Simon Chatelain, Anne-Laure Fougères, J. Nešlehová
Archimax copula models can account for any type of asymptotic dependence between extremes and at the same time capture joint risks at medium levels. An Archimax copula is characterized by two functional parameters, the stable tail dependence function `, and the Archimedean generator ψ which distorts the extreme-value dependence structure. This article develops semiparametric inference for Archimax copulas: a nonparametric estimator of ` and a momentbased estimator of ψ assuming the latter belongs to a parametric family. Conditions under which ψ and ` are identifiable are derived. The asymptotic behavior of the estimators is then established under broad regularity conditions; performance in small samples is assessed through a comprehensive simulation study. The Archimax copula model with the Clayton generator is then used to analyze monthly rainfall maxima at three stations in French Brittany. The model is seen to fit the data very well, both in the lower and in the upper tail. The nonparametric estimator of ` reveals asymmetric extremal dependence between the stations, which reflects heavy precipitation patterns in the area. Technical proofs, simulation results and R code are provided in the Online Supplement.
{"title":"Inference for Archimax copulas","authors":"Simon Chatelain, Anne-Laure Fougères, J. Nešlehová","doi":"10.1214/19-aos1836","DOIUrl":"https://doi.org/10.1214/19-aos1836","url":null,"abstract":"Archimax copula models can account for any type of asymptotic dependence between extremes and at the same time capture joint risks at medium levels. An Archimax copula is characterized by two functional parameters, the stable tail dependence function `, and the Archimedean generator ψ which distorts the extreme-value dependence structure. This article develops semiparametric inference for Archimax copulas: a nonparametric estimator of ` and a momentbased estimator of ψ assuming the latter belongs to a parametric family. Conditions under which ψ and ` are identifiable are derived. The asymptotic behavior of the estimators is then established under broad regularity conditions; performance in small samples is assessed through a comprehensive simulation study. The Archimax copula model with the Clayton generator is then used to analyze monthly rainfall maxima at three stations in French Brittany. The model is seen to fit the data very well, both in the lower and in the upper tail. The nonparametric estimator of ` reveals asymmetric extremal dependence between the stations, which reflects heavy precipitation patterns in the area. Technical proofs, simulation results and R code are provided in the Online Supplement.","PeriodicalId":8032,"journal":{"name":"Annals of Statistics","volume":"48 1","pages":"1025-1051"},"PeriodicalIF":4.5,"publicationDate":"2020-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44067400","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Hurst function estimation","authors":"Jinqi Shen, T. Hsing","doi":"10.1214/19-aos1825","DOIUrl":"https://doi.org/10.1214/19-aos1825","url":null,"abstract":"","PeriodicalId":8032,"journal":{"name":"Annals of Statistics","volume":"48 1","pages":"838-862"},"PeriodicalIF":4.5,"publicationDate":"2020-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46240529","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
A. Pananjady, Cheng Mao, Vidya Muthukumar, M. Wainwright, T. Courtade
Pairwise comparison data arises in many domains, including tournament rankings, web search, and preference elicitation. Given noisy comparisons of a fixed subset of pairs of items, we study the problem of estimating the underlying comparison probabilities under the assumption of strong stochastic transitivity (SST). We also consider the noisy sorting subclass of the SST model. We show that when the assignment of items to the topology is arbitrary, these permutationbased models, unlike their parametric counterparts, do not admit consistent estimation for most comparison topologies used in practice. We then demonstrate that consistent estimation is possible when the assignment of items to the topology is randomized, thus establishing a dichotomy between worst-case and average-case designs. We propose two computationally efficient estimators in the average-case setting and analyze their risk, showing that it depends on the comparison topology only through the degree sequence of the topology. We also provide explicit classes of graphs for which the rates achieved by these estimators are optimal. Our results are corroborated by simulations on multiple comparison topologies.
{"title":"Worst-case versus average-case design for estimation from partial pairwise comparisons","authors":"A. Pananjady, Cheng Mao, Vidya Muthukumar, M. Wainwright, T. Courtade","doi":"10.1214/19-aos1838","DOIUrl":"https://doi.org/10.1214/19-aos1838","url":null,"abstract":"Pairwise comparison data arises in many domains, including tournament rankings, web search, and preference elicitation. Given noisy comparisons of a fixed subset of pairs of items, we study the problem of estimating the underlying comparison probabilities under the assumption of strong stochastic transitivity (SST). We also consider the noisy sorting subclass of the SST model. We show that when the assignment of items to the topology is arbitrary, these permutationbased models, unlike their parametric counterparts, do not admit consistent estimation for most comparison topologies used in practice. We then demonstrate that consistent estimation is possible when the assignment of items to the topology is randomized, thus establishing a dichotomy between worst-case and average-case designs. We propose two computationally efficient estimators in the average-case setting and analyze their risk, showing that it depends on the comparison topology only through the degree sequence of the topology. We also provide explicit classes of graphs for which the rates achieved by these estimators are optimal. Our results are corroborated by simulations on multiple comparison topologies.","PeriodicalId":8032,"journal":{"name":"Annals of Statistics","volume":"48 1","pages":"1072-1097"},"PeriodicalIF":4.5,"publicationDate":"2020-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49323852","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Prediction error after model search","authors":"Xiaoying Tian","doi":"10.1214/19-AOS1818","DOIUrl":"https://doi.org/10.1214/19-AOS1818","url":null,"abstract":"","PeriodicalId":8032,"journal":{"name":"Annals of Statistics","volume":"48 1","pages":"763-784"},"PeriodicalIF":4.5,"publicationDate":"2020-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45252274","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Bootstrap confidence regions based on M-estimators under nonstandard conditions","authors":"Stephen M. S. Lee, Puyudi Yang","doi":"10.1214/18-aos1803","DOIUrl":"https://doi.org/10.1214/18-aos1803","url":null,"abstract":"","PeriodicalId":8032,"journal":{"name":"Annals of Statistics","volume":"48 1","pages":"274-299"},"PeriodicalIF":4.5,"publicationDate":"2020-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45596271","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Two-step semiparametric empirical likelihood inference","authors":"Francesco Bravo, J. Escanciano, I. Keilegom","doi":"10.1214/18-AOS1788","DOIUrl":"https://doi.org/10.1214/18-AOS1788","url":null,"abstract":"","PeriodicalId":8032,"journal":{"name":"Annals of Statistics","volume":"48 1","pages":"1-26"},"PeriodicalIF":4.5,"publicationDate":"2020-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46503774","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
This article considers simultaneous variable selection and parameter estimation as well as hypothesis testing in censored regression models with unspecified parametric likelihood. For the problem, we utilize certain growing dimensional general estimating equations and propose a penalized generalized empirical likelihood using the folded concave penalties. We first construct general estimating equations attaining the semiparametric efficiency bound with censored regression data and then establish the consistency and oracle properties of the penalized generalized empirical likelihood estimators. Furthermore, we show that the penalized generalized empirical likelihood ratio test statistic has an asymptotic standard central chi-squared distribution. The conditions of local and restricted global optimality of weighted penalized generalized empirical likelihood estimators are also discussed. We present an two-layer iterative algorithm for efficient implementation, and rigorously investigate its convergence property. The good performance of the proposed methods are demonstrated by extensive simulation studies and a real data example is provided for illustration.
{"title":"Penalized generalized empirical likelihood with a diverging number of general estimating equations for censored data","authors":"Nian-Sheng Tang, Xiaodong Yan, Xingqiu Zhao","doi":"10.1214/19-aos1870","DOIUrl":"https://doi.org/10.1214/19-aos1870","url":null,"abstract":"This article considers simultaneous variable selection and parameter estimation as well as hypothesis testing in censored regression models with unspecified parametric likelihood. For the problem, we utilize certain growing dimensional general estimating equations and propose a penalized generalized empirical likelihood using the folded concave penalties. We first construct general estimating equations attaining the semiparametric efficiency bound with censored regression data and then establish the consistency and oracle properties of the penalized generalized empirical likelihood estimators. Furthermore, we show that the penalized generalized empirical likelihood ratio test statistic has an asymptotic standard central chi-squared distribution. The conditions of local and restricted global optimality of weighted penalized generalized empirical likelihood estimators are also discussed. We present an two-layer iterative algorithm for efficient implementation, and rigorously investigate its convergence property. The good performance of the proposed methods are demonstrated by extensive simulation studies and a real data example is provided for illustration.","PeriodicalId":8032,"journal":{"name":"Annals of Statistics","volume":"48 1","pages":"607-627"},"PeriodicalIF":4.5,"publicationDate":"2020-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44061735","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Sliced inverse regression (SIR) is an innovative and effective method for sufficient dimension reduction and data visualization. Recently, an impressive range of penalized SIR methods has been proposed to estimate the central subspace in a sparse fashion. Nonetheless, few of them considered the sparse sufficient dimension reduction from a decision-theoretic point of view. To address this issue, we in this paper establish the minimax rates of convergence for estimating the sparse SIR directions under various commonly used loss functions in the literature of sufficient dimension reduction. We also discover the possible trade-off between statistical guarantee and computational performance for sparse SIR. We finally propose an adaptive estimation scheme for sparse SIR which is computationally tractable and rate optimal. Numerical studies are carried out to confirm the theoretical properties of our proposed methods.
{"title":"Sparse SIR: Optimal rates and adaptive estimation","authors":"Kai Tan, Lei Shi, Zhou Yu","doi":"10.1214/18-aos1791","DOIUrl":"https://doi.org/10.1214/18-aos1791","url":null,"abstract":"Sliced inverse regression (SIR) is an innovative and effective method for sufficient dimension reduction and data visualization. Recently, an impressive range of penalized SIR methods has been proposed to estimate the central subspace in a sparse fashion. Nonetheless, few of them considered the sparse sufficient dimension reduction from a decision-theoretic point of view. To address this issue, we in this paper establish the minimax rates of convergence for estimating the sparse SIR directions under various commonly used loss functions in the literature of sufficient dimension reduction. We also discover the possible trade-off between statistical guarantee and computational performance for sparse SIR. We finally propose an adaptive estimation scheme for sparse SIR which is computationally tractable and rate optimal. Numerical studies are carried out to confirm the theoretical properties of our proposed methods.","PeriodicalId":8032,"journal":{"name":"Annals of Statistics","volume":"48 1","pages":"64-85"},"PeriodicalIF":4.5,"publicationDate":"2020-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43689262","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"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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