{"title":"Characterization of the solutions set of the generalized LASSO problems for non-full rank cases","authors":"Jaesung Hwang, Joong-Yeon Won, Yongdai Kim","doi":"10.1214/23-ejs2138","DOIUrl":"https://doi.org/10.1214/23-ejs2138","url":null,"abstract":"","PeriodicalId":49272,"journal":{"name":"Electronic Journal of Statistics","volume":" ","pages":""},"PeriodicalIF":1.1,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46041984","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The composite quantile regression (CQR) was introduced by Zou and Yuan [Ann. Statist. 36 (2008) 1108–1126] as a robust regression method for linear models with heavy-tailed errors while achieving high efficiency. Its penalized counterpart for high-dimensional sparse models was recently studied in Gu and Zou [IEEE Trans. Inf. Theory 66 (2020) 7132–7154], along with a specialized optimization algorithm based on the alternating direct method of multipliers (ADMM). Compared to the various first-order algorithms for penalized least squares, ADMM-based algorithms are not well-adapted to large-scale problems. To overcome this computational hardness, in this paper we employ a convolution-smoothed technique to CQR, complemented with iteratively reweighted ℓ1-regularization. The smoothed composite loss function is convex, twice continuously differentiable, and locally strong convex with high probability. We propose a gradient-based algorithm for penalized smoothed CQR via a variant of the majorize-minimization principal, which gains substantial computational efficiency over ADMM. Theoretically, we show that the iteratively reweighted ℓ1-penalized smoothed CQR estimator achieves near-minimax optimal convergence rate under heavy-tailed errors without any moment constraint, and further achieves near-oracle convergence rate under a weaker minimum signal strength condition than needed in Gu and Zou (2020). Numerical studies demonstrate that the proposed method exhibits significant computational advantages without compromising statistical performance compared to two state-of-the-art methods that achieve robustness and high efficiency simultaneously.
综合分位数回归(CQR)是由邹和袁[Ann]提出的。统计学家。36(2008)1108-1126]作为具有重尾误差的线性模型的鲁棒回归方法，同时实现了高效率。最近，Gu和Zou [IEEE Trans]研究了高维稀疏模型的惩罚对应物。Inf. Theory 66(2020) 7132-7154]，以及基于乘数交替直接法(ADMM)的专门优化算法。与各种一阶惩罚最小二乘算法相比，基于admm的算法不太适合大规模问题。为了克服这种计算困难，在本文中，我们对CQR采用了卷积平滑技术，并辅以迭代重加权的1-正则化。光滑复合损失函数是凸的、两次连续可微的、高概率的局部强凸。我们提出了一种基于梯度的惩罚平滑CQR算法，该算法通过最大-最小原则的变体获得了比ADMM更高的计算效率。理论上，我们证明了迭代重加权的1-惩罚光滑CQR估计器在没有任何矩约束的情况下在重尾误差下实现了近极小极大最优收敛速率，并且在较弱的最小信号强度条件下实现了比Gu和Zou(2020)所需的近oracle收敛速率。数值研究表明，与同时实现鲁棒性和高效率的两种最先进的方法相比，该方法在不影响统计性能的情况下具有显著的计算优势。
{"title":"High-dimensional composite quantile regression: Optimal statistical guarantees and fast algorithms","authors":"Haeseong Moon, Wen-Xin Zhou","doi":"10.1214/23-ejs2147","DOIUrl":"https://doi.org/10.1214/23-ejs2147","url":null,"abstract":"The composite quantile regression (CQR) was introduced by Zou and Yuan [Ann. Statist. 36 (2008) 1108–1126] as a robust regression method for linear models with heavy-tailed errors while achieving high efficiency. Its penalized counterpart for high-dimensional sparse models was recently studied in Gu and Zou [IEEE Trans. Inf. Theory 66 (2020) 7132–7154], along with a specialized optimization algorithm based on the alternating direct method of multipliers (ADMM). Compared to the various first-order algorithms for penalized least squares, ADMM-based algorithms are not well-adapted to large-scale problems. To overcome this computational hardness, in this paper we employ a convolution-smoothed technique to CQR, complemented with iteratively reweighted ℓ1-regularization. The smoothed composite loss function is convex, twice continuously differentiable, and locally strong convex with high probability. We propose a gradient-based algorithm for penalized smoothed CQR via a variant of the majorize-minimization principal, which gains substantial computational efficiency over ADMM. Theoretically, we show that the iteratively reweighted ℓ1-penalized smoothed CQR estimator achieves near-minimax optimal convergence rate under heavy-tailed errors without any moment constraint, and further achieves near-oracle convergence rate under a weaker minimum signal strength condition than needed in Gu and Zou (2020). Numerical studies demonstrate that the proposed method exhibits significant computational advantages without compromising statistical performance compared to two state-of-the-art methods that achieve robustness and high efficiency simultaneously.","PeriodicalId":49272,"journal":{"name":"Electronic Journal of Statistics","volume":"38 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135911568","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"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 present and analyze random number generators for the Poisson Kernel-Based Distribution (PKBD) on the sphere. We show that the only currently available sampling scheme presented in Golzy and Markatou (2020) can be improved by a better selection of hyper-parameters but still yields an unbounded rejection constant as the concentration parameter approaches 1. Furthermore, we introduce two additional and superior sampling methods for which boundedness in the above mentioned case can be obtained. The first method proposes initial draws from angular central Gaussian distribution and offers uniformly bounded rejection constants for a significant part of the PKBD parameter space. The second method uses adaptive rejection sampling and the results of Ulrich (1984) to sample from the projected Saw distribution (Saw, 1978). Finally, both new methods are compared in a simulation study.
{"title":"Efficient sampling from the PKBD distribution","authors":"Lukas Sablica, Kurt Hornik, Josef Leydold","doi":"10.1214/23-ejs2149","DOIUrl":"https://doi.org/10.1214/23-ejs2149","url":null,"abstract":"In this paper we present and analyze random number generators for the Poisson Kernel-Based Distribution (PKBD) on the sphere. We show that the only currently available sampling scheme presented in Golzy and Markatou (2020) can be improved by a better selection of hyper-parameters but still yields an unbounded rejection constant as the concentration parameter approaches 1. Furthermore, we introduce two additional and superior sampling methods for which boundedness in the above mentioned case can be obtained. The first method proposes initial draws from angular central Gaussian distribution and offers uniformly bounded rejection constants for a significant part of the PKBD parameter space. The second method uses adaptive rejection sampling and the results of Ulrich (1984) to sample from the projected Saw distribution (Saw, 1978). Finally, both new methods are compared in a simulation study.","PeriodicalId":49272,"journal":{"name":"Electronic Journal of Statistics","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135911577","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Recent works have proposed regression models which are invariant across data collection environments [24, 20, 11, 16, 8]. These estimators often have a causal interpretation under conditions on the environments and type of invariance imposed. One recent example, the Causal Dantzig (CD), is consistent under hidden confounding and represents an alternative to classical instrumental variable estimators such as Two Stage Least Squares (TSLS). In this work we derive the CD as a generalized method of moments (GMM) estimator. The GMM representation leads to several practical results, including 1) creation of the Generalized Causal Dantzig (GCD) estimator which can be applied to problems with continuous environments where the CD cannot be fit 2) a Hybrid (GCD-TSLS combination) estimator which has properties superior to GCD or TSLS alone 3) straightforward asymptotic results for all methods using GMM theory. We compare the CD, GCD, TSLS, and Hybrid estimators in simulations and an application to a Flow Cytometry data set. The newly proposed GCD and Hybrid estimators have superior performance to existing methods in many settings.
{"title":"Estimating causal effects with hidden confounding using instrumental variables and environments","authors":"James P. Long, Hongxu Zhu, Kim-Anh Do, Min Jin Ha","doi":"10.1214/23-ejs2160","DOIUrl":"https://doi.org/10.1214/23-ejs2160","url":null,"abstract":"Recent works have proposed regression models which are invariant across data collection environments [24, 20, 11, 16, 8]. These estimators often have a causal interpretation under conditions on the environments and type of invariance imposed. One recent example, the Causal Dantzig (CD), is consistent under hidden confounding and represents an alternative to classical instrumental variable estimators such as Two Stage Least Squares (TSLS). In this work we derive the CD as a generalized method of moments (GMM) estimator. The GMM representation leads to several practical results, including 1) creation of the Generalized Causal Dantzig (GCD) estimator which can be applied to problems with continuous environments where the CD cannot be fit 2) a Hybrid (GCD-TSLS combination) estimator which has properties superior to GCD or TSLS alone 3) straightforward asymptotic results for all methods using GMM theory. We compare the CD, GCD, TSLS, and Hybrid estimators in simulations and an application to a Flow Cytometry data set. The newly proposed GCD and Hybrid estimators have superior performance to existing methods in many settings.","PeriodicalId":49272,"journal":{"name":"Electronic Journal of Statistics","volume":"4 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135610266","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Given a zero-mean Gaussian random field with a covariance function that belongs to a parametric family of covariance functions, we introduce a new notion of likelihood approximations, termed truncated-likelihood functions. Truncated-likelihood functions are based on direct functional approximations of the presumed family of covariance functions. For compactly supported covariance functions, within an increasing-domain asymptotic framework, we provide sufficient conditions under which consistency and asymptotic normality of estimators based on truncated-likelihood functions are preserved. We apply our result to the family of generalized Wendland covariance functions and discuss several examples of Wendland approximations. For families of covariance functions that are not compactly supported, we combine our results with the covariance tapering approach and show that ML estimators, based on truncated-tapered likelihood functions, asymptotically minimize the Kullback-Leibler divergence, when the taper range is fixed.
{"title":"Asymptotic analysis of ML-covariance parameter estimators based on covariance approximations","authors":"Reinhard Furrer, Michael Hediger","doi":"10.1214/23-ejs2170","DOIUrl":"https://doi.org/10.1214/23-ejs2170","url":null,"abstract":"Given a zero-mean Gaussian random field with a covariance function that belongs to a parametric family of covariance functions, we introduce a new notion of likelihood approximations, termed truncated-likelihood functions. Truncated-likelihood functions are based on direct functional approximations of the presumed family of covariance functions. For compactly supported covariance functions, within an increasing-domain asymptotic framework, we provide sufficient conditions under which consistency and asymptotic normality of estimators based on truncated-likelihood functions are preserved. We apply our result to the family of generalized Wendland covariance functions and discuss several examples of Wendland approximations. For families of covariance functions that are not compactly supported, we combine our results with the covariance tapering approach and show that ML estimators, based on truncated-tapered likelihood functions, asymptotically minimize the Kullback-Leibler divergence, when the taper range is fixed.","PeriodicalId":49272,"journal":{"name":"Electronic Journal of Statistics","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135662412","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Covariance discriminative power of kernel clustering methods","authors":"A. Kammoun, Romain Couillet","doi":"10.1214/23-ejs2107","DOIUrl":"https://doi.org/10.1214/23-ejs2107","url":null,"abstract":"","PeriodicalId":49272,"journal":{"name":"Electronic Journal of Statistics","volume":" ","pages":""},"PeriodicalIF":1.1,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48173445","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In the famous least sum of trimmed squares (LTS) estimator [21], residuals are first squared and then trimmed. In this article, we first trim residuals – using a depth trimming scheme – and then square the remaining of residuals. The estimator that minimizes the sum of trimmed and squared residuals, is called an LST estimator. Not only is the LST a robust alternative to the classic least sum of squares (LS) estimator. It also has a high finite sample breakdown point-and can resist, asymptotically, up to 50% contamination without breakdown – in sharp contrast to the 0% of the LS estimator. The population version of the LST is Fisher consistent, and the sample version is strong, root-n consistent, and asymptotically normal. We propose approximate algorithms for computing the LST and test on synthetic and real data sets. Despite being approximate, one of the algorithms compute the LST estimator quickly with relatively small variances in contrast to the famous LTS estimator. Thus, evidence suggests the LST serves as a robust alternative to the LS estimator and is feasible even in high dimension data sets with contamination and outliers.
{"title":"Least sum of squares of trimmed residuals regression","authors":"Yijun Zuo, Hanwen Zuo","doi":"10.1214/23-ejs2164","DOIUrl":"https://doi.org/10.1214/23-ejs2164","url":null,"abstract":"In the famous least sum of trimmed squares (LTS) estimator [21], residuals are first squared and then trimmed. In this article, we first trim residuals – using a depth trimming scheme – and then square the remaining of residuals. The estimator that minimizes the sum of trimmed and squared residuals, is called an LST estimator. Not only is the LST a robust alternative to the classic least sum of squares (LS) estimator. It also has a high finite sample breakdown point-and can resist, asymptotically, up to 50% contamination without breakdown – in sharp contrast to the 0% of the LS estimator. The population version of the LST is Fisher consistent, and the sample version is strong, root-n consistent, and asymptotically normal. We propose approximate algorithms for computing the LST and test on synthetic and real data sets. Despite being approximate, one of the algorithms compute the LST estimator quickly with relatively small variances in contrast to the famous LTS estimator. Thus, evidence suggests the LST serves as a robust alternative to the LS estimator and is feasible even in high dimension data sets with contamination and outliers.","PeriodicalId":49272,"journal":{"name":"Electronic Journal of Statistics","volume":"2010 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136202186","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
This paper studies a class of plug-in estimators of the stationary density of an autoregressive model with autoregression parameter 0<ϱ<1. These use two types of estimator of the innovation density, a standard kernel estimator and a weighted kernel estimator with weights chosen to mimic the condition that the innovation density has mean zero. Bahadur expansions are obtained for this class of estimators in L1, the space of integrable functions. These stochastic expansions establish root-n consistency in the L1-norm. It is shown that the density estimators based on the weighted kernel estimators are asymptotically efficient if an asymptotically efficient estimator of the autoregression parameter is used. Here asymptotic efficiency is understood in the sense of the Hájek–Le Cam convolution theorem.
{"title":"Efficient density estimation in an AR(1) model","authors":"Anton Schick, Wolfgang Wefelmeyer","doi":"10.1214/23-ejs2166","DOIUrl":"https://doi.org/10.1214/23-ejs2166","url":null,"abstract":"This paper studies a class of plug-in estimators of the stationary density of an autoregressive model with autoregression parameter 0<ϱ<1. These use two types of estimator of the innovation density, a standard kernel estimator and a weighted kernel estimator with weights chosen to mimic the condition that the innovation density has mean zero. Bahadur expansions are obtained for this class of estimators in L1, the space of integrable functions. These stochastic expansions establish root-n consistency in the L1-norm. It is shown that the density estimators based on the weighted kernel estimators are asymptotically efficient if an asymptotically efficient estimator of the autoregression parameter is used. Here asymptotic efficiency is understood in the sense of the Hájek–Le Cam convolution theorem.","PeriodicalId":49272,"journal":{"name":"Electronic Journal of Statistics","volume":"358 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135610486","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Binary data with high-dimensional covariates have become more and more common in many disciplines. In this paper we consider the maximum likelihood estimation for logistic regression models with a diverging number of covariates. Under mild conditions we establish the asymptotic normality of the maximum likelihood estimate when the number of covariates p goes to infinity with the sample size n in the order of p = o(n). This remarkably improves the existing results that can only allow p growing in an order of o(nα) with α ∈ [1/5, 1/2] [12, 14]. A major innovation in our proof is the use of the injective function. AMS 2000 subject classifications: Primary 62F12; secondary 62J12.
{"title":"Corrigendum to “Maximum likelihood estimation in logistic regression models with a diverging number of covariates”","authors":"Hua Liang, Pang Du","doi":"10.1214/12-EJS731","DOIUrl":"https://doi.org/10.1214/12-EJS731","url":null,"abstract":"Binary data with high-dimensional covariates have become more and more common in many disciplines. In this paper we consider the maximum likelihood estimation for logistic regression models with a diverging number of covariates. Under mild conditions we establish the asymptotic normality of the maximum likelihood estimate when the number of covariates p goes to infinity with the sample size n in the order of p = o(n). This remarkably improves the existing results that can only allow p growing in an order of o(nα) with α ∈ [1/5, 1/2] [12, 14]. A major innovation in our proof is the use of the injective function. AMS 2000 subject classifications: Primary 62F12; secondary 62J12.","PeriodicalId":49272,"journal":{"name":"Electronic Journal of Statistics","volume":" ","pages":""},"PeriodicalIF":1.1,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1214/12-EJS731","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48042414","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"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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