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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
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(x≤l). 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.
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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
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.
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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
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.
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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
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.
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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
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.
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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
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.
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13 articles
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