{"title":"Universal regression with adversarial responses","authors":"Moise Blanchard, P. Jaillet","doi":"10.1214/23-aos2299","DOIUrl":"https://doi.org/10.1214/23-aos2299","url":null,"abstract":"","PeriodicalId":22375,"journal":{"name":"The Annals of Statistics","volume":"30 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2023-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"72809724","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":"Minimax rate of distribution estimation on unknown submanifolds under adversarial losses","authors":"Rong Tang, Yun Yang","doi":"10.1214/23-aos2291","DOIUrl":"https://doi.org/10.1214/23-aos2291","url":null,"abstract":"","PeriodicalId":22375,"journal":{"name":"The Annals of Statistics","volume":"144 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2023-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"85272876","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":"AutoRegressive approximations to nonstationary time series with inference and applications","authors":"Xiucai Ding, Zhou Zhou","doi":"10.1214/23-aos2288","DOIUrl":"https://doi.org/10.1214/23-aos2288","url":null,"abstract":"","PeriodicalId":22375,"journal":{"name":"The Annals of Statistics","volume":"260 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2023-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76481151","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":"Nonlinear independent component analysis for discrete-time and continuous-time signals","authors":"A. Schell, Harald Oberhauser","doi":"10.1214/23-aos2256","DOIUrl":"https://doi.org/10.1214/23-aos2256","url":null,"abstract":"","PeriodicalId":22375,"journal":{"name":"The Annals of Statistics","volume":"45 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2023-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"72981122","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}
Developments in genome-wide association studies and the increasing availability of summary genetic association data have made the application of two-sample Mendelian Randomization (MR) with summary data increasingly popular. Conventional two-sample MR methods often employ the same sample for selecting relevant genetic variants and for constructing final causal estimates. Such a practice often leads to biased causal effect estimates due to the well known"winner's curse"phenomenon. To address this fundamental challenge, we first examine its consequence on causal effect estimation both theoretically and empirically. We then propose a novel framework that systematically breaks the winner's curse, leading to unbiased association effect estimates for the selected genetic variants. Building upon the proposed framework, we introduce a novel rerandomized inverse variance weighted estimator that is consistent when selection and parameter estimation are conducted on the same sample. Under appropriate conditions, we show that the proposed RIVW estimator for the causal effect converges to a normal distribution asymptotically and its variance can be well estimated. We illustrate the finite-sample performance of our approach through Monte Carlo experiments and two empirical examples.
{"title":"Breaking the winner’s curse in Mendelian randomization: Rerandomized inverse variance weighted estimator","authors":"Xinwei Ma, Jingshen Wang, Chong Wu","doi":"10.1214/22-aos2247","DOIUrl":"https://doi.org/10.1214/22-aos2247","url":null,"abstract":"Developments in genome-wide association studies and the increasing availability of summary genetic association data have made the application of two-sample Mendelian Randomization (MR) with summary data increasingly popular. Conventional two-sample MR methods often employ the same sample for selecting relevant genetic variants and for constructing final causal estimates. Such a practice often leads to biased causal effect estimates due to the well known\"winner's curse\"phenomenon. To address this fundamental challenge, we first examine its consequence on causal effect estimation both theoretically and empirically. We then propose a novel framework that systematically breaks the winner's curse, leading to unbiased association effect estimates for the selected genetic variants. Building upon the proposed framework, we introduce a novel rerandomized inverse variance weighted estimator that is consistent when selection and parameter estimation are conducted on the same sample. Under appropriate conditions, we show that the proposed RIVW estimator for the causal effect converges to a normal distribution asymptotically and its variance can be well estimated. We illustrate the finite-sample performance of our approach through Monte Carlo experiments and two empirical examples.","PeriodicalId":22375,"journal":{"name":"The Annals of Statistics","volume":"17 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2023-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86399291","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 a general class of regression models with normally distributed covariates, and the associated nonconvex problem of fitting these models from data. We develop a general recipe for analyzing the convergence of iterative algorithms for this task from a random initialization. In particular, provided each iteration can be written as the solution to a convex optimization problem satisfying some natural conditions, we leverage Gaussian comparison theorems to derive a deterministic sequence that provides sharp upper and lower bounds on the error of the algorithm with sample-splitting. Crucially, this deterministic sequence accurately captures both the convergence rate of the algorithm and the eventual error floor in the finite-sample regime, and is distinct from the commonly used “population” sequence that results from taking the infinite-sample limit. We apply our general framework to derive several concrete consequences for parameter estimation in popular statistical models including phase retrieval and mixtures of regressions. Provided the sample size scales near-linearly in the dimension, we show sharp global convergence rates for both higher-order algorithms based on alternating updates and first-order algorithms based on subgradient descent. These corollaries, in turn, reveal multiple nonstandard phenomena that are then corroborated by extensive numerical experiments.
{"title":"Sharp global convergence guarantees for iterative nonconvex optimization with random data","authors":"Kabir Chandrasekher, Ashwin Pananjady, Christos Thrampoulidis","doi":"10.1214/22-aos2246","DOIUrl":"https://doi.org/10.1214/22-aos2246","url":null,"abstract":"We consider a general class of regression models with normally distributed covariates, and the associated nonconvex problem of fitting these models from data. We develop a general recipe for analyzing the convergence of iterative algorithms for this task from a random initialization. In particular, provided each iteration can be written as the solution to a convex optimization problem satisfying some natural conditions, we leverage Gaussian comparison theorems to derive a deterministic sequence that provides sharp upper and lower bounds on the error of the algorithm with sample-splitting. Crucially, this deterministic sequence accurately captures both the convergence rate of the algorithm and the eventual error floor in the finite-sample regime, and is distinct from the commonly used “population” sequence that results from taking the infinite-sample limit. We apply our general framework to derive several concrete consequences for parameter estimation in popular statistical models including phase retrieval and mixtures of regressions. Provided the sample size scales near-linearly in the dimension, we show sharp global convergence rates for both higher-order algorithms based on alternating updates and first-order algorithms based on subgradient descent. These corollaries, in turn, reveal multiple nonstandard phenomena that are then corroborated by extensive numerical experiments.","PeriodicalId":22375,"journal":{"name":"The Annals of Statistics","volume":"54 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2023-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80800901","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 the question of estimating the drift for a large class of ergodic multivariate and possibly nonreversible diffusion processes, based on continuous observations, in sup -norm loss. Nonparametric classes of smooth functions of unknown order are considered, and we suggest an adaptive approach which allows to construct drift estimators attaining optimal sup -norm rates of convergence. Reversibility structures and related functional inequalities are known to be key tools for these estimation problems. We can discard such restrictions by making use of mixing properties which are satisfied for the very general class of processes under consideration. Analysing diffusions, the scalar case is very distinct from the general multivariate setting. Therefore, we treat scalar and multivariate processes separately which leads to in several aspects improved univariate results. While we consider drift estimation on bounded domains for exponentially β -mixing multivariate processes, for scalar diffusion processes we work under minimal assumptions that allow estimation of unbounded drift terms over the entire real line, and we provide classical minimax results (including lower bounds) which cannot be obtained under state-of-the-art conditions in the multivariate case. In addition, we prove a Donsker theorem for the classical kernel estimator of the invariant density in the scalar setting and establish its semiparametric efficiency.
{"title":"Sup-norm adaptive drift estimation for multivariate nonreversible diffusions","authors":"Cathrine Aeckerle-Willems, C. Strauch","doi":"10.1214/22-aos2237","DOIUrl":"https://doi.org/10.1214/22-aos2237","url":null,"abstract":"We consider the question of estimating the drift for a large class of ergodic multivariate and possibly nonreversible diffusion processes, based on continuous observations, in sup -norm loss. Nonparametric classes of smooth functions of unknown order are considered, and we suggest an adaptive approach which allows to construct drift estimators attaining optimal sup -norm rates of convergence. Reversibility structures and related functional inequalities are known to be key tools for these estimation problems. We can discard such restrictions by making use of mixing properties which are satisfied for the very general class of processes under consideration. Analysing diffusions, the scalar case is very distinct from the general multivariate setting. Therefore, we treat scalar and multivariate processes separately which leads to in several aspects improved univariate results. While we consider drift estimation on bounded domains for exponentially β -mixing multivariate processes, for scalar diffusion processes we work under minimal assumptions that allow estimation of unbounded drift terms over the entire real line, and we provide classical minimax results (including lower bounds) which cannot be obtained under state-of-the-art conditions in the multivariate case. In addition, we prove a Donsker theorem for the classical kernel estimator of the invariant density in the scalar setting and establish its semiparametric efficiency.","PeriodicalId":22375,"journal":{"name":"The Annals of Statistics","volume":"12 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2022-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87453902","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}
This paper suggests a multiplicative volatility model where volatility is decomposed into a stationary and a non-stationary persistent part. We provide a testing procedure to determine which type of volatility is prevalent in the data. The persistent part of volatility is associated with a nonstationary persistent process satisfying some smoothness and moment conditions. The stationary part is related to stationary conditional heteroskedasticity. We outline theory and conditions that allow the extraction of the persistent part from the data and enable standard conditional heteroskedasticity tests to detect stationary volatility after persistent volatility is taken into account. Monte Carlo results support the testing strategy in small samples. The empirical application of the theory supports the persistent volatility paradigm, suggesting that stationary conditional heteroskedasticity is considerably less pronounced than previously thought.
{"title":"Choosing between persistent and stationary volatility","authors":"Ilias Chronopoulos, L. Giraitis, G. Kapetanios","doi":"10.1214/22-aos2236","DOIUrl":"https://doi.org/10.1214/22-aos2236","url":null,"abstract":"This paper suggests a multiplicative volatility model where volatility is decomposed into a stationary and a non-stationary persistent part. We provide a testing procedure to determine which type of volatility is prevalent in the data. The persistent part of volatility is associated with a nonstationary persistent process satisfying some smoothness and moment conditions. The stationary part is related to stationary conditional heteroskedasticity. We outline theory and conditions that allow the extraction of the persistent part from the data and enable standard conditional heteroskedasticity tests to detect stationary volatility after persistent volatility is taken into account. Monte Carlo results support the testing strategy in small samples. The empirical application of the theory supports the persistent volatility paradigm, suggesting that stationary conditional heteroskedasticity is considerably less pronounced than previously thought.","PeriodicalId":22375,"journal":{"name":"The Annals of Statistics","volume":"23 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2022-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"89429894","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 the nonparametric multivariate isotonic regression problem, where the regression function is assumed to be nondecreasing with respect to each predictor. Our goal is to construct a Bayesian credible interval for the function value at a given interior point with assured limiting frequentist coverage. We put a prior on unrestricted step-functions, but make inference using the induced posterior measure by an"immersion map"from the space of unrestricted functions to that of multivariate monotone functions. This allows maintaining the natural conjugacy for posterior sampling. A natural immersion map to use is a projection via a distance, but in the present context, a block isotonization map is found to be more useful. The approach of using the induced"immersion posterior"measure instead of the original posterior to make inference provides a useful extension of the Bayesian paradigm, particularly helpful when the model space is restricted by some complex relations. We establish a key weak convergence result for the posterior distribution of the function at a point in terms of some functional of a multi-indexed Gaussian process that leads to an expression for the limiting coverage of the Bayesian credible interval. Analogous to a recent result for univariate monotone functions, we find that the limiting coverage is slightly higher than the credibility, the opposite of a phenomenon observed in smoothing problems. Interestingly, the relation between credibility and limiting coverage does not involve any unknown parameter. Hence by a recalibration procedure, we can get a predetermined asymptotic coverage by choosing a suitable credibility level smaller than the targeted coverage, and thus also shorten the credible intervals.
{"title":"Coverage of credible intervals in Bayesian multivariate isotonic regression","authors":"Kangkang Wang, S. Ghosal","doi":"10.1214/23-aos2298","DOIUrl":"https://doi.org/10.1214/23-aos2298","url":null,"abstract":"We consider the nonparametric multivariate isotonic regression problem, where the regression function is assumed to be nondecreasing with respect to each predictor. Our goal is to construct a Bayesian credible interval for the function value at a given interior point with assured limiting frequentist coverage. We put a prior on unrestricted step-functions, but make inference using the induced posterior measure by an\"immersion map\"from the space of unrestricted functions to that of multivariate monotone functions. This allows maintaining the natural conjugacy for posterior sampling. A natural immersion map to use is a projection via a distance, but in the present context, a block isotonization map is found to be more useful. The approach of using the induced\"immersion posterior\"measure instead of the original posterior to make inference provides a useful extension of the Bayesian paradigm, particularly helpful when the model space is restricted by some complex relations. We establish a key weak convergence result for the posterior distribution of the function at a point in terms of some functional of a multi-indexed Gaussian process that leads to an expression for the limiting coverage of the Bayesian credible interval. Analogous to a recent result for univariate monotone functions, we find that the limiting coverage is slightly higher than the credibility, the opposite of a phenomenon observed in smoothing problems. Interestingly, the relation between credibility and limiting coverage does not involve any unknown parameter. Hence by a recalibration procedure, we can get a predetermined asymptotic coverage by choosing a suitable credibility level smaller than the targeted coverage, and thus also shorten the credible intervals.","PeriodicalId":22375,"journal":{"name":"The Annals of Statistics","volume":"152 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2022-11-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"77051651","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}
Motivated by crowd-sourcing applications, we consider a model where we have partial observations from a bivariate isotonic n x d matrix with an unknown permutation $pi$ * acting on its rows. Focusing on the twin problems of recovering the permutation $pi$ * and estimating the unknown matrix, we introduce a polynomial-time procedure achieving the minimax risk for these two problems, this for all possible values of n, d, and all possible sampling efforts. Along the way, we establish that, in some regimes, recovering the unknown permutation $pi$ * is considerably simpler than estimating the matrix.
受众包应用程序的激励，我们考虑一个模型，其中我们有来自二元等渗n x d矩阵的部分观测值，该矩阵具有未知排列$pi$ *作用于其行。关注恢复排列$pi$ *和估计未知矩阵的孪生问题，我们引入了一个多项式时间过程来实现这两个问题的最小最大风险，这适用于所有可能的n, d值和所有可能的采样努力。在此过程中，我们建立了，在某些情况下，恢复未知的排列$pi$ *比估计矩阵要简单得多。
{"title":"Optimal Permutation Estimation in CrowdSourcing problems","authors":"Emmanuel Pilliat, A. Carpentier, N. Verzelen","doi":"10.1214/23-aos2271","DOIUrl":"https://doi.org/10.1214/23-aos2271","url":null,"abstract":"Motivated by crowd-sourcing applications, we consider a model where we have partial observations from a bivariate isotonic n x d matrix with an unknown permutation $pi$ * acting on its rows. Focusing on the twin problems of recovering the permutation $pi$ * and estimating the unknown matrix, we introduce a polynomial-time procedure achieving the minimax risk for these two problems, this for all possible values of n, d, and all possible sampling efforts. Along the way, we establish that, in some regimes, recovering the unknown permutation $pi$ * is considerably simpler than estimating the matrix.","PeriodicalId":22375,"journal":{"name":"The Annals of Statistics","volume":"77 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2022-11-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"81295858","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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