Yuting Lan

In this article, we propose a novel variable screening method for linear models named as conditional screening via ordinary least squares projection (COLP). COLP can take advantage of prior knowledge concerning certain active predictors by eliminating the adverse impact of their coefficients in the estimation of remaining ones and thus significantly enhance the screening accuracy. We prove its sure-screening property under reasonable assumptions and demonstrate its utility in an application to a leukemia dataset. Moreover, based on the conditional approach, we introduce an iterative algorithm named as forward screening via ordinary least squares projection (FOLP), which not only could exploit the prior information more effectively, but also has promising performance when no prior knowledge is available using a data-driven conditioning set. Extensive simulation studies are carried out to demonstrate the competence of both proposed methods.

Fan and Lv (2008) proposed the path-breaking theory of sure independence screening (SIS) and an iterative algorithm (ISIS) to effectively reduce the predictor dimension for further variable selection approaches. Fan et al. (2009) extended ISIS to generalized linear models and introduced the Vanilla ISIS (Van-ISIS) algorithm, allowing selected predictors to be screened out in upcoming iterations. The success of SIS depends on its sure screening property, which was obtained by Fan and Lv (2008) under the marginal correlation assumption. However, despite wide applications of ISIS and Van-ISIS in various scientific fields, their sure screening properties have not been proved during the past decade. To fill this gap, we prove the sure screening properties of three different types of iterative algorithms for linear models without relying on the marginal correlation assumption, where ISIS and Van-ISIS can be regarded as two special cases of them.

Under non‐additive probabilities, cluster points of the empirical average have been proved to quasi-surely fall into the interval constructed by either the lower and upper expectations or the lower and upper Choquet expectations. In this paper, based on the initiated notion of independence, we obtain a different Marcinkiewicz-Zygmund type strong law of large numbers. Then the Kolmogorov type strong law of large numbers can be derived from it directly, stating that the closed interval between the lower and upper expectations is the smallest one that covers cluster points of the empirical average quasi-surely.

In this paper, the notion of negatively associated random variables is introduced under nonlinear probabilities, which is weaker than some existing conditions of independence. Within this framework, a law of the logarithm for arrays of rowwise negatively associated random variables is established by investigating theirconvergence properties. In addition, a weak law of the logarithm with the convergence of capacity is obtained.

In this paper, on the sublinear expectation space, we establish a comparison theorem between independent and convolutionary random vectors, which states that the partial sums of those two sequences of random vectors are identically distributed. Under the sublinear framework, through the comparison theorem, several fundamental limit theorems for convolutionary random vectors are obtained, including the law of large numbers, the central limit theorem and the law of iterated logarithm.

In this paper, based on the initiation of the notion of negatively associated random variables under nonlinear probability, a strong limit theorem for weighted sums of random variables within the same frame is achieved without assumptions of independence and identical distribution, from which the Marcinkiewich-Zygmund type and Kolmogorov type strong laws of large numbers are derived. In addition, as applications of our results, Stranssen type invariance principles of negatively associated random variables and vertically independent random variables are proposed respectively.