Pub Date : 2025-02-06DOI: 10.1016/j.jspi.2025.106275
Jingying Zhou , Hui Jiang , Weigang Wang
In this paper, under discrete observations, we study the asymptotic consistency, asymptotic normality and Cramér-type moderate deviations of Yule’s nonsense correlation statistic for two Ornstein–Uhlenbeck processes. As applications, the global and local powers of the hypothesis testing for the independence between two Ornstein–Uhlenbeck processes are shown to approach one at exponential rates. Simulation experiments are conducted to confirm the theoretical results. Moreover, empirical applications illustrate the usefulness of the above mentioned statistic and the asymptotic theory. The main methods consist of the deviation inequalities and Cramér-type moderate deviations for multiple Wiener–Itô integrals and asymptotic analysis techniques.
{"title":"Asymptotic normality and Cramér-type moderate deviations of Yule’s nonsense correlation statistic for Ornstein–Uhlenbeck processes","authors":"Jingying Zhou , Hui Jiang , Weigang Wang","doi":"10.1016/j.jspi.2025.106275","DOIUrl":"10.1016/j.jspi.2025.106275","url":null,"abstract":"<div><div>In this paper, under discrete observations, we study the asymptotic consistency, asymptotic normality and Cramér-type moderate deviations of Yule’s nonsense correlation statistic for two Ornstein–Uhlenbeck processes. As applications, the global and local powers of the hypothesis testing for the independence between two Ornstein–Uhlenbeck processes are shown to approach one at exponential rates. Simulation experiments are conducted to confirm the theoretical results. Moreover, empirical applications illustrate the usefulness of the above mentioned statistic and the asymptotic theory. The main methods consist of the deviation inequalities and Cramér-type moderate deviations for multiple Wiener–Itô integrals and asymptotic analysis techniques.</div></div>","PeriodicalId":50039,"journal":{"name":"Journal of Statistical Planning and Inference","volume":"238 ","pages":"Article 106275"},"PeriodicalIF":0.8,"publicationDate":"2025-02-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143349639","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 : 2025-02-06DOI: 10.1016/j.jspi.2025.106273
Ansgar Steland
Gumbel-type extreme value theory for arrays of discrete Gaussian random fields is studied and applied to some classes of discretely sampled approximately locally self-similar Gaussian processes, especially micro-noise models. Non-Gaussian discrete random fields are handled by considering the maximum of local averages of raw data or residuals. Based on some novel weak approximations with rate for (weighted) partial sums for spatial linear processes including results under a class of local alternatives, sufficient conditions for Gumbel-type asymptotics of maximum-type detection rules to detect peaks and suspicious areas in image data and, more generally, random field data, are established. The results are examined by simulations and illustrated by analyzing CT brain image data.
{"title":"Detection of suspicious areas in non-stationary Gaussian fields and locally averaged non-Gaussian linear fields","authors":"Ansgar Steland","doi":"10.1016/j.jspi.2025.106273","DOIUrl":"10.1016/j.jspi.2025.106273","url":null,"abstract":"<div><div>Gumbel-type extreme value theory for arrays of discrete Gaussian random fields is studied and applied to some classes of discretely sampled approximately locally self-similar Gaussian processes, especially micro-noise models. Non-Gaussian discrete random fields are handled by considering the maximum of local averages of raw data or residuals. Based on some novel weak approximations with rate for (weighted) partial sums for spatial linear processes including results under a class of local alternatives, sufficient conditions for Gumbel-type asymptotics of maximum-type detection rules to detect peaks and suspicious areas in image data and, more generally, random field data, are established. The results are examined by simulations and illustrated by analyzing CT brain image data.</div></div>","PeriodicalId":50039,"journal":{"name":"Journal of Statistical Planning and Inference","volume":"238 ","pages":"Article 106273"},"PeriodicalIF":0.8,"publicationDate":"2025-02-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143349644","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 : 2025-01-28DOI: 10.1016/j.jspi.2025.106272
Riddhiman Saha , Priyam Das , Nilanjana Laha
In this paper, we consider the two-sample location shift model, a classic semiparametric model introduced by Stein(1956). This model is known for its adaptive nature, enabling nonparametric estimation with full parametric efficiency. Existing nonparametric estimators of the location shift often depend on external tuning parameters, which restricts their practical applicability Vanet al. (1998). We demonstrate that introducing an additional assumption of log-concavity on the underlying density can alleviate the need for tuning parameters. We propose a one step estimator for location shift estimation, utilizing log-concave density estimation techniques to facilitate tuning-free estimation of the efficient influence function. While we use a truncated version of the one step estimator to theoretically demonstrate adaptivity, our simulations indicate that the one step estimators perform best with zero truncation, eliminating the need for tuning during practical implementation. Notably, the efficiency of the truncated one step estimators steadily increases as the truncation level decreases, and those with low levels of truncation exhibit nearly identical empirical performance to the estimator with zero truncation. We apply our method to investigate the location shift in the distribution of Spanish annual household incomes following the 2008 financial crisis.
{"title":"The two-sample location shift model under log-concavity","authors":"Riddhiman Saha , Priyam Das , Nilanjana Laha","doi":"10.1016/j.jspi.2025.106272","DOIUrl":"10.1016/j.jspi.2025.106272","url":null,"abstract":"<div><div>In this paper, we consider the two-sample location shift model, a classic semiparametric model introduced by Stein(1956). This model is known for its adaptive nature, enabling nonparametric estimation with full parametric efficiency. Existing nonparametric estimators of the location shift often depend on external tuning parameters, which restricts their practical applicability Vanet al. (1998). We demonstrate that introducing an additional assumption of log-concavity on the underlying density can alleviate the need for tuning parameters. We propose a one step estimator for location shift estimation, utilizing log-concave density estimation techniques to facilitate tuning-free estimation of the efficient influence function. While we use a truncated version of the one step estimator to theoretically demonstrate adaptivity, our simulations indicate that the one step estimators perform best with zero truncation, eliminating the need for tuning during practical implementation. Notably, the efficiency of the truncated one step estimators steadily increases as the truncation level decreases, and those with low levels of truncation exhibit nearly identical empirical performance to the estimator with zero truncation. We apply our method to investigate the location shift in the distribution of Spanish annual household incomes following the 2008 financial crisis.</div></div>","PeriodicalId":50039,"journal":{"name":"Journal of Statistical Planning and Inference","volume":"238 ","pages":"Article 106272"},"PeriodicalIF":0.8,"publicationDate":"2025-01-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143150096","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 : 2025-01-25DOI: 10.1016/j.jspi.2025.106271
Jian Zhang , Tong Wang
Skew normal model suffers from inferential drawbacks, namely singular Fisher information when it is close to symmetry and diverging of maximum likelihood estimation. This causes a large variation of the conventional maximum likelihood estimate. To address the above drawbacks, Azzalini and Arellano-Valle (2013) introduced maximum penalised likelihood estimation (MPLE) by subtracting a penalty function from the log-likelihood function with a pre-specified penalty coefficient. Here, we propose a cross-validated MPLE to improve its performance when the underlying model is close to symmetry. We develop a theory for MPLE, where an asymptotic rate for the cross-validated penalty coefficient is derived. We further show that the proposed cross-validated MPLE is asymptotically efficient under certain conditions. In simulation studies and a real data application, we demonstrate that the proposed estimator can outperform the conventional MPLE when the model is close to symmetry.
{"title":"On cross-validated estimation of skew normal model","authors":"Jian Zhang , Tong Wang","doi":"10.1016/j.jspi.2025.106271","DOIUrl":"10.1016/j.jspi.2025.106271","url":null,"abstract":"<div><div>Skew normal model suffers from inferential drawbacks, namely singular Fisher information when it is close to symmetry and diverging of maximum likelihood estimation. This causes a large variation of the conventional maximum likelihood estimate. To address the above drawbacks, Azzalini and Arellano-Valle (2013) introduced maximum penalised likelihood estimation (MPLE) by subtracting a penalty function from the log-likelihood function with a pre-specified penalty coefficient. Here, we propose a cross-validated MPLE to improve its performance when the underlying model is close to symmetry. We develop a theory for MPLE, where an asymptotic rate for the cross-validated penalty coefficient is derived. We further show that the proposed cross-validated MPLE is asymptotically efficient under certain conditions. In simulation studies and a real data application, we demonstrate that the proposed estimator can outperform the conventional MPLE when the model is close to symmetry.</div></div>","PeriodicalId":50039,"journal":{"name":"Journal of Statistical Planning and Inference","volume":"238 ","pages":"Article 106271"},"PeriodicalIF":0.8,"publicationDate":"2025-01-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143150094","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}
Pub Date : 2025-01-06DOI: 10.1016/j.jspi.2024.106260
Xiaoguang Wang , Rong Hu , Mengyu Li
It is a fundamental task to predict patients’ survival outcomes in clinical research. As an extension of the Cox proportional hazards model, the time-dependent coefficient Cox model is typically utilized for time-to-event data with time-dependent effects. When the number of covariates is large, the curse of dimensionality emerges for most existing methods. To overcome the limitation and improve predictive performance, a semiparametric model averaging approach is proposed for the time-dependent coefficient Cox model. We introduce a novel criterion to estimate model weights and demonstrate its theoretical properties. Extensive simulation studies are conducted to compare the proposed technique with existing competitive methods. A real clinical data set is also analyzed to illustrate the advantages of our approach.
{"title":"Model averaging prediction for survival data with time-dependent effects","authors":"Xiaoguang Wang , Rong Hu , Mengyu Li","doi":"10.1016/j.jspi.2024.106260","DOIUrl":"10.1016/j.jspi.2024.106260","url":null,"abstract":"<div><div>It is a fundamental task to predict patients’ survival outcomes in clinical research. As an extension of the Cox proportional hazards model, the time-dependent coefficient Cox model is typically utilized for time-to-event data with time-dependent effects. When the number of covariates is large, the curse of dimensionality emerges for most existing methods. To overcome the limitation and improve predictive performance, a semiparametric model averaging approach is proposed for the time-dependent coefficient Cox model. We introduce a novel criterion to estimate model weights and demonstrate its theoretical properties. Extensive simulation studies are conducted to compare the proposed technique with existing competitive methods. A real clinical data set is also analyzed to illustrate the advantages of our approach.</div></div>","PeriodicalId":50039,"journal":{"name":"Journal of Statistical Planning and Inference","volume":"238 ","pages":"Article 106260"},"PeriodicalIF":0.8,"publicationDate":"2025-01-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143150095","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 : 2024-12-30DOI: 10.1016/j.jspi.2024.106261
Bingjing Tang , Vinayak Rao
Nonparametric Bayesian models are used routinely as flexible and powerful models of complex data. In many situations, an applied scientist may have additional informative beliefs about the data distribution of interest, for instance, the distribution of its mean or a subset components. This often will not be compatible with the nonparametric prior. An important challenge is then to incorporate this partial prior belief into nonparametric Bayesian models. In this paper, we are motivated by settings where practitioners have additional distributional information about a subset of the coordinates of the observations being modeled. Our approach links this problem to that of conditional density modeling. Our main idea is a novel constrained Bayesian model, based on a perturbation of a parametric distribution with a transformed Gaussian process prior on the perturbation function. We develop a corresponding posterior sampling method based on data augmentation. We illustrate the efficacy of our proposed constrained nonparametric Bayesian model in a variety of real-world scenarios including modeling environmental and earthquake data.
{"title":"Marginally constrained nonparametric Bayesian inference through Gaussian processes","authors":"Bingjing Tang , Vinayak Rao","doi":"10.1016/j.jspi.2024.106261","DOIUrl":"10.1016/j.jspi.2024.106261","url":null,"abstract":"<div><div>Nonparametric Bayesian models are used routinely as flexible and powerful models of complex data. In many situations, an applied scientist may have additional informative beliefs about the data distribution of interest, for instance, the distribution of its mean or a subset components. This often will not be compatible with the nonparametric prior. An important challenge is then to incorporate this partial prior belief into nonparametric Bayesian models. In this paper, we are motivated by settings where practitioners have additional distributional information about a subset of the coordinates of the observations being modeled. Our approach links this problem to that of conditional density modeling. Our main idea is a novel constrained Bayesian model, based on a perturbation of a parametric distribution with a transformed Gaussian process prior on the perturbation function. We develop a corresponding posterior sampling method based on data augmentation. We illustrate the efficacy of our proposed constrained nonparametric Bayesian model in a variety of real-world scenarios including modeling environmental and earthquake data.</div></div>","PeriodicalId":50039,"journal":{"name":"Journal of Statistical Planning and Inference","volume":"237 ","pages":"Article 106261"},"PeriodicalIF":0.8,"publicationDate":"2024-12-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143133631","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 : 2024-12-28DOI: 10.1016/j.jspi.2024.106262
Liuping Hu , Kashinath Chatterjee , Jianhui Ning , Hong Qin
Asymmetrical (mixed-level) uniform designs are useful for both computer and physical experiments. However, constructing these designs is often challenging due to their complex asymmetrical structure. In this paper, we propose novel methods for constructing uniform designs with mixed two-, three-, and four/nine-levels. Our construction methods are deterministic, allowing us to circumvent the complexity associated with stochastic algorithms. We evaluate uniformity using the wrap-around - and Lee discrepancies. We establish useful analytic relationships between uniformity and aberration, and derive new general lower bounds for discrepancies that are tighter than those currently available in the literature. These new benchmarks can effectively measure the uniformity of asymmetrical designs. Additionally, we provide examples demonstrating the efficacy of our construction methods and the relevance of the newly obtained lower bounds. Finally, through simulations, we show that the designs produced using our methods perform well in constructing statistical surrogate models.
{"title":"Deterministic construction methods for asymmetrical uniform designs","authors":"Liuping Hu , Kashinath Chatterjee , Jianhui Ning , Hong Qin","doi":"10.1016/j.jspi.2024.106262","DOIUrl":"10.1016/j.jspi.2024.106262","url":null,"abstract":"<div><div>Asymmetrical (mixed-level) uniform designs are useful for both computer and physical experiments. However, constructing these designs is often challenging due to their complex asymmetrical structure. In this paper, we propose novel methods for constructing uniform designs with mixed two-, three-, and four/nine-levels. Our construction methods are deterministic, allowing us to circumvent the complexity associated with stochastic algorithms. We evaluate uniformity using the wrap-around <span><math><msub><mrow><mi>L</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span>- and Lee discrepancies. We establish useful analytic relationships between uniformity and aberration, and derive new general lower bounds for discrepancies that are tighter than those currently available in the literature. These new benchmarks can effectively measure the uniformity of asymmetrical designs. Additionally, we provide examples demonstrating the efficacy of our construction methods and the relevance of the newly obtained lower bounds. Finally, through simulations, we show that the designs produced using our methods perform well in constructing statistical surrogate models.</div></div>","PeriodicalId":50039,"journal":{"name":"Journal of Statistical Planning and Inference","volume":"237 ","pages":"Article 106262"},"PeriodicalIF":0.8,"publicationDate":"2024-12-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143133632","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 : 2024-12-18DOI: 10.1016/j.jspi.2024.106252
Kazuhiko Hayakawa, Boyan Yin
This paper proposes the maximum likelihood (ML) estimator for a short panel autoregressive model with a flexible form of observed factors as well as unknown interactive fixed effects. We show that the ML estimator is consistent and asymptotically normally distributed as the number of cross-sectional units increases with the number of time periods being fixed. It should be noted that this asymptotic result holds uniformly for the autoregressive coefficient less than, equal to, or greater than one, in sharp contrast to existing estimators. Monte Carlo simulation results show that the ML estimator has desirable finite sample properties.
{"title":"Maximum likelihood estimation of short panel autoregressive models with flexible form of fixed effects","authors":"Kazuhiko Hayakawa, Boyan Yin","doi":"10.1016/j.jspi.2024.106252","DOIUrl":"10.1016/j.jspi.2024.106252","url":null,"abstract":"<div><div>This paper proposes the maximum likelihood (ML) estimator for a short panel autoregressive model with a flexible form of observed factors as well as unknown interactive fixed effects. We show that the ML estimator is consistent and asymptotically normally distributed as the number of cross-sectional units increases with the number of time periods being fixed. It should be noted that this asymptotic result holds uniformly for the autoregressive coefficient less than, equal to, or greater than one, in sharp contrast to existing estimators. Monte Carlo simulation results show that the ML estimator has desirable finite sample properties.</div></div>","PeriodicalId":50039,"journal":{"name":"Journal of Statistical Planning and Inference","volume":"237 ","pages":"Article 106252"},"PeriodicalIF":0.8,"publicationDate":"2024-12-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143133630","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 : 2024-12-04DOI: 10.1016/j.jspi.2024.106253
Jie Yin , Jieli Ding , Changming Yang
In order to break the constraints and barriers caused by limited computing power in processing massive datasets, we propose an outcome dependent subsampling divide and conquer strategy in this paper. The proposed strategy can process data on multiple blocks in parallel and concentrate the computing resources of each block on regions with the most information. We develop a distributed statistical inference method and propose a computation-efficient algorithm in the generalized linear models for massive data. The proposed method only need to preserve some summary statistics from each data block and then use them to directly construct the proposed estimator. The asymptotic properties of the proposed method are established. Simulation studies and real data analysis are conducted to illustrate the merits of the proposed method.
{"title":"Outcome dependent subsampling divide and conquer in generalized linear models for massive data","authors":"Jie Yin , Jieli Ding , Changming Yang","doi":"10.1016/j.jspi.2024.106253","DOIUrl":"10.1016/j.jspi.2024.106253","url":null,"abstract":"<div><div>In order to break the constraints and barriers caused by limited computing power in processing massive datasets, we propose an outcome dependent subsampling divide and conquer strategy in this paper. The proposed strategy can process data on multiple blocks in parallel and concentrate the computing resources of each block on regions with the most information. We develop a distributed statistical inference method and propose a computation-efficient algorithm in the generalized linear models for massive data. The proposed method only need to preserve some summary statistics from each data block and then use them to directly construct the proposed estimator. The asymptotic properties of the proposed method are established. Simulation studies and real data analysis are conducted to illustrate the merits of the proposed method.</div></div>","PeriodicalId":50039,"journal":{"name":"Journal of Statistical Planning and Inference","volume":"237 ","pages":"Article 106253"},"PeriodicalIF":0.8,"publicationDate":"2024-12-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143133629","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 : 2024-11-28DOI: 10.1016/j.jspi.2024.106251
Alicja Jokiel-Rokita, Sylwester Pia̧tek
Classical inequality curves and inequality measures are defined for distributions with finite mean value. Moreover, their empirical counterparts are not resistant to outliers. For these reasons, quantile versions of known inequality curves such as the Lorenz, Bonferroni, Zenga and curves, and quantile versions of inequality measures such as the Gini, Bonferroni, Zenga and indices have been proposed in the literature. We propose various nonparametric estimators of quantile versions of inequality curves and inequality measures, prove their consistency, and compare their accuracy in a simulation study. We also give examples of the use of quantile versions of inequality measures in real data analysis.
{"title":"Nonparametric estimators of inequality curves and inequality measures","authors":"Alicja Jokiel-Rokita, Sylwester Pia̧tek","doi":"10.1016/j.jspi.2024.106251","DOIUrl":"10.1016/j.jspi.2024.106251","url":null,"abstract":"<div><div>Classical inequality curves and inequality measures are defined for distributions with finite mean value. Moreover, their empirical counterparts are not resistant to outliers. For these reasons, quantile versions of known inequality curves such as the Lorenz, Bonferroni, Zenga and <span><math><mi>D</mi></math></span> curves, and quantile versions of inequality measures such as the Gini, Bonferroni, Zenga and <span><math><mi>D</mi></math></span> indices have been proposed in the literature. We propose various nonparametric estimators of quantile versions of inequality curves and inequality measures, prove their consistency, and compare their accuracy in a simulation study. We also give examples of the use of quantile versions of inequality measures in real data analysis.</div></div>","PeriodicalId":50039,"journal":{"name":"Journal of Statistical Planning and Inference","volume":"237 ","pages":"Article 106251"},"PeriodicalIF":0.8,"publicationDate":"2024-11-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143133628","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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