Abstract Matching a nonprobability sample to a probability sample is one strategy both for selecting the nonprobability units and for weighting them. This approach has been employed in the past to select subsamples of persons from a large panel of volunteers. One method of weighting, introduced here, is to assign a unit in the nonprobability sample the weight from its matched case in the probability sample. The properties of resulting estimators depend on whether the probability sample weights are inverses of selection probabilities or are calibrated. In addition, imperfect matching can cause estimates from the matched sample to be biased so that its weights need to be adjusted, especially when the size of the volunteer panel is small. Calibration weighting combined with matching is one approach to correct bias and reduce variances. We explore the theoretical properties of the matched and matched, calibrated estimators with respect to a quasirandomization distribution that is assumed to describe how units in the nonprobability sample are observed, a superpopulation model for analysis variables collected in the nonprobability sample, and the randomization distribution for the probability sample. Numerical studies using simulated and real data from the 2015 US Behavioral Risk Factor Surveillance Survey are conducted to examine the performance of the alternative estimators.
{"title":"Investigating an Alternative for Estimation from a Nonprobability Sample: Matching plus Calibration","authors":"Zhanxu Liu, R. Valliant","doi":"10.2478/jos-2023-0003","DOIUrl":"https://doi.org/10.2478/jos-2023-0003","url":null,"abstract":"Abstract Matching a nonprobability sample to a probability sample is one strategy both for selecting the nonprobability units and for weighting them. This approach has been employed in the past to select subsamples of persons from a large panel of volunteers. One method of weighting, introduced here, is to assign a unit in the nonprobability sample the weight from its matched case in the probability sample. The properties of resulting estimators depend on whether the probability sample weights are inverses of selection probabilities or are calibrated. In addition, imperfect matching can cause estimates from the matched sample to be biased so that its weights need to be adjusted, especially when the size of the volunteer panel is small. Calibration weighting combined with matching is one approach to correct bias and reduce variances. We explore the theoretical properties of the matched and matched, calibrated estimators with respect to a quasirandomization distribution that is assumed to describe how units in the nonprobability sample are observed, a superpopulation model for analysis variables collected in the nonprobability sample, and the randomization distribution for the probability sample. Numerical studies using simulated and real data from the 2015 US Behavioral Risk Factor Surveillance Survey are conducted to examine the performance of the alternative estimators.","PeriodicalId":51092,"journal":{"name":"Journal of Official Statistics","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2021-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48839103","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 Indisputable figures on income and wealth inequality are indispensable for politics, society and science. Although the Gini coefficient is the most common measure of inequality, the straightforward concept of the Robin Hood index (namely, the income share that has to be transferred from the rich to the poor to make everyone equally well off) makes it a more attractive measure for the general public. In a distribution with many negative values – particularly wealth distributions – the Robin Hood index can take on values larger than 1, indicating an intuitively impossible income transfer of more than 100%. This article proposes a method to normalise the Robin Hood index. In contrast to the original index, the normalised Robin Hood index always takes on values between 0 and 1 and ends up as the original index in a distribution without negatives. As inequality measures are commonly applied to equivalised income, we also introduce a method for adequately transferring equivalised incomes from the rich to the poor within the framework of the (normalised) Robin Hood index. An empirical application shows the effect of normalisation for the Robin Hood index, and compares it to the normalisation of the Gini coefficient from previous research.
{"title":"The Robin Hood Index Adjusted for Negatives and Equivalised Incomes","authors":"Marion van den Brakel, R. Lok","doi":"10.2478/jos-2021-0044","DOIUrl":"https://doi.org/10.2478/jos-2021-0044","url":null,"abstract":"Abstract Indisputable figures on income and wealth inequality are indispensable for politics, society and science. Although the Gini coefficient is the most common measure of inequality, the straightforward concept of the Robin Hood index (namely, the income share that has to be transferred from the rich to the poor to make everyone equally well off) makes it a more attractive measure for the general public. In a distribution with many negative values – particularly wealth distributions – the Robin Hood index can take on values larger than 1, indicating an intuitively impossible income transfer of more than 100%. This article proposes a method to normalise the Robin Hood index. In contrast to the original index, the normalised Robin Hood index always takes on values between 0 and 1 and ends up as the original index in a distribution without negatives. As inequality measures are commonly applied to equivalised income, we also introduce a method for adequately transferring equivalised incomes from the rich to the poor within the framework of the (normalised) Robin Hood index. An empirical application shows the effect of normalisation for the Robin Hood index, and compares it to the normalisation of the Gini coefficient from previous research.","PeriodicalId":51092,"journal":{"name":"Journal of Official Statistics","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2021-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43899828","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 draw attention to how, in the name of protecting the confidentiality of personal data, national statistical agencies have limited public access to spatial data on COVID-19. We also draw attention to large disparities in the way that access has been limited. In doing so, we distinguish between absolute confidentiality in which the probability of detection is 1, relative confidentiality where this probability is less than 1, and collective confidentiality, which refers to the probability of detection of at least one person. In spatial data, the probability of personal detection is less than 1, and the probability of collective detection varies directly with this probability and COVID-19 morbidity. Statistical agencies have been concerned with relative and collective confidentiality, which they implement using the techniques of truncation, where spatial data are not made public for zones with small populations, and censoring, where exact data are not made public for zones where morbidity is small. Granular spatial data are essential for epidemiological research into COVID-19. We argue that in their reluctance to make these data available to the public, data security officers (DSO) have unreasonably prioritized data protection over freedom of information. We also argue that by attaching importance to relative and collective confidentiality, they have over-indulged in data truncation and censoring. We highlight the need for legislation concerning relative and collective confidentiality, and regulation of DSO practices regarding data truncation and censoring.
{"title":"Freedom of Information and Personal Confidentiality in Spatial COVID-19 Data","authors":"M. Beenstock, D. Felsenstein","doi":"10.2478/jos-2021-0035","DOIUrl":"https://doi.org/10.2478/jos-2021-0035","url":null,"abstract":"Abstract We draw attention to how, in the name of protecting the confidentiality of personal data, national statistical agencies have limited public access to spatial data on COVID-19. We also draw attention to large disparities in the way that access has been limited. In doing so, we distinguish between absolute confidentiality in which the probability of detection is 1, relative confidentiality where this probability is less than 1, and collective confidentiality, which refers to the probability of detection of at least one person. In spatial data, the probability of personal detection is less than 1, and the probability of collective detection varies directly with this probability and COVID-19 morbidity. Statistical agencies have been concerned with relative and collective confidentiality, which they implement using the techniques of truncation, where spatial data are not made public for zones with small populations, and censoring, where exact data are not made public for zones where morbidity is small. Granular spatial data are essential for epidemiological research into COVID-19. We argue that in their reluctance to make these data available to the public, data security officers (DSO) have unreasonably prioritized data protection over freedom of information. We also argue that by attaching importance to relative and collective confidentiality, they have over-indulged in data truncation and censoring. We highlight the need for legislation concerning relative and collective confidentiality, and regulation of DSO practices regarding data truncation and censoring.","PeriodicalId":51092,"journal":{"name":"Journal of Official Statistics","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2021-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41691355","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 Misclassified frame records (also called stratum jumpers) and low response rates are characteristic for business surveys. In the context of estimation of the domain parameters, jumpers may contribute to extreme variation in sample weights and skewed sampling distributions of the estimators, especially for domains with a small number of observations. There is limited literature about the extent to which these problems may affect the performance of the ratio estimators with nonresponse-adjusted weights. To address this gap, we designed a simulation study to explore the properties of the Horvitz-Thompson type ratio estimators, with and without smoothing of the weights, under different scenarios. The ratio estimator with propensity-adjusted weights showed satisfactory performance in all scenarios with a high response rate. For scenarios with a low response rate, the performance of this estimator improved with an increase in the proportion of jumpers in the domain. The smoothed estimators that we studied performed well in scenarios with non-informative weights, but can become markedly biased when the weights are informative, irrespective of response rate. We also studied the performance of the ’doubled half’ bootstrap method for variance estimation. We illustrated an application of the methods in a real business survey.
{"title":"Estimation of Domain Means from Business Surveys in the Presence of Stratum Jumpers and Nonresponse","authors":"Mengxuan Xu, V. Landsman, B. Graubard","doi":"10.2478/jos-2021-0045","DOIUrl":"https://doi.org/10.2478/jos-2021-0045","url":null,"abstract":"Abstract Misclassified frame records (also called stratum jumpers) and low response rates are characteristic for business surveys. In the context of estimation of the domain parameters, jumpers may contribute to extreme variation in sample weights and skewed sampling distributions of the estimators, especially for domains with a small number of observations. There is limited literature about the extent to which these problems may affect the performance of the ratio estimators with nonresponse-adjusted weights. To address this gap, we designed a simulation study to explore the properties of the Horvitz-Thompson type ratio estimators, with and without smoothing of the weights, under different scenarios. The ratio estimator with propensity-adjusted weights showed satisfactory performance in all scenarios with a high response rate. For scenarios with a low response rate, the performance of this estimator improved with an increase in the proportion of jumpers in the domain. The smoothed estimators that we studied performed well in scenarios with non-informative weights, but can become markedly biased when the weights are informative, irrespective of response rate. We also studied the performance of the ’doubled half’ bootstrap method for variance estimation. We illustrated an application of the methods in a real business survey.","PeriodicalId":51092,"journal":{"name":"Journal of Official Statistics","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2021-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44958616","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 the Netherlands, very precise and detailed statistical information on labour force participation is derived from registers. A drawback of this data source is that it is not timely since definitive versions typically become available with a delay of two years. More timely information on labour force participation can be derived from the Labour Force Survey (LFS). Quarterly figures, for example, become available six weeks after the calendar quarter. A well-known drawback of this data source is the uncertainty due to sampling error. In this article, a nowcast method is proposed to produce preliminary but timely nowcasts for the register labour force participation on a quarterly frequency at the level of municipalities and neighbourhoods, using the data from the LFS. As a first step, small area estimates for quarterly municipal figures on labour force participation are obtained using the LFS data and the unit-level modelling approach of Battese, Harter and Fuller (1988). Subsequently, time series of these small area estimates at the municipal level are combined with time series on register labour force participation in a bivariate structural time series model in order to nowcast the register labour force participation at the level of municipalities and neighbourhoods.
{"title":"Nowcasting Register Labour Force Participation Rates in Municipal Districts Using Survey Data","authors":"Jan van den Brakel, J. Michiels","doi":"10.2478/jos-2021-0043","DOIUrl":"https://doi.org/10.2478/jos-2021-0043","url":null,"abstract":"Abstract In the Netherlands, very precise and detailed statistical information on labour force participation is derived from registers. A drawback of this data source is that it is not timely since definitive versions typically become available with a delay of two years. More timely information on labour force participation can be derived from the Labour Force Survey (LFS). Quarterly figures, for example, become available six weeks after the calendar quarter. A well-known drawback of this data source is the uncertainty due to sampling error. In this article, a nowcast method is proposed to produce preliminary but timely nowcasts for the register labour force participation on a quarterly frequency at the level of municipalities and neighbourhoods, using the data from the LFS. As a first step, small area estimates for quarterly municipal figures on labour force participation are obtained using the LFS data and the unit-level modelling approach of Battese, Harter and Fuller (1988). Subsequently, time series of these small area estimates at the municipal level are combined with time series on register labour force participation in a bivariate structural time series model in order to nowcast the register labour force participation at the level of municipalities and neighbourhoods.","PeriodicalId":51092,"journal":{"name":"Journal of Official Statistics","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2021-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41956383","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 Previous research is inconclusive regarding the effects of paper and web surveys on response burdens. We conducted an establishment survey with random assignment to paper and web modes to examine this issue. We compare how the actual and perceived response burdens differ when respondents complete a survey in the paper mode, in the web mode and when they are allowed to choose between the two modes. Our results show that in the web mode, respondents have a lower estimated time to complete the questionnaire, while we do not find differences between paper and the web on the perceived response time and perceived burden. Even though the response burden in the web mode is lower, our study finds no evidence of an increased response burden when moving an establishment survey from paper to the web.
{"title":"Comparing the Response Burden between Paper and Web Modes in Establishment Surveys","authors":"G. Haas, S. Eckman, Ruben L. Bach","doi":"10.2478/jos-2021-0039","DOIUrl":"https://doi.org/10.2478/jos-2021-0039","url":null,"abstract":"Abstract Previous research is inconclusive regarding the effects of paper and web surveys on response burdens. We conducted an establishment survey with random assignment to paper and web modes to examine this issue. We compare how the actual and perceived response burdens differ when respondents complete a survey in the paper mode, in the web mode and when they are allowed to choose between the two modes. Our results show that in the web mode, respondents have a lower estimated time to complete the questionnaire, while we do not find differences between paper and the web on the perceived response time and perceived burden. Even though the response burden in the web mode is lower, our study finds no evidence of an increased response burden when moving an establishment survey from paper to the web.","PeriodicalId":51092,"journal":{"name":"Journal of Official Statistics","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2021-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48497043","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}
Corinna König, J. Sakshaug, Jens Stegmaier, S. Kohaut
Abstract Evidence from the household survey literature shows a declining response rate trend in recent decades, but whether a similar trend exists for voluntary establishment surveys is an understudied issue. This article examines trends in nonresponse rates and nonresponse bias over a period of 17 years in the annual cross-sectional refreshment samples of the IAB Establishment Panel in Germany. In addition, rich administrative data about the establishment and employee composition are used to examine changes in nonresponse bias and its two main components, refusal and noncontact, over time. Our findings show that response rates dropped by nearly a third: from 50.2% in 2001 to 34.5% in 2017. Simultaneously, nonresponse bias increased over this period, which was mainly driven by increasing refusal bias whereas noncontact bias fluctuated relatively evenly over the same period. Nonresponse biases for individual establishment and employee characteristics did not show a distinct pattern over time with few exceptions. Notably, larger establishments participated less frequently than smaller establishments over the entire period. This implies that survey organizations may need to put more effort into recruiting larger establishments to counteract nonresponse bias.
{"title":"Trends in Establishment Survey Nonresponse Rates and Nonresponse Bias: Evidence from the 2001-2017 IAB Establishment Panel","authors":"Corinna König, J. Sakshaug, Jens Stegmaier, S. Kohaut","doi":"10.2478/jos-2021-0040","DOIUrl":"https://doi.org/10.2478/jos-2021-0040","url":null,"abstract":"Abstract Evidence from the household survey literature shows a declining response rate trend in recent decades, but whether a similar trend exists for voluntary establishment surveys is an understudied issue. This article examines trends in nonresponse rates and nonresponse bias over a period of 17 years in the annual cross-sectional refreshment samples of the IAB Establishment Panel in Germany. In addition, rich administrative data about the establishment and employee composition are used to examine changes in nonresponse bias and its two main components, refusal and noncontact, over time. Our findings show that response rates dropped by nearly a third: from 50.2% in 2001 to 34.5% in 2017. Simultaneously, nonresponse bias increased over this period, which was mainly driven by increasing refusal bias whereas noncontact bias fluctuated relatively evenly over the same period. Nonresponse biases for individual establishment and employee characteristics did not show a distinct pattern over time with few exceptions. Notably, larger establishments participated less frequently than smaller establishments over the entire period. This implies that survey organizations may need to put more effort into recruiting larger establishments to counteract nonresponse bias.","PeriodicalId":51092,"journal":{"name":"Journal of Official Statistics","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2021-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45795458","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 Small area estimation is receiving considerable attention due to the high demand for small area statistics. Small area estimators of means and totals have been widely studied in the literature. Moreover, in the last years also small area estimators of quantiles and poverty indicators have been studied. In contrast, small area estimators of inequality indicators, which are often used in socio-economic studies, have received less attention. In this article, we propose a robust method based on the M-quantile regression model for small area estimation of the Theil index and the Gini coefficient, two popular inequality measures. To estimate the mean squared error a non-parametric bootstrap is adopted. A robust approach is used because often inequality is measured using income or consumption data, which are often non-normal and affected by outliers. The proposed methodology is applied to income data to estimate the Theil index and the Gini coefficient for small domains in Tuscany (provinces by age groups), using survey and Census micro-data as auxiliary variables. In addition, a design-based simulation is carried out to study the behaviour of the proposed robust estimators. The performance of the bootstrap mean squared error estimator is also investigated in the simulation study.
{"title":"Robust Estimation of the Theil Index and the Gini Coeffient for Small Areas","authors":"S. Marchetti, N. Tzavidis","doi":"10.2478/jos-2021-0041","DOIUrl":"https://doi.org/10.2478/jos-2021-0041","url":null,"abstract":"Abstract Small area estimation is receiving considerable attention due to the high demand for small area statistics. Small area estimators of means and totals have been widely studied in the literature. Moreover, in the last years also small area estimators of quantiles and poverty indicators have been studied. In contrast, small area estimators of inequality indicators, which are often used in socio-economic studies, have received less attention. In this article, we propose a robust method based on the M-quantile regression model for small area estimation of the Theil index and the Gini coefficient, two popular inequality measures. To estimate the mean squared error a non-parametric bootstrap is adopted. A robust approach is used because often inequality is measured using income or consumption data, which are often non-normal and affected by outliers. The proposed methodology is applied to income data to estimate the Theil index and the Gini coefficient for small domains in Tuscany (provinces by age groups), using survey and Census micro-data as auxiliary variables. In addition, a design-based simulation is carried out to study the behaviour of the proposed robust estimators. The performance of the bootstrap mean squared error estimator is also investigated in the simulation study.","PeriodicalId":51092,"journal":{"name":"Journal of Official Statistics","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2021-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47611919","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 Coding respondent occupation is one of the most challenging aspects of survey data collection. Traditionally performed manually by office coders post-interview, previous research has acknowledged the advantages of coding occupation during the interview, including reducing costs, processing time and coding uncertainties that are more difficult to address post-interview. However, a number of concerns have been raised as well, including the potential for interviewer effects, the challenge of implementing the coding system in a web survey, in which respondents perform the coding procedure themselves, or the feasibility of implementing the same standardized coding system in a mixed-mode self- and interviewer-administered survey. This study sheds light on these issues by presenting an evaluation of a new occupation coding method administered during the interview in a large-scale sequential mixed-mode (web, telephone, face-to-face) cohort study of young adults in the UK. Specifically, we assess the take-up rates of this new coding method across the different modes and report on several other performance measures thought to impact the quality of the collected occupation data. Furthermore, we identify factors that affect the coding of occupation during the interview, including interviewer effects. The results carry several implications for survey practice and directions for future research.
{"title":"Occupation Coding During the Interview in a Web-First Sequential Mixed-Mode Survey","authors":"D. Peycheva, J. Sakshaug, L. Calderwood","doi":"10.2478/jos-2021-0042","DOIUrl":"https://doi.org/10.2478/jos-2021-0042","url":null,"abstract":"Abstract Coding respondent occupation is one of the most challenging aspects of survey data collection. Traditionally performed manually by office coders post-interview, previous research has acknowledged the advantages of coding occupation during the interview, including reducing costs, processing time and coding uncertainties that are more difficult to address post-interview. However, a number of concerns have been raised as well, including the potential for interviewer effects, the challenge of implementing the coding system in a web survey, in which respondents perform the coding procedure themselves, or the feasibility of implementing the same standardized coding system in a mixed-mode self- and interviewer-administered survey. This study sheds light on these issues by presenting an evaluation of a new occupation coding method administered during the interview in a large-scale sequential mixed-mode (web, telephone, face-to-face) cohort study of young adults in the UK. Specifically, we assess the take-up rates of this new coding method across the different modes and report on several other performance measures thought to impact the quality of the collected occupation data. Furthermore, we identify factors that affect the coding of occupation during the interview, including interviewer effects. The results carry several implications for survey practice and directions for future research.","PeriodicalId":51092,"journal":{"name":"Journal of Official Statistics","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2021-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44009760","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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