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":null,"pages":null},"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}
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":null,"pages":null},"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}
Pub Date : 2023-01-01Epub Date: 2023-09-03DOI: 10.1214/23-ejs2151
Hongxiang Qiu, Alex Luedtke
Bayes estimators are well known to provide a means to incorporate prior knowledge that can be expressed in terms of a single prior distribution. However, when this knowledge is too vague to express with a single prior, an alternative approach is needed. Gamma-minimax estimators provide such an approach. These estimators minimize the worst-case Bayes risk over a set of prior distributions that are compatible with the available knowledge. Traditionally, Gamma-minimaxity is defined for parametric models. In this work, we define Gamma-minimax estimators for general models and propose adversarial meta-learning algorithms to compute them when the set of prior distributions is constrained by generalized moments. Accompanying convergence guarantees are also provided. We also introduce a neural network class that provides a rich, but finite-dimensional, class of estimators from which a Gamma-minimax estimator can be selected. We illustrate our method in two settings, namely entropy estimation and a prediction problem that arises in biodiversity studies.
{"title":"Adversarial meta-learning of Gamma-minimax estimators that leverage prior knowledge.","authors":"Hongxiang Qiu, Alex Luedtke","doi":"10.1214/23-ejs2151","DOIUrl":"10.1214/23-ejs2151","url":null,"abstract":"<p><p>Bayes estimators are well known to provide a means to incorporate prior knowledge that can be expressed in terms of a single prior distribution. However, when this knowledge is too vague to express with a single prior, an alternative approach is needed. Gamma-minimax estimators provide such an approach. These estimators minimize the worst-case Bayes risk over a set <math><mi>Γ</mi></math> of prior distributions that are compatible with the available knowledge. Traditionally, Gamma-minimaxity is defined for parametric models. In this work, we define Gamma-minimax estimators for general models and propose adversarial meta-learning algorithms to compute them when the set of prior distributions is constrained by generalized moments. Accompanying convergence guarantees are also provided. We also introduce a neural network class that provides a rich, but finite-dimensional, class of estimators from which a Gamma-minimax estimator can be selected. We illustrate our method in two settings, namely entropy estimation and a prediction problem that arises in biodiversity studies.</p>","PeriodicalId":49272,"journal":{"name":"Electronic Journal of Statistics","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC10923594/pdf/","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41630279","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","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":null,"pages":null},"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}
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":null,"pages":null},"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}
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":null,"pages":null},"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}
{"title":"Sieve estimation of semiparametric accelerated mean models with panel count data","authors":"Xiangbin Hu, Wen Su, Xingqiu Zhao","doi":"10.1214/23-ejs2128","DOIUrl":"https://doi.org/10.1214/23-ejs2128","url":null,"abstract":"","PeriodicalId":49272,"journal":{"name":"Electronic Journal of Statistics","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45925239","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":null,"pages":null},"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}
Nonregular designs are attractive, as compared with regular designs, not just because they have flexible run sizes but also because of their performances in terms of generalized resolution, projectivity, and hidden projection property. In this paper, we conduct a comprehensive study on three classes of designs that are obtained from Paley’s two constructions of Hadamard matrices. In terms of generalized resolution, we complete the study of Shi and Tang [15] on strength-two designs by adding results on strength-three designs. In terms of projectivty and hidden projection property, our results substantially expand those of Bulutoglu and Cheng [2]. For the purpose of practical applications, we conduct an extensive search of minimum G-aberration designs from those with maximum generalized resolutions and results are obtained for strength-two designs with 36, 44, 48, 52, 60, 64, 96 and 128 runs and strength-three designs with 72, 88 and 120 runs.
{"title":"Nonregular designs from Paley’s Hadamard matrices: Generalized resolution, projectivity and hidden projection property","authors":"Guanzhou Chen, Chenlu Shi, Boxin Tang","doi":"10.1214/23-ejs2148","DOIUrl":"https://doi.org/10.1214/23-ejs2148","url":null,"abstract":"Nonregular designs are attractive, as compared with regular designs, not just because they have flexible run sizes but also because of their performances in terms of generalized resolution, projectivity, and hidden projection property. In this paper, we conduct a comprehensive study on three classes of designs that are obtained from Paley’s two constructions of Hadamard matrices. In terms of generalized resolution, we complete the study of Shi and Tang [15] on strength-two designs by adding results on strength-three designs. In terms of projectivty and hidden projection property, our results substantially expand those of Bulutoglu and Cheng [2]. For the purpose of practical applications, we conduct an extensive search of minimum G-aberration designs from those with maximum generalized resolutions and results are obtained for strength-two designs with 36, 44, 48, 52, 60, 64, 96 and 128 runs and strength-three designs with 72, 88 and 120 runs.","PeriodicalId":49272,"journal":{"name":"Electronic Journal of Statistics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135911387","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}
We investigate the frequentist guarantees of the variational sparse Gaussian process regression model. In the theoretical analysis, we focus on the variational approach with spectral features as inducing variables. We derive guarantees and limitations for the frequentist coverage of the resulting variational credible sets. We also derive sufficient and necessary lower bounds for the number of inducing variables required to achieve minimax posterior contraction rates. The implications of these results are demonstrated for different choices of priors. In a numerical analysis we consider a wider range of inducing variable methods and observe similar phenomena beyond the scope of our theoretical findings.
{"title":"Uncertainty quantification for sparse spectral variational approximations in Gaussian process regression","authors":"Dennis Nieman, Botond Szabo, Harry van Zanten","doi":"10.1214/23-ejs2155","DOIUrl":"https://doi.org/10.1214/23-ejs2155","url":null,"abstract":"We investigate the frequentist guarantees of the variational sparse Gaussian process regression model. In the theoretical analysis, we focus on the variational approach with spectral features as inducing variables. We derive guarantees and limitations for the frequentist coverage of the resulting variational credible sets. We also derive sufficient and necessary lower bounds for the number of inducing variables required to achieve minimax posterior contraction rates. The implications of these results are demonstrated for different choices of priors. In a numerical analysis we consider a wider range of inducing variable methods and observe similar phenomena beyond the scope of our theoretical findings.","PeriodicalId":49272,"journal":{"name":"Electronic Journal of Statistics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135952877","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}
京公网安备 11010802042870号
京ICP备2023020795号-1
扫码关注我们
求助内容：
应助结果提醒方式：