Stirring-exclusion processes are exclusion processes with particles being stirred. We investigate a tagged particle among a Bernoulli product environment measure on the lattice ℤ ^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.

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.

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.

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.

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 change-loss 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.

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.

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.

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 for M-estimators with differentiable loss functions on ℝ or ℝ\{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.