We derive approximations to the bias and squared bias with errors of order where is the sample size. Our results hold for a large class of estimators, including quantiles, transformations of unbiased estimators, maximum likelihood estimators in (possibly) incorrectly specified models, and functions thereof. Furthermore, we use the approximations to derive estimators of the mean squared error (MSE) which are correct to order . Since the variance of many estimators is of order , this level of precision is needed for the MSE estimator to properly take the variance into account. We also formulate a new focused information criterion (FIC) for model selection based on the estimators of the squared bias. Lastly, we illustrate the methods on data containing the number of battle deaths in all major inter-state wars between 1823 and the present day. The application illustrates the potentially large impact of using a less-accurate estimator of the squared bias.
{"title":"Accurate bias estimation with applications to focused model selection","authors":"Ingrid Dæhlen, Nils Lid Hjort, Ingrid Hobæk Haff","doi":"10.1111/sjos.12696","DOIUrl":"https://doi.org/10.1111/sjos.12696","url":null,"abstract":"We derive approximations to the bias and squared bias with errors of order <math altimg=\"urn:x-wiley:sjos:media:sjos12696:sjos12696-math-0001\" display=\"inline\" location=\"graphic/sjos12696-math-0001.png\" overflow=\"scroll\">\u0000<semantics>\u0000<mrow>\u0000<mi>o</mi>\u0000<mo stretchy=\"false\">(</mo>\u0000<mn>1</mn>\u0000<mo stretchy=\"false\">/</mo>\u0000<mi>n</mi>\u0000<mo stretchy=\"false\">)</mo>\u0000</mrow>\u0000$$ oleft(1/nright) $$</annotation>\u0000</semantics></math> where <math altimg=\"urn:x-wiley:sjos:media:sjos12696:sjos12696-math-0002\" display=\"inline\" location=\"graphic/sjos12696-math-0002.png\" overflow=\"scroll\">\u0000<semantics>\u0000<mrow>\u0000<mi>n</mi>\u0000</mrow>\u0000$$ n $$</annotation>\u0000</semantics></math> is the sample size. Our results hold for a large class of estimators, including quantiles, transformations of unbiased estimators, maximum likelihood estimators in (possibly) incorrectly specified models, and functions thereof. Furthermore, we use the approximations to derive estimators of the mean squared error (MSE) which are correct to order <math altimg=\"urn:x-wiley:sjos:media:sjos12696:sjos12696-math-0003\" display=\"inline\" location=\"graphic/sjos12696-math-0003.png\" overflow=\"scroll\">\u0000<semantics>\u0000<mrow>\u0000<mi>o</mi>\u0000<mo stretchy=\"false\">(</mo>\u0000<mn>1</mn>\u0000<mo stretchy=\"false\">/</mo>\u0000<mi>n</mi>\u0000<mo stretchy=\"false\">)</mo>\u0000</mrow>\u0000$$ oleft(1/nright) $$</annotation>\u0000</semantics></math>. Since the variance of many estimators is of order <math altimg=\"urn:x-wiley:sjos:media:sjos12696:sjos12696-math-0004\" display=\"inline\" location=\"graphic/sjos12696-math-0004.png\" overflow=\"scroll\">\u0000<semantics>\u0000<mrow>\u0000<mi>O</mi>\u0000<mo stretchy=\"false\">(</mo>\u0000<mn>1</mn>\u0000<mo stretchy=\"false\">/</mo>\u0000<mi>n</mi>\u0000<mo stretchy=\"false\">)</mo>\u0000</mrow>\u0000$$ Oleft(1/nright) $$</annotation>\u0000</semantics></math>, this level of precision is needed for the MSE estimator to properly take the variance into account. We also formulate a new focused information criterion (FIC) for model selection based on the estimators of the squared bias. Lastly, we illustrate the methods on data containing the number of battle deaths in all major inter-state wars between 1823 and the present day. The application illustrates the potentially large impact of using a less-accurate estimator of the squared bias.","PeriodicalId":49567,"journal":{"name":"Scandinavian Journal of Statistics","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2023-11-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138526435","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}
Jinyuan Liu, Xinlian Zhang, Tuo Lin, Ruohui Chen, Yuan Zhong, Tian Chen, Tsungchin Wu, Chenyu Liu, Anna Huang, Tanya T. Nguyen, Ellen E. Lee, Dilip V. Jeste, Xin M. Tu
Abstract This article proposes a distance‐based framework incentivized by the paradigm shift towards feature aggregation for high‐dimensional data, which does not rely on the sparse‐feature assumption or the permutation‐based inference. Focusing on distance‐based outcomes that preserve information without truncating any features, a class of semiparametric regression has been developed, which encapsulates multiple sources of high‐dimensional variables using pairwise outcomes of between‐subject attributes. Further, we propose a strategy to address the interlocking correlations among pairs via the U‐statistics‐based estimating equations (UGEE), which correspond to their unique efficient influence function (EIF). Hence, the resulting semiparametric estimators are robust to distributional misspecification while enjoying root‐n consistency and asymptotic optimality to facilitate inference. In essence, the proposed approach not only circumvents information loss due to feature selection but also improves the model's interpretability and computational feasibility. Simulation studies and applications to the human microbiome and wearables data are provided, where the feature dimensions are tens of thousands. This article is protected by copyright. All rights reserved.
{"title":"A New Paradigm for High‐dimensional Data: Distance‐Based Semiparametric Feature Aggregation Framework via Between‐Subject Attributes","authors":"Jinyuan Liu, Xinlian Zhang, Tuo Lin, Ruohui Chen, Yuan Zhong, Tian Chen, Tsungchin Wu, Chenyu Liu, Anna Huang, Tanya T. Nguyen, Ellen E. Lee, Dilip V. Jeste, Xin M. Tu","doi":"10.1111/sjos.12695","DOIUrl":"https://doi.org/10.1111/sjos.12695","url":null,"abstract":"Abstract This article proposes a distance‐based framework incentivized by the paradigm shift towards feature aggregation for high‐dimensional data, which does not rely on the sparse‐feature assumption or the permutation‐based inference. Focusing on distance‐based outcomes that preserve information without truncating any features, a class of semiparametric regression has been developed, which encapsulates multiple sources of high‐dimensional variables using pairwise outcomes of between‐subject attributes. Further, we propose a strategy to address the interlocking correlations among pairs via the U‐statistics‐based estimating equations (UGEE), which correspond to their unique efficient influence function (EIF). Hence, the resulting semiparametric estimators are robust to distributional misspecification while enjoying root‐n consistency and asymptotic optimality to facilitate inference. In essence, the proposed approach not only circumvents information loss due to feature selection but also improves the model's interpretability and computational feasibility. Simulation studies and applications to the human microbiome and wearables data are provided, where the feature dimensions are tens of thousands. This article is protected by copyright. All rights reserved.","PeriodicalId":49567,"journal":{"name":"Scandinavian Journal of Statistics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-11-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135342278","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}
Abstract We give a thorough description of the asymptotic property of the maximum likelihood estimator (MLE) of the skewness parameter of a Skew Brownian Motion (SBM). Thanks to recent results on the Central Limit Theorem of the rate of convergence of estimators for the SBM, we prove a conjecture left open that the MLE has asymptotically a mixed normal distribution involving the local time with a rate of convergence of order . We also give a series expansion of the MLE and study the asymptotic behavior of the score and its derivatives, as well as their variation with the skewness parameter. In particular, we exhibit a specific behavior when the SBM is actually a Brownian motion, and quantify the explosion of the coefficients of the expansion when the skewness parameter is close to or 1.
{"title":"Maximum likelihood estimator for skew Brownian motion: the convergence rate","authors":"Antoine Lejay, Sara Mazzonetto","doi":"10.1111/sjos.12694","DOIUrl":"https://doi.org/10.1111/sjos.12694","url":null,"abstract":"Abstract We give a thorough description of the asymptotic property of the maximum likelihood estimator (MLE) of the skewness parameter of a Skew Brownian Motion (SBM). Thanks to recent results on the Central Limit Theorem of the rate of convergence of estimators for the SBM, we prove a conjecture left open that the MLE has asymptotically a mixed normal distribution involving the local time with a rate of convergence of order . We also give a series expansion of the MLE and study the asymptotic behavior of the score and its derivatives, as well as their variation with the skewness parameter. In particular, we exhibit a specific behavior when the SBM is actually a Brownian motion, and quantify the explosion of the coefficients of the expansion when the skewness parameter is close to or 1.","PeriodicalId":49567,"journal":{"name":"Scandinavian Journal of Statistics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-11-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135874775","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}
Abstract In this paper, we modify the Bayes risk for the expectile, the so‐called variantile risk measure, to better capture extreme risks. The modified risk measure is called the adjusted standard‐deviatile. First, we derive the asymptotic expansions of the adjusted standard‐deviatile. Next, based on the first‐order asymptotic expansion, we propose two efficient estimation methods for the adjusted standard‐deviatile at intermediate and extreme levels. By using techniques from extreme value theory, the asymptotic normality is proved for both estimators for independent and identically distributed observations and for ‐mixing time series, respectively. Simulations and real data applications are conducted to examine the performance of the proposed estimators.
{"title":"Estimation of the Adjusted Standard‐deviatile for Extreme Risks","authors":"Haoyu Chen, Tiantian Mao, Fan Yang","doi":"10.1111/sjos.12693","DOIUrl":"https://doi.org/10.1111/sjos.12693","url":null,"abstract":"Abstract In this paper, we modify the Bayes risk for the expectile, the so‐called variantile risk measure, to better capture extreme risks. The modified risk measure is called the adjusted standard‐deviatile. First, we derive the asymptotic expansions of the adjusted standard‐deviatile. Next, based on the first‐order asymptotic expansion, we propose two efficient estimation methods for the adjusted standard‐deviatile at intermediate and extreme levels. By using techniques from extreme value theory, the asymptotic normality is proved for both estimators for independent and identically distributed observations and for ‐mixing time series, respectively. Simulations and real data applications are conducted to examine the performance of the proposed estimators.","PeriodicalId":49567,"journal":{"name":"Scandinavian Journal of Statistics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-10-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135461959","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}
Abstract This paper introduces a Nearly Unstable INteger‐valued AutoRegressive Conditional Heteroscedastic (NU‐INARCH) process for dealing with count time series data. It is proved that a proper normalization of the NU‐INARCH process weakly converges to a Cox–Ingersoll–Ross diffusion in the Skorohod topology. The asymptotic distribution of the conditional least squares estimator of the correlation parameter is established as a functional of certain stochastic integrals. Numerical experiments based on Monte Carlo simulations are provided to verify the behavior of the asymptotic distribution under finite samples. These simulations reveal that the nearly unstable approach provides satisfactory and better results than those based on the stationarity assumption even when the true process is not that close to nonstationarity. A unit root test is proposed and its Type‐I error and power are examined via Monte Carlo simulations. As an illustration, the proposed methodology is applied to the daily number of deaths due to COVID‐19 in the United Kingdom.
{"title":"Nearly Unstable Integer‐Valued ARCH Process and Unit Root Testing","authors":"Wagner Barreto-Souza, Ngai Hang Chan","doi":"10.1111/sjos.12689","DOIUrl":"https://doi.org/10.1111/sjos.12689","url":null,"abstract":"Abstract This paper introduces a Nearly Unstable INteger‐valued AutoRegressive Conditional Heteroscedastic (NU‐INARCH) process for dealing with count time series data. It is proved that a proper normalization of the NU‐INARCH process weakly converges to a Cox–Ingersoll–Ross diffusion in the Skorohod topology. The asymptotic distribution of the conditional least squares estimator of the correlation parameter is established as a functional of certain stochastic integrals. Numerical experiments based on Monte Carlo simulations are provided to verify the behavior of the asymptotic distribution under finite samples. These simulations reveal that the nearly unstable approach provides satisfactory and better results than those based on the stationarity assumption even when the true process is not that close to nonstationarity. A unit root test is proposed and its Type‐I error and power are examined via Monte Carlo simulations. As an illustration, the proposed methodology is applied to the daily number of deaths due to COVID‐19 in the United Kingdom.","PeriodicalId":49567,"journal":{"name":"Scandinavian Journal of Statistics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-10-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135666555","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}
Abstract This study intends to introduce kernel mean embedding of probability measures over infinite‐dimensional separable Hilbert spaces induced by functional response statistical models. The embedded function represents the concentration of probability measures in small open neighborhoods, which identifies a pseudo‐likelihood and fosters a rich framework for statistical inference. Utilizing Maximum Mean Discrepancy, we devise new tests in functional response models. The performance of new derived tests is evaluated against competitors in three major problems in functional data analysis including function‐on‐scalar regression, functional one‐way ANOVA, and equality of covariance operators. This article is protected by copyright. All rights reserved.
{"title":"Kernel Mean Embedding of Probability Measures and its Applications to Functional Data Analysis","authors":"Saeed Hayati, Kenji Fukumizu, Afshin Parvardeh","doi":"10.1111/sjos.12691","DOIUrl":"https://doi.org/10.1111/sjos.12691","url":null,"abstract":"Abstract This study intends to introduce kernel mean embedding of probability measures over infinite‐dimensional separable Hilbert spaces induced by functional response statistical models. The embedded function represents the concentration of probability measures in small open neighborhoods, which identifies a pseudo‐likelihood and fosters a rich framework for statistical inference. Utilizing Maximum Mean Discrepancy, we devise new tests in functional response models. The performance of new derived tests is evaluated against competitors in three major problems in functional data analysis including function‐on‐scalar regression, functional one‐way ANOVA, and equality of covariance operators. This article is protected by copyright. All rights reserved.","PeriodicalId":49567,"journal":{"name":"Scandinavian Journal of Statistics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-10-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135923550","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}
Abstract A constrained multivariate linear model is a multivariate linear model with the columns of its coefficient matrix constrained to lie in a known subspace. This class of models includes those typically used to study growth curves and longitudinal data. Envelope methods have been proposed to improve the estimation efficiency in unconstrained multivariate linear models, but have not yet been developed for constrained models. We pursue that development in this article. We first compare the standard envelope estimator with the standard estimator arising from a constrained multivariate model in terms of bias and efficiency. To further improve efficiency, we propose a novel envelope estimator based on a constrained multivariate model. We show the advantage of our proposals by simulations and by studying the probiotic capacity to reduced Salmonella infection. This article is protected by copyright. All rights reserved.
{"title":"Envelopes for multivariate linear regression with linearly constrained coefficients","authors":"Dennis Cook, Liliana Forzani, Lan Liu","doi":"10.1111/sjos.12690","DOIUrl":"https://doi.org/10.1111/sjos.12690","url":null,"abstract":"Abstract A constrained multivariate linear model is a multivariate linear model with the columns of its coefficient matrix constrained to lie in a known subspace. This class of models includes those typically used to study growth curves and longitudinal data. Envelope methods have been proposed to improve the estimation efficiency in unconstrained multivariate linear models, but have not yet been developed for constrained models. We pursue that development in this article. We first compare the standard envelope estimator with the standard estimator arising from a constrained multivariate model in terms of bias and efficiency. To further improve efficiency, we propose a novel envelope estimator based on a constrained multivariate model. We show the advantage of our proposals by simulations and by studying the probiotic capacity to reduced Salmonella infection. This article is protected by copyright. All rights reserved.","PeriodicalId":49567,"journal":{"name":"Scandinavian Journal of Statistics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-10-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135923147","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}
Abstract We consider the problem of clustering functional data according to their covariance structure. We contribute a soft clustering methodology based on the Wasserstein‐Procrustes distance, where the in‐between cluster variability is penalised by a term proportional to the entropy of the partition matrix. In this way, each covariance operator can be partially classified into more than one group. Such soft classification allows for clusters to overlap, and arises naturally in situations where the separation between all or some of the clusters is not well‐defined. We also discuss how to estimate the number of groups and to test for the presence of any cluster structure. The algorithm is illustrated using simulated and real data. An R implementation is available in the Supplementary materials. This article is protected by copyright. All rights reserved.
{"title":"Covariance‐based soft clustering of functional data based on the Wasserstein‐Procrustes metric","authors":"Valentina Masarotto, Guido Masarotto","doi":"10.1111/sjos.12692","DOIUrl":"https://doi.org/10.1111/sjos.12692","url":null,"abstract":"Abstract We consider the problem of clustering functional data according to their covariance structure. We contribute a soft clustering methodology based on the Wasserstein‐Procrustes distance, where the in‐between cluster variability is penalised by a term proportional to the entropy of the partition matrix. In this way, each covariance operator can be partially classified into more than one group. Such soft classification allows for clusters to overlap, and arises naturally in situations where the separation between all or some of the clusters is not well‐defined. We also discuss how to estimate the number of groups and to test for the presence of any cluster structure. The algorithm is illustrated using simulated and real data. An R implementation is available in the Supplementary materials. This article is protected by copyright. All rights reserved.","PeriodicalId":49567,"journal":{"name":"Scandinavian Journal of Statistics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-10-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135482582","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":"Greenland, S. (2023). Divergence vs. decision P‐values: A distinction worth making in theory and keeping in practice. <i>Scandinavian Journal of Statistics</i>, 50, 1–35, https://onlinelibrary.wiley.com/doi/10.1111/sjos.12625","authors":"","doi":"10.1111/sjos.12687","DOIUrl":"https://doi.org/10.1111/sjos.12687","url":null,"abstract":"","PeriodicalId":49567,"journal":{"name":"Scandinavian Journal of Statistics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-10-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135744625","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}
Abstract We address the problem of estimating the mixing time of a Markov chain from a single trajectory of observations. Unlike most previous works which employed Hilbert space methods to estimate spectral gaps, we opt for an approach based on contraction with respect to total variation. Specifically, we estimate the contraction coefficient introduced in Wolfer (2020), inspired from Dobrushin's. This quantity, unlike the spectral gap, controls the mixing time up to strong universal constants and remains applicable to nonreversible chains. We improve existing fully data‐dependent confidence intervals around this contraction coefficient, which are both easier to compute and thinner than spectral counterparts. Furthermore, we introduce a novel analysis beyond the worst‐case scenario by leveraging additional information about the transition matrix. This allows us to derive instance‐dependent rates for estimating the matrix with respect to the induced uniform norm, and some of its mixing properties.
{"title":"Empirical and Instance–Dependent Estimation of Markov Chain and Mixing Time","authors":"Geoffrey Wolfer","doi":"10.1111/sjos.12686","DOIUrl":"https://doi.org/10.1111/sjos.12686","url":null,"abstract":"Abstract We address the problem of estimating the mixing time of a Markov chain from a single trajectory of observations. Unlike most previous works which employed Hilbert space methods to estimate spectral gaps, we opt for an approach based on contraction with respect to total variation. Specifically, we estimate the contraction coefficient introduced in Wolfer (2020), inspired from Dobrushin's. This quantity, unlike the spectral gap, controls the mixing time up to strong universal constants and remains applicable to nonreversible chains. We improve existing fully data‐dependent confidence intervals around this contraction coefficient, which are both easier to compute and thinner than spectral counterparts. Furthermore, we introduce a novel analysis beyond the worst‐case scenario by leveraging additional information about the transition matrix. This allows us to derive instance‐dependent rates for estimating the matrix with respect to the induced uniform norm, and some of its mixing properties.","PeriodicalId":49567,"journal":{"name":"Scandinavian Journal of Statistics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-10-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135790340","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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