22-aos2242","url":null,"abstract":"Block-based resampling estimators have been intensively investigated for weakly dependent time processes, which has helped to inform implementation (e.g., best block sizes). However, little is known about resampling performance and block sizes under strong or long-range dependence. To establish guideposts in block selection, we consider a broad class of strongly dependent time processes, formed by a transformation of a stationary long-memory Gaussian series, and examine block-based resampling estimators for the variance of the prototypical sample mean; extensions to general statistical functionals are also considered. Unlike weak dependence, the properties of resampling estimators under strong dependence are shown to depend intricately on the nature of non-linearity in the time series (beyond Hermite ranks) in addition the long-memory coefficient and block size. Additionally, the intuition has often been that optimal block sizes should be larger under strong dependence (say $O(n^{1/2})$ for a sample size $n$) than the optimal order $O(n^{1/3})$ known under weak dependence. This intuition turns out to be largely incorrect, though a block order $O(n^{1/2})$ may be reasonable (and even optimal) in many cases, owing to non-linearity in a long-memory time series. While optimal block sizes are more complex under long-range dependence compared to short-range, we provide a consistent data-driven rule for block selection, and numerical studies illustrate that the guides for block selection perform well in other block-based problems with long-memory time series, such as distribution estimation and strategies for testing Hermite rank.","PeriodicalId":22375,"journal":{"name":"The Annals of Statistics","volume":"117 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2022-08-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"81076550","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 signal detection by likelihood ratio tests in a number of spiked random matrix models, including but not limited to Gaussian mixtures and spiked Wishart covariance matrices. We work directly with multi-spiked cases in these models and with flexible priors on signal components that allow dependence across spikes. We derive asymptotic normality for the log-likelihood ratios when the signal-tonoise ratios are below certain bounds. In addition, the log-likelihood ratios can be asymptotically decomposed as weighted sums of a collection of statistics which we call bipartite signed cycles. Based on this decomposition, we show that below the bounds we could always achieve the asymptotically optimal powers of likelihood ratio tests via tests based on linear spectral statistics which have polynomial time complexity.
{"title":"Optimal signal detection in some spiked random matrix models: Likelihood ratio tests and linear spectral statistics","authors":"Debapratim Banerjee, Zongming Ma","doi":"10.1214/21-aos2150","DOIUrl":"https://doi.org/10.1214/21-aos2150","url":null,"abstract":"We study signal detection by likelihood ratio tests in a number of spiked random matrix models, including but not limited to Gaussian mixtures and spiked Wishart covariance matrices. We work directly with multi-spiked cases in these models and with flexible priors on signal components that allow dependence across spikes. We derive asymptotic normality for the log-likelihood ratios when the signal-tonoise ratios are below certain bounds. In addition, the log-likelihood ratios can be asymptotically decomposed as weighted sums of a collection of statistics which we call bipartite signed cycles. Based on this decomposition, we show that below the bounds we could always achieve the asymptotically optimal powers of likelihood ratio tests via tests based on linear spectral statistics which have polynomial time complexity.","PeriodicalId":22375,"journal":{"name":"The Annals of Statistics","volume":"52 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2022-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87239089","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 consider d -dimensional stochastic continuum-armed bandits with the expected reward function being additive β -H¨older with sparsity s for 0 < β < ∞ and 1 ≤ s ≤ d . The rate of convergence ˜ O ( s · T β +1 2 β +1 ) for the minimax regret is established where T is the number of rounds. In particular, the minimax regret does not depend on d and is linear in s . A novel algorithm is proposed and is shown to be rate-optimal, up to a logarithmic factor of T . The problem of adaptivity is also studied. A lower bound on the cost of adaptation to the smoothness is obtained and the result implies that adaptation for free is impossible in general without further structural assumptions. We then consider adaptive additive SCAB under an additional self-similarity assumption. An adaptive procedure is constructed and is shown to simultaneously achieve the minimax regret for a range of smoothness levels.
{"title":"Stochastic continuum-armed bandits with additive models: Minimax regrets and adaptive algorithm","authors":"T. Cai, Hongming Pu","doi":"10.1214/22-aos2182","DOIUrl":"https://doi.org/10.1214/22-aos2182","url":null,"abstract":"We consider d -dimensional stochastic continuum-armed bandits with the expected reward function being additive β -H¨older with sparsity s for 0 < β < ∞ and 1 ≤ s ≤ d . The rate of convergence ˜ O ( s · T β +1 2 β +1 ) for the minimax regret is established where T is the number of rounds. In particular, the minimax regret does not depend on d and is linear in s . A novel algorithm is proposed and is shown to be rate-optimal, up to a logarithmic factor of T . The problem of adaptivity is also studied. A lower bound on the cost of adaptation to the smoothness is obtained and the result implies that adaptation for free is impossible in general without further structural assumptions. We then consider adaptive additive SCAB under an additional self-similarity assumption. An adaptive procedure is constructed and is shown to simultaneously achieve the minimax regret for a range of smoothness levels.","PeriodicalId":22375,"journal":{"name":"The Annals of Statistics","volume":"46 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2022-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"84781752","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 distributional symmetry is invariance of a distribution under a group of transformations. Exchangeability and stationarity are examples. We explain that a result of ergodic theory provides a law of large numbers: If the group satisfies suitable conditions, expectations can be estimated by averaging over subsets of transformations, and these estimators are strongly consistent. We show that, if a mixing condition holds, the averages also satisfy a central limit theorem, a Berry-Esseen bound, and concentration. These are extended further to apply to triangular arrays, to randomly subsampled averages, and to a generalization of U-statistics. As applications, we obtain new results on exchangeability, random fields, network models, and a class of marked point processes. We also establish asymptotic normality of the empirical entropy for a large class of processes. Some known results are recovered as special cases, and can hence be interpreted as an outcome of symmetry. The proofs adapt Stein’s method.
{"title":"Limit theorems for distributions invariant under groups of transformations","authors":"Morgane Austern, Peter Orbanz","doi":"10.1214/21-aos2165","DOIUrl":"https://doi.org/10.1214/21-aos2165","url":null,"abstract":"A distributional symmetry is invariance of a distribution under a group of transformations. Exchangeability and stationarity are examples. We explain that a result of ergodic theory provides a law of large numbers: If the group satisfies suitable conditions, expectations can be estimated by averaging over subsets of transformations, and these estimators are strongly consistent. We show that, if a mixing condition holds, the averages also satisfy a central limit theorem, a Berry-Esseen bound, and concentration. These are extended further to apply to triangular arrays, to randomly subsampled averages, and to a generalization of U-statistics. As applications, we obtain new results on exchangeability, random fields, network models, and a class of marked point processes. We also establish asymptotic normality of the empirical entropy for a large class of processes. Some known results are recovered as special cases, and can hence be interpreted as an outcome of symmetry. The proofs adapt Stein’s method.","PeriodicalId":22375,"journal":{"name":"The Annals of Statistics","volume":"238 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2022-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"77012912","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}
Jere Koskela, P. A. Jenkins, A. M. Johansen, Dario Spanò
∗Supported by EPSRC grant EP/R044732/1. †Supported in part by funding from the Lloyd’s Register Foundation – Alan Turing Institute Programme on Data-Centric Engineering, and by EPSRC grants EP/R034710/1 and EP/T004134/1. ‡Also at the Department of Computer Science, University of Warwick. §Also at the Alan Turing Institute. MSC 2010 subject classifications: Primary 60E15; secondary 60G99, 62E20
{"title":"Erratum: Asymptotic genealogies of interacting particle systems with an application to sequential Monte Carlo","authors":"Jere Koskela, P. A. Jenkins, A. M. Johansen, Dario Spanò","doi":"10.1214/21-aos2135","DOIUrl":"https://doi.org/10.1214/21-aos2135","url":null,"abstract":"∗Supported by EPSRC grant EP/R044732/1. †Supported in part by funding from the Lloyd’s Register Foundation – Alan Turing Institute Programme on Data-Centric Engineering, and by EPSRC grants EP/R034710/1 and EP/T004134/1. ‡Also at the Department of Computer Science, University of Warwick. §Also at the Alan Turing Institute. MSC 2010 subject classifications: Primary 60E15; secondary 60G99, 62E20","PeriodicalId":22375,"journal":{"name":"The Annals of Statistics","volume":"280 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2022-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"85222770","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":"Generalization error bounds of dynamic treatment regimes in penalized regression-based learning","authors":"E. J. Oh, Min Qian, Y. Cheung","doi":"10.1214/22-aos2171","DOIUrl":"https://doi.org/10.1214/22-aos2171","url":null,"abstract":"","PeriodicalId":22375,"journal":{"name":"The Annals of Statistics","volume":"30 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2022-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"84365667","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}
Distributed estimation of a Gaussian mean with unknown variance under communication constraints is studied. Necessary and sufficient communication costs under different types of distributed protocols are derived for any estimator that is adaptively rate-optimal over a range of possible values for the variance. Communication-efficient and statistically optimal procedures are developed. The analysis reveals an interesting and important distinction among different types of distributed protocols: compared to the independent protocols, interactive protocols such as the sequential and blackboard protocols require less communication costs for rate-optimal adaptive Gaussian mean estimation. The lower bound techniques developed in the present paper are novel and can be of independent interest. in this supplement the detailed proofs of Lemmas in the paper “Distributed Adaptive Gaussian Mean Estimation with Unknown Variance: Interactive Protocol Helps Adaptation”.
{"title":"Distributed adaptive Gaussian mean estimation with unknown variance: Interactive protocol helps adaptation","authors":"T. Cai, Hongjie Wei","doi":"10.1214/21-aos2167","DOIUrl":"https://doi.org/10.1214/21-aos2167","url":null,"abstract":"Distributed estimation of a Gaussian mean with unknown variance under communication constraints is studied. Necessary and sufficient communication costs under different types of distributed protocols are derived for any estimator that is adaptively rate-optimal over a range of possible values for the variance. Communication-efficient and statistically optimal procedures are developed. The analysis reveals an interesting and important distinction among different types of distributed protocols: compared to the independent protocols, interactive protocols such as the sequential and blackboard protocols require less communication costs for rate-optimal adaptive Gaussian mean estimation. The lower bound techniques developed in the present paper are novel and can be of independent interest. in this supplement the detailed proofs of Lemmas in the paper “Distributed Adaptive Gaussian Mean Estimation with Unknown Variance: Interactive Protocol Helps Adaptation”.","PeriodicalId":22375,"journal":{"name":"The Annals of Statistics","volume":"238 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2022-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79125682","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 estimation of the reach, an ubiquitous regularity parameter in manifold estimation and geometric data analysis. Given an i.i.d. sample over an unknown $d$-dimensional $mathcal{C}^k$-smooth submanifold of $mathbb{R}^D$, we provide optimal nonasymptotic bounds for the estimation of its reach. We build upon a formulation of the reach in terms of maximal curvature on one hand, and geodesic metric distortion on the other hand. The derived rates are adaptive, with rates depending on whether the reach of $M$ arises from curvature or from a bottleneck structure. In the process, we derive optimal geodesic metric estimation bounds.
{"title":"Optimal reach estimation and metric learning","authors":"Eddie Aamari, Cl'ement Berenfeld, Clément Levrard","doi":"10.1214/23-aos2281","DOIUrl":"https://doi.org/10.1214/23-aos2281","url":null,"abstract":"We study the estimation of the reach, an ubiquitous regularity parameter in manifold estimation and geometric data analysis. Given an i.i.d. sample over an unknown $d$-dimensional $mathcal{C}^k$-smooth submanifold of $mathbb{R}^D$, we provide optimal nonasymptotic bounds for the estimation of its reach. We build upon a formulation of the reach in terms of maximal curvature on one hand, and geodesic metric distortion on the other hand. The derived rates are adaptive, with rates depending on whether the reach of $M$ arises from curvature or from a bottleneck structure. In the process, we derive optimal geodesic metric estimation bounds.","PeriodicalId":22375,"journal":{"name":"The Annals of Statistics","volume":"1 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2022-07-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"88306929","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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