Frontiers of Mathematics in China

ISSN 1673-3452

ISSN 1673-3576(Online)

CN 11-5739/O1

Postal Subscription Code 80-964

2018 Impact Factor: 0.565

   Online First

Administered by

, Volume 8 Issue 3

For Selected: View Abstracts Toggle Thumbnails
EDITORIAL
RESEARCH ARTICLE
Limit theorems for the position of a tagged particle in the stirring-exclusion process
Peng CHEN, Fuxi ZHANG
Front Math Chin. 2013, 8 (3): 479-496.  
https://doi.org/10.1007/s11464-013-0283-0

Abstract   HTML   PDF (167KB)

Stirring-exclusion processes are exclusion processes with particles being stirred. We investigate a tagged particle among a Bernoulli product environment measure on the lattice ?d.We show the strong law of large numbers and the central limit theorem for the tagged particle. The proof of the central limit theorem is based on the method of martingale decomposition with a sector condition.

References | Related Articles | Metrics
Frequentist model averaging for linear mixed-effects models
Xinjie CHEN, Guohua ZOU, Xinyu ZHANG
Front Math Chin. 2013, 8 (3): 497-515.  
https://doi.org/10.1007/s11464-012-0254-x

Abstract   HTML   PDF (160KB)

Linear mixed-effects models are a powerful tool for the analysis of longitudinal data. The aim of this paper is to study model averaging for linear mixed-effects models. The asymptotic distribution of the frequentist model average estimator is derived, and a confidence interval procedure with an actual coverage probability that tends to the nominal level in large samples is developed. The two confidence intervals based on the model averaging and based on the full model are shown to be asymptotically equivalent. A simulation study shows good finite sample performance of the model average estimators.

References | Related Articles | Metrics
Identifiability of intermediate variables on causal paths
Wanlu DENG, Zhi GENG, Peng LUO
Front Math Chin. 2013, 8 (3): 517-539.  
https://doi.org/10.1007/s11464-013-0270-5

Abstract   HTML   PDF (195KB)

We discuss the discovery of causal mechanisms and identifiability of intermediate variables on a causal path. Different from variable selection, we try to distinguish intermediate variables on the causal path from other variables. It is also different from ordinary model selection approaches which do not concern the causal relationships and do not contain unobserved variables. We propose an approach for selecting a causal mechanism depicted by a directed acyclic graph (DAG) with an unobserved variable. We consider several causal networks, and discuss their identifiability by observed data. We show that causal mechanisms of linear structural equation models are not identifiable. Furthermore, we present that causal mechanisms of nonlinear models are identifiable, and we demonstrate the identifiability of causal mechanisms of quadratic equation models. Sensitivity analysis is conducted for the identifiability.

References | Related Articles | Metrics
Variable selection for single-index varying-coefficient model
Sanying FENG, Liugen XUE
Front Math Chin. 2013, 8 (3): 541-565.  
https://doi.org/10.1007/s11464-013-0284-z

Abstract   HTML   PDF (231KB)

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.

References | Related Articles | Metrics
A computational algebraic-geometry method for conditional-independence inference
Benchong LI, Shoufeng CAI, Jianhua GUO
Front Math Chin. 2013, 8 (3): 567-582.  
https://doi.org/10.1007/s11464-013-0295-9

Abstract   HTML   PDF (155KB)

We consider the problems of semi-graphoid inference and of independence implication from a set of conditional-independence statements. Based on ideas from R. Hemmecke et al. [Combin. Probab. Comput., 2008, 17: 239-257], we present algebraic-geometry characterizations of these two problems, and propose two corresponding algorithms. These algorithms can be realized with any computer algebra system when the number of variables is small.

References | Related Articles | Metrics
VaR Criteria for optimal limited changeloss and truncated change-loss reinsurance
Xiaojing MA, Lan WU
Front Math Chin. 2013, 8 (3): 583-608.  
https://doi.org/10.1007/s11464-013-0278-x

Abstract   HTML   PDF (260KB)

Reinsurance can provide an effective way for insurer to manage its risk exposure. In this paper, we further analyze the optimal reinsurance models recently proposed by J. Cai and K. S. Tan [Astin Bulletin, 2007, 37(1): 93-112]. With the criteria of minimizing the value-at-risk (VaR) risk measure of insurer’s total loss exposure, we derive the optimal values of sharing proportion a, retention d, and layer l of two reinsurance treaties: the limited changeloss f(x) = a{(x - d)+ - (x - l)+} and the truncated change-loss f(x) = a(x-d)+I(xl). Both of the reinsurance plans have been considered to be more realistic and practical in the real business. Our solutions have several appealing features: (i) there is only one condition to verify for the existence of optimal limited change-loss reinsurance while there always exists an optimal truncated change-loss reinsurance, (ii) the resulting optimal parameters have simple analytic forms which depend only on assumed loss distribution, reinsurer’s safety loading, and insurer’s risk tolerance, (iii) the optimal retention d for limited change-loss reinsurance is the same as that by Cai and Tan while the optimal value is smaller for truncated change-loss, (iv) the optimal sharing proportion and layer are always the same for both reinsurance plans, (v) minimized VaR are strictly lower than the values derived by Cai and Tan, (vi) the optimization results reveal possible drawbacks of VaR-based risk management that a heavy tail risk exposure may be expressed by lower VaR.

References | Related Articles | Metrics
Fluctuations of deformed Wigner random matrices
Zhonggen SU
Front Math Chin. 2013, 8 (3): 609-641.  
https://doi.org/10.1007/s11464-012-0259-5

Abstract   HTML   PDF (224KB)

Let Xn be a standard real symmetric (complex Hermitian) Wigner matrix, y1, y2, . . . , yn a sequence of independent real random variables independent of Xn. Consider the deformed Wigner matrix Hn,α = n-1/2Xn + n-α/2 diag (y1, . . . , yn), where 0<α<1. It is well known that the average spectral distribution is the classical Wigner semicircle law, i.e., the Stieltjes transform mn,α(z) converges in probability to the corresponding Stieltjes transform m(z). In this paper, we shall give the asymptotic estimate for the expectation Emn,α(z) and varianceVar(mn,α(z)), and establish the central limit theorem for linear statistics with sufficiently regular test function. A basic tool in the study is Stein’s equation and its generalization which naturally leads to a certain recursive equation.

References | Related Articles | Metrics
Discovering causes and effects of a given node in Bayesian networks
Changzhang WANG, You ZHOU, Zhi GENG
Front Math Chin. 2013, 8 (3): 643-663.  
https://doi.org/10.1007/s11464-013-0285-y

Abstract   HTML   PDF (183KB)

Causal relationships among variables can be depicted by a causal network of these variables. We propose a local structure learning approach for discovering the direct causes and the direct effects of a given target variable. In the approach, we first find the variable set of parents, children, and maybe some descendants (PCD) of the target variable, but generally we cannot distinguish the parents from the children in the PCD of the target variable. Next, to distinguish the causes from the effects of the target variable, we find the PCD of each variable in the PCD of the target variable, and we repeat the process of finding PCDs along the paths starting from the target variable. Without constructing a whole network over all variables, we find only a local structure around the target variable. Theoretically, we show the correctness of the proposed approach under the assumptions of faithfulness, causal sufficiency, and that conditional independencies are correctly checked.

References | Related Articles | Metrics
Probability density estimation with surrogate data and validation sample
Qihua WANG, Wenquan CUI
Front Math Chin. 2013, 8 (3): 665-694.  
https://doi.org/10.1007/s11464-013-0267-0

Abstract   HTML   PDF (254KB)

The probability density estimation problem with surrogate data and validation sample is considered. A regression calibration kernel density estimator is defined to incorporate the information contained in both surrogate variates and validation sample. Also, we define two weighted estimators which have less asymptotic variances but have bigger biases than the regression calibration kernel density estimator. All the proposed estimators are proved to be asymptotically normal. And the asymptotic representations for the mean squared error and mean integrated square error of the proposed estimators are established, respectively. A simulation study is conducted to compare the finite sample behaviors of the proposed estimators.

References | Related Articles | Metrics
General relative error criterion and M-estimation
Ying YANG, Fei YE
Front Math Chin. 2013, 8 (3): 695-715.  
https://doi.org/10.1007/s11464-013-0286-x

Abstract   HTML   PDF (176KB)

Relative error rather than the error itself is of the main interest in many practical applications. Criteria based on minimizing the sum of absolute relative errors (MRE) and the sum of squared relative errors (RLS) were proposed in the different areas. Motivated by K. Chen et al.’s recent work [J. Amer. Statist. Assoc., 2010, 105: 1104-1112] on the least absolute relative error (LARE) estimation for the accelerated failure time (AFT) model, in this paper, we establish the connection between relative error estimators and the M-estimation in the linear model. This connection allows us to deduce the asymptotic properties of many relative error estimators (e.g., LARE) by the well-developed M-estimation theories. On the other hand, the asymptotic properties of some important estimators (e.g., MRE and RLS) cannot be established directly. In this paper, we propose a general relative error criterion (GREC) for estimating the unknown parameter in the AFT model. Then we develop the approaches to deal with the asymptotic normalities forM-estimators with differentiable loss functions on ? or ?\{0} in the linear model. The simulation studies are conducted to evaluate the performance of the proposed estimates for the different scenarios. Illustration with a real data example is also provided.

References | Related Articles | Metrics
A two-stage variable selection strategy for supersaturated designs with multiple responses
Yuhui YIN, Qiaozhen ZHANG, Min-Qian LIU
Front Math Chin. 2013, 8 (3): 717-730.  
https://doi.org/10.1007/s11464-012-0255-9

Abstract   HTML   PDF (132KB)

A supersaturated design (SSD), whose run size is not enough for estimating all the main effects, is commonly used in screening experiments. It offers a potential useful tool to investigate a large number of factors with only a few experimental runs. The associated analysis methods have been proposed by many authors to identify active effects in situations where only one response is considered. However, there are often situations where two or more responses are observed simultaneously in one screening experiment, and the analysis of SSDs with multiple responses is thus needed. In this paper, we propose a two-stage variable selection strategy, called the multivariate partial least squares-stepwise regression (MPLS-SR) method, which uses the multivariate partial least squares regression in conjunction with the stepwise regression procedure to select true active effects in SSDs with multiple responses. Simulation studies show that the MPLS-SR method performs pretty good and is easy to understand and implement.

References | Related Articles | Metrics
A nonparametric regression method for multiple longitudinal phenotypes using multivariate adaptive splines
Wensheng ZHU, Heping ZHANG
Front Math Chin. 2013, 8 (3): 731-743.  
https://doi.org/10.1007/s11464-012-0256-8

Abstract   HTML   PDF (124KB)

In genetic studies of complex diseases, particularly mental illnesses, and behavior disorders, two distinct characteristics have emerged in some data sets. First, genetic data sets are collected with a large number of phenotypes that are potentially related to the complex disease under study. Second, each phenotype is collected from the same subject repeatedly over time. In this study, we present a nonparametric regression approach to study multivariate and time-repeated phenotypes together by using the technique of the multivariate adaptive regression splines for analysis of longitudinal data (MASAL), which makes it possible to identify genes, gene-gene and gene-environment, including time, interactions associated with the phenotypes of interest. Furthermore, we propose a permutation test to assess the associations between the phenotypes and selected markers. Through simulation, we demonstrate that our proposed approach has advantages over the existing methods that examine each longitudinal phenotype separately or analyze the summarized values of phenotypes by compressing them into one-time-point phenotypes. Application of the proposed method to the Framingham Heart Study illustrates that the use of multivariate longitudinal phenotypes enhanced the significance of the association test.

References | Related Articles | Metrics
13 articles