In this article, a partially linear single-index model /or longitudinal data is investigated. The generalized penalized spline least squares estimates of the unknown parameters are suggested. All parameters can be est...In this article, a partially linear single-index model /or longitudinal data is investigated. The generalized penalized spline least squares estimates of the unknown parameters are suggested. All parameters can be estimated simultaneously by the proposed method while the feature of longitudinal data is considered. The existence, strong consistency and asymptotic normality of the estimators are proved under suitable conditions. A simulation study is conducted to investigate the finite sample performance of the proposed method. Our approach can also be used to study the pure single-index model for longitudinal data.展开更多
The single-index model with monotonic link function is investigated. Firstly, it is showed that the link function h(.) can be viewed by a graphic method. That is, the plot with the fitted response y on the horizonta...The single-index model with monotonic link function is investigated. Firstly, it is showed that the link function h(.) can be viewed by a graphic method. That is, the plot with the fitted response y on the horizontal axis and the observed y on the vertical axis can be used to visualize the link function. It is pointed out that this graphic approach is also applicable even when the link function is not monotonic. Note that many existing nonparametric smoothers can also be used to assess h(.). Therefore, the I-spline approximation of the link function via maximizing the covariance function with a penalty function is investigated in the present work. The consistency of the criterion is constructed. A small simulation is carried out to evidence the efficiency of the approach proposed in the paper.展开更多
Varying-coefficient single-index model( VCSIM) avoids the so-called "curse of dimensionality " and is flexible enough to include several important statistical models. This paper considers statistical diagnos...Varying-coefficient single-index model( VCSIM) avoids the so-called "curse of dimensionality " and is flexible enough to include several important statistical models. This paper considers statistical diagnosis for VCSIM. First,the parametric estimation equation is established based on empirical likelihood. Then,some diagnosis statistics are defined. At last, an example is given to illustrate all the results.展开更多
In this article, we study the variable selection of partially linear single-index model(PLSIM). Based on the minimized average variance estimation, the variable selection of PLSIM is done by minimizing average varianc...In this article, we study the variable selection of partially linear single-index model(PLSIM). Based on the minimized average variance estimation, the variable selection of PLSIM is done by minimizing average variance with adaptive l1 penalty. Implementation algorithm is given. Under some regular conditions, we demonstrate the oracle properties of aLASSO procedure for PLSIM. Simulations are used to investigate the effectiveness of the proposed method for variable selection of PLSIM.展开更多
We consider the problem of variable selection for the single-index random effects models with longitudinal data. An automatic variable selection procedure is developed using smooth-threshold. The proposed method share...We consider the problem of variable selection for the single-index random effects models with longitudinal data. An automatic variable selection procedure is developed using smooth-threshold. The proposed method shares some of the desired features of existing variable selection methods: the resulting estimator enjoys the oracle property;the proposed procedure avoids the convex optimization problem and is flexible and easy to implement. Moreover, we use the penalized weighted deviance criterion for a data-driven choice of the tuning parameters. Simulation studies are carried out to assess the performance of our method, and a real dataset is analyzed for further illustration.展开更多
In many applications a heterogeneous population consists of several subpopulations. When each subpopulation can be adequately modeled by a heteroscedastic single-index model, the whole population is characterized by a...In many applications a heterogeneous population consists of several subpopulations. When each subpopulation can be adequately modeled by a heteroscedastic single-index model, the whole population is characterized by a finite mixture of heteroscedastic single-index models. In this article, we propose an estimation algorithm for fitting this model, and discuss the implementation in detail. Simulation studies are used to demonstrate the performance of the algorithm, and a real example is used to illustrate the application of the model.展开更多
This paper studies the inference problem of index coefficient in single-index models under massive dataset.Analysis of massive dataset is challenging owing to formidable computational costs or memory requirements.A na...This paper studies the inference problem of index coefficient in single-index models under massive dataset.Analysis of massive dataset is challenging owing to formidable computational costs or memory requirements.A natural method is the averaging divide-and-conquer approach,which splits data into several blocks,obtains the estimators for each block and then aggregates the estimators via averaging.However,there is a restriction on the number of blocks.To overcome this limitation,this paper proposed a computationally efficient method,which only requires an initial estimator and then successively refines the estimator via multiple rounds of aggregations.The proposed estimator achieves the optimal convergence rate without any restriction on the number of blocks.We present both theoretical analysis and experiments to explore the property of the proposed method.展开更多
In this paper, a partially linear single-index model is investigated, and three empirical log-likelihood ratio statistics for the unknown parameters in the model are suggested. It is proved that the proposed statistic...In this paper, a partially linear single-index model is investigated, and three empirical log-likelihood ratio statistics for the unknown parameters in the model are suggested. It is proved that the proposed statistics are asymptotically standard chi-square under some suitable conditions, and hence can be used to construct the confidence regions of the parameters. Our methods can also deal with the confidence region construction for the index in the pure single-index model. A simulation study indicates that, in terms of coverage probabilities and average areas of the confidence regions, the proposed methods perform better than the least-squares method.展开更多
Statistical inference on parametric part for the partially linear single-index model (PLSIM) is considered in this paper. A profile least-squares technique for estimating the parametric part is proposed and the asympt...Statistical inference on parametric part for the partially linear single-index model (PLSIM) is considered in this paper. A profile least-squares technique for estimating the parametric part is proposed and the asymptotic normality of the profile least-squares estimator is given. Based on the estimator, a generalized likelihood ratio (GLR) test is proposed to test whether parameters on linear part for the model is under a contain linear restricted condition. Under the null model, the proposed GLR statistic follows asymptotically the χ2-distribution with the scale constant and degree of freedom independent of the nuisance parameters, known as Wilks phenomenon. Both simulated and real data examples are used to illustrate our proposed methods.展开更多
We consider the problem of variable selection for single-index varying-coefficient model, and present a regularized variable selection procedure by combining basis function approximations with SCAD penalty. The propos...We consider the problem of variable selection for single-index varying-coefficient model, and present a regularized variable selection procedure by combining basis function approximations with SCAD penalty. The proposed procedure simultaneously selects significant covariates with functional coefficients and local significant variables with parametric coefficients. With appropriate selection of the tuning parameters, the consistency of the variable selection procedure and the oracle property of the estimators are established. The proposed method can naturally be applied to deal with pure single-index model and varying-coefficient model. Finite sample performances of the proposed method are illustrated by a simulation study and the real data analysis.展开更多
The varying-coefficient single-index model(VCSIM)is widely used in economics,statistics and biology.A model averaging method for VCSIM based on a Mallows-type criterion is proposed to improve prodictive capacity,which...The varying-coefficient single-index model(VCSIM)is widely used in economics,statistics and biology.A model averaging method for VCSIM based on a Mallows-type criterion is proposed to improve prodictive capacity,which allows the number of candidate models to diverge with sample size.Under model misspecification,the asymptotic optimality is derived in the sense of achieving the lowest possible squared errors.The authors compare the proposed model averaging method with several other classical model selection methods by simulations and the corresponding results show that the model averaging estimation has a outstanding performance.The authors also apply the method to a real dataset.展开更多
We propose a robust estimation procedure based on local Walsh-average regression(LWR) for single-index models. Our novel method provides a root-n consistent estimate of the single-index parameter under some mild regul...We propose a robust estimation procedure based on local Walsh-average regression(LWR) for single-index models. Our novel method provides a root-n consistent estimate of the single-index parameter under some mild regularity conditions;the estimate of the unknown link function converges at the usual rate for the nonparametric estimation of a univariate covariate. We theoretically demonstrate that the new estimators show significant efficiency gain across a wide spectrum of non-normal error distributions and have almost no loss of efficiency for the normal error. Even in the worst case, the asymptotic relative efficiency(ARE) has a lower bound compared with the least squares(LS) estimates;the lower bounds of the AREs are 0.864 and 0.8896 for the single-index parameter and nonparametric function, respectively. Moreover, the ARE of the proposed LWR-based approach versus the ARE of the LS-based method has an expression that is closely related to the ARE of the signed-rank Wilcoxon test as compared with the t-test. In addition, to obtain a sparse estimate of the single-index parameter, we develop a variable selection procedure by combining the estimation method with smoothly clipped absolute deviation penalty;this procedure is shown to possess the oracle property. We also propose a Bayes information criterion(BIC)-type criterion for selecting the tuning parameter and further prove its ability to consistently identify the true model. We conduct some Monte Carlo simulations and a real data analysis to illustrate the finite sample performance of the proposed methods.展开更多
In this paper, we study the sure independence screening of ultrahigh-dimensional censored data with varying coefficient single-index model. This general model framework covers a large number of commonly used survival ...In this paper, we study the sure independence screening of ultrahigh-dimensional censored data with varying coefficient single-index model. This general model framework covers a large number of commonly used survival models. The property that the proposed method is not derived for a specific model is appealing in ultrahigh dimensional regressions, as it is difficult to specify a correct model for ultrahigh dimensional predictors.Once the assuming data generating process does not meet the actual one, the screening method based on the model will be problematic. We establish the sure screening property and consistency in ranking property of the proposed method. Simulations are conducted to study the finite sample performances, and the results demonstrate that the proposed method is competitive compared with the existing methods. We also illustrate the results via the analysis of data from The National Alzheimers Coordinating Center(NACC).展开更多
The purpose of this paper is two fold.First,the authors investigate quantile regression(QR)estimation for single-index QR models when the response is subject to random left truncation.The random weights are introduced...The purpose of this paper is two fold.First,the authors investigate quantile regression(QR)estimation for single-index QR models when the response is subject to random left truncation.The random weights are introduced to deal with left truncated data and the associated iteration estimation method is proposed.The asymptotic properties for the proposed QR estimates of the index parameter and unknown link function are both obtained.Further,by combining the QR loss function and the adaptive LASSO penalization,a variable selection procedure for the index parameter is introduced and its oracle property is established.Second,a weighted empirical log-likelihood ratio of the index parameter based on the QR method is introduced and is proved to be asymptotic standard chi-square distribution.Furthermore,confidence regions of the index parameter can be constructed.The finite sample performance of the proposed methods are demonstrated.A real data analysis is also conducted to show the usefulness of the proposed approaches.展开更多
The partially linear single-index model(PLSIM) is a flexible and powerful model for analyzing the relationship between the response and the multivariate covariates. This paper considers the PLSIM with measurement erro...The partially linear single-index model(PLSIM) is a flexible and powerful model for analyzing the relationship between the response and the multivariate covariates. This paper considers the PLSIM with measurement error possibly in all the variables. The authors propose a new efficient estimation procedure based on the local linear smoothing and the simulation-extrapolation method,and further establish the asymptotic normality of the proposed estimators for both the index parameter and nonparametric link function. The authors also carry out extensive Monte Carlo simulation studies to evaluate the finite sample performance of the new method, and apply it to analyze the osteoporosis prevention data.展开更多
In this article, we consider a class of seemingly unrelated single-index regression models. By taking the contemporaneous correlation among equations into account we construct the weighted estimators (WEs) for unkno...In this article, we consider a class of seemingly unrelated single-index regression models. By taking the contemporaneous correlation among equations into account we construct the weighted estimators (WEs) for unknown parameters of the coefficients and the improved local polynomial estimators for the unknown functions, respectively. We establish the asymptotic normalities of these estimators, and show both of them are more asymptotically efficient than those ignoring the contemporaneous correlation. The performances of the proposed procedures are evaluated through simulation studies.展开更多
We propose the single-index hazards model for censored survival data.As an extension of the Cox model and many transformation models,this model allows nonparametric modeling of covariate effects in a parsimonious way ...We propose the single-index hazards model for censored survival data.As an extension of the Cox model and many transformation models,this model allows nonparametric modeling of covariate effects in a parsimonious way via a single index.In addition,the relative importance of covariates can be assessed via this model.We consider two commonly used profile likelihood methods for parameter estimation:the local profile likelihood method and the stratified profile likelihood method.It is shown that both methods may give consistent estimators under certain restrictive conditions,but in general they can yield biased estimation.Simulation studies are also conducted to demonstrate these bias phenomena.The existence and nature of the failures of these two commonly used approaches is somewhat surprising.展开更多
Estimating treatment effects has always been one of the hot issues in empirical research.It brings great challenges to estimating treatment effects because heterogeneity exists in the distribution of covariates betwee...Estimating treatment effects has always been one of the hot issues in empirical research.It brings great challenges to estimating treatment effects because heterogeneity exists in the distribution of covariates between treated and controlled groups.Propensity score methods have been widely used to adjust for heterogeneity in observational studies.However,the propensity score is usually unknown and needs to be estimated.In this article,we propose a generalized single-index model to estimate the propensity score and use the propensity score residuals to reduce the estimation bias.The finite-sample performance of the proposed method is evaluated through simulation stud-ies.We use the proposed method to evaluate the policy of"Sunshine Running"and find that the physical test scores of college students par-ticipating in the"Sunshine Running"can be improved by 3.72 points.展开更多
基金Supported by the National Natural Science Foundation of China (10571008)the Natural Science Foundation of Henan (092300410149)the Core Teacher Foundationof Henan (2006141)
文摘In this article, a partially linear single-index model /or longitudinal data is investigated. The generalized penalized spline least squares estimates of the unknown parameters are suggested. All parameters can be estimated simultaneously by the proposed method while the feature of longitudinal data is considered. The existence, strong consistency and asymptotic normality of the estimators are proved under suitable conditions. A simulation study is conducted to investigate the finite sample performance of the proposed method. Our approach can also be used to study the pure single-index model for longitudinal data.
基金Supported by the National Natural science Foundation of China(10701035)ChenGuang Project of Shang-hai Education Development Foundation(2007CG33)a Special Fund for Young Teachers in Shanghai Universities(79001320)
文摘The single-index model with monotonic link function is investigated. Firstly, it is showed that the link function h(.) can be viewed by a graphic method. That is, the plot with the fitted response y on the horizontal axis and the observed y on the vertical axis can be used to visualize the link function. It is pointed out that this graphic approach is also applicable even when the link function is not monotonic. Note that many existing nonparametric smoothers can also be used to assess h(.). Therefore, the I-spline approximation of the link function via maximizing the covariance function with a penalty function is investigated in the present work. The consistency of the criterion is constructed. A small simulation is carried out to evidence the efficiency of the approach proposed in the paper.
文摘Varying-coefficient single-index model( VCSIM) avoids the so-called "curse of dimensionality " and is flexible enough to include several important statistical models. This paper considers statistical diagnosis for VCSIM. First,the parametric estimation equation is established based on empirical likelihood. Then,some diagnosis statistics are defined. At last, an example is given to illustrate all the results.
文摘In this article, we study the variable selection of partially linear single-index model(PLSIM). Based on the minimized average variance estimation, the variable selection of PLSIM is done by minimizing average variance with adaptive l1 penalty. Implementation algorithm is given. Under some regular conditions, we demonstrate the oracle properties of aLASSO procedure for PLSIM. Simulations are used to investigate the effectiveness of the proposed method for variable selection of PLSIM.
文摘We consider the problem of variable selection for the single-index random effects models with longitudinal data. An automatic variable selection procedure is developed using smooth-threshold. The proposed method shares some of the desired features of existing variable selection methods: the resulting estimator enjoys the oracle property;the proposed procedure avoids the convex optimization problem and is flexible and easy to implement. Moreover, we use the penalized weighted deviance criterion for a data-driven choice of the tuning parameters. Simulation studies are carried out to assess the performance of our method, and a real dataset is analyzed for further illustration.
文摘In many applications a heterogeneous population consists of several subpopulations. When each subpopulation can be adequately modeled by a heteroscedastic single-index model, the whole population is characterized by a finite mixture of heteroscedastic single-index models. In this article, we propose an estimation algorithm for fitting this model, and discuss the implementation in detail. Simulation studies are used to demonstrate the performance of the algorithm, and a real example is used to illustrate the application of the model.
基金the Fundamental Research Funds for the Central Universities of China(No.2232020D-43).
文摘This paper studies the inference problem of index coefficient in single-index models under massive dataset.Analysis of massive dataset is challenging owing to formidable computational costs or memory requirements.A natural method is the averaging divide-and-conquer approach,which splits data into several blocks,obtains the estimators for each block and then aggregates the estimators via averaging.However,there is a restriction on the number of blocks.To overcome this limitation,this paper proposed a computationally efficient method,which only requires an initial estimator and then successively refines the estimator via multiple rounds of aggregations.The proposed estimator achieves the optimal convergence rate without any restriction on the number of blocks.We present both theoretical analysis and experiments to explore the property of the proposed method.
基金supported by the Natural Science Foundation of Beijing City(Grant No.1042002)Technology Development Plan Project of Beijing Education Committee(Grant No.KM2005 10005009)+1 种基金the Special Grants of Beijing for Talents(Grant No.20041D0501515)supported by a grant from the Research Grants Council of Hong Kong,Hong Kong(Grant No.HKU7060/04P).
文摘In this paper, a partially linear single-index model is investigated, and three empirical log-likelihood ratio statistics for the unknown parameters in the model are suggested. It is proved that the proposed statistics are asymptotically standard chi-square under some suitable conditions, and hence can be used to construct the confidence regions of the parameters. Our methods can also deal with the confidence region construction for the index in the pure single-index model. A simulation study indicates that, in terms of coverage probabilities and average areas of the confidence regions, the proposed methods perform better than the least-squares method.
基金supported by National Natural Science Foundation of China (Grant No. 10871072)Natural Science Foundation of Shanxi Province of China (Grant No. 2007011014)PhD Program Scholarship Fund of ECNU 2009
文摘Statistical inference on parametric part for the partially linear single-index model (PLSIM) is considered in this paper. A profile least-squares technique for estimating the parametric part is proposed and the asymptotic normality of the profile least-squares estimator is given. Based on the estimator, a generalized likelihood ratio (GLR) test is proposed to test whether parameters on linear part for the model is under a contain linear restricted condition. Under the null model, the proposed GLR statistic follows asymptotically the χ2-distribution with the scale constant and degree of freedom independent of the nuisance parameters, known as Wilks phenomenon. Both simulated and real data examples are used to illustrate our proposed methods.
文摘We consider the problem of variable selection for single-index varying-coefficient model, and present a regularized variable selection procedure by combining basis function approximations with SCAD penalty. The proposed procedure simultaneously selects significant covariates with functional coefficients and local significant variables with parametric coefficients. With appropriate selection of the tuning parameters, the consistency of the variable selection procedure and the oracle property of the estimators are established. The proposed method can naturally be applied to deal with pure single-index model and varying-coefficient model. Finite sample performances of the proposed method are illustrated by a simulation study and the real data analysis.
基金supported by the National Nature Science Foundation of Chinaunder Grant Nos.12001559and 11971324+1 种基金the Ministry of Education of Humanities and Social Science projectunder Grant No.19YJC910008。
文摘The varying-coefficient single-index model(VCSIM)is widely used in economics,statistics and biology.A model averaging method for VCSIM based on a Mallows-type criterion is proposed to improve prodictive capacity,which allows the number of candidate models to diverge with sample size.Under model misspecification,the asymptotic optimality is derived in the sense of achieving the lowest possible squared errors.The authors compare the proposed model averaging method with several other classical model selection methods by simulations and the corresponding results show that the model averaging estimation has a outstanding performance.The authors also apply the method to a real dataset.
基金partially supported by National Natural Science Foundation of China(Grant Nos.11801168,11801169,11571055 and 11671059)the Natural Science Foundation of Hunan Province(Grant No.2018JJ3322)
文摘We propose a robust estimation procedure based on local Walsh-average regression(LWR) for single-index models. Our novel method provides a root-n consistent estimate of the single-index parameter under some mild regularity conditions;the estimate of the unknown link function converges at the usual rate for the nonparametric estimation of a univariate covariate. We theoretically demonstrate that the new estimators show significant efficiency gain across a wide spectrum of non-normal error distributions and have almost no loss of efficiency for the normal error. Even in the worst case, the asymptotic relative efficiency(ARE) has a lower bound compared with the least squares(LS) estimates;the lower bounds of the AREs are 0.864 and 0.8896 for the single-index parameter and nonparametric function, respectively. Moreover, the ARE of the proposed LWR-based approach versus the ARE of the LS-based method has an expression that is closely related to the ARE of the signed-rank Wilcoxon test as compared with the t-test. In addition, to obtain a sparse estimate of the single-index parameter, we develop a variable selection procedure by combining the estimation method with smoothly clipped absolute deviation penalty;this procedure is shown to possess the oracle property. We also propose a Bayes information criterion(BIC)-type criterion for selecting the tuning parameter and further prove its ability to consistently identify the true model. We conduct some Monte Carlo simulations and a real data analysis to illustrate the finite sample performance of the proposed methods.
基金Supported by the National Natural Science Foundation of China(No.11801567)
文摘In this paper, we study the sure independence screening of ultrahigh-dimensional censored data with varying coefficient single-index model. This general model framework covers a large number of commonly used survival models. The property that the proposed method is not derived for a specific model is appealing in ultrahigh dimensional regressions, as it is difficult to specify a correct model for ultrahigh dimensional predictors.Once the assuming data generating process does not meet the actual one, the screening method based on the model will be problematic. We establish the sure screening property and consistency in ranking property of the proposed method. Simulations are conducted to study the finite sample performances, and the results demonstrate that the proposed method is competitive compared with the existing methods. We also illustrate the results via the analysis of data from The National Alzheimers Coordinating Center(NACC).
基金supported by the National Social Science Foundation of China under Grant No.21BTJ038。
文摘The purpose of this paper is two fold.First,the authors investigate quantile regression(QR)estimation for single-index QR models when the response is subject to random left truncation.The random weights are introduced to deal with left truncated data and the associated iteration estimation method is proposed.The asymptotic properties for the proposed QR estimates of the index parameter and unknown link function are both obtained.Further,by combining the QR loss function and the adaptive LASSO penalization,a variable selection procedure for the index parameter is introduced and its oracle property is established.Second,a weighted empirical log-likelihood ratio of the index parameter based on the QR method is introduced and is proved to be asymptotic standard chi-square distribution.Furthermore,confidence regions of the index parameter can be constructed.The finite sample performance of the proposed methods are demonstrated.A real data analysis is also conducted to show the usefulness of the proposed approaches.
基金the National Natural Science Foundation of China under Grant Nos. 11971171,11971300, 11901286, 12071267 and 12171310the Shanghai Natural Science Foundation under Grant No.20ZR1421800+2 种基金the Open Research Fund of Key Laboratory of Advanced Theory and Application in Statistics and Data Science (East China Normal University)the General Research Fund (HKBU12303421, HKBU12303918)the Initiation Grant for Faculty Niche Research Areas (RC-FNRA-IG/20-21/SCI/03) of Hong Kong Baptist University。
文摘The partially linear single-index model(PLSIM) is a flexible and powerful model for analyzing the relationship between the response and the multivariate covariates. This paper considers the PLSIM with measurement error possibly in all the variables. The authors propose a new efficient estimation procedure based on the local linear smoothing and the simulation-extrapolation method,and further establish the asymptotic normality of the proposed estimators for both the index parameter and nonparametric link function. The authors also carry out extensive Monte Carlo simulation studies to evaluate the finite sample performance of the new method, and apply it to analyze the osteoporosis prevention data.
基金Supported by the National Natural Science Foundation of China(No.11471140)
文摘In this article, we consider a class of seemingly unrelated single-index regression models. By taking the contemporaneous correlation among equations into account we construct the weighted estimators (WEs) for unknown parameters of the coefficients and the improved local polynomial estimators for the unknown functions, respectively. We establish the asymptotic normalities of these estimators, and show both of them are more asymptotically efficient than those ignoring the contemporaneous correlation. The performances of the proposed procedures are evaluated through simulation studies.
文摘We propose the single-index hazards model for censored survival data.As an extension of the Cox model and many transformation models,this model allows nonparametric modeling of covariate effects in a parsimonious way via a single index.In addition,the relative importance of covariates can be assessed via this model.We consider two commonly used profile likelihood methods for parameter estimation:the local profile likelihood method and the stratified profile likelihood method.It is shown that both methods may give consistent estimators under certain restrictive conditions,but in general they can yield biased estimation.Simulation studies are also conducted to demonstrate these bias phenomena.The existence and nature of the failures of these two commonly used approaches is somewhat surprising.
基金Supported by 2020 Guilin University of Aerospace Technology Teaching Group Construction Project(2020JXTD19)2021 Guangxi Philosophy and Social Science Research Project(21FTY012)2022 Guangxi Higher Education Undergraduate Teaching Reform Project(2022JGA358)。
文摘Estimating treatment effects has always been one of the hot issues in empirical research.It brings great challenges to estimating treatment effects because heterogeneity exists in the distribution of covariates between treated and controlled groups.Propensity score methods have been widely used to adjust for heterogeneity in observational studies.However,the propensity score is usually unknown and needs to be estimated.In this article,we propose a generalized single-index model to estimate the propensity score and use the propensity score residuals to reduce the estimation bias.The finite-sample performance of the proposed method is evaluated through simulation stud-ies.We use the proposed method to evaluate the policy of"Sunshine Running"and find that the physical test scores of college students par-ticipating in the"Sunshine Running"can be improved by 3.72 points.