To estimate the nonparametric regression we
To estimate the nonparametric regression, we use a B-spline basis. Let be the space of polynomial splines of degree and denote a normalized B-spline basis with and , where is the supremum norm. For any and , we have for some coefficients . Here we allow to increase with and differ for different because different coefficient functions may have different smoothness. Under some conditions, the nonparametric coefficient functions can be well approximated by functions in .
Denote and , and define similarly to . Substituting (3) into (2), the maximum partial likelihood estimate of (2) is to maximize with respect to . We next propose a feature screening procedure based on (4).
Discussion Theorem 1 ensures the ascent property of the proposed algorithm under certain conditions, but it does not imply that the proposed algorithm converges to the global optimizer. If the proposed algorithm converges to a global maximizer of (5), then Theorem 2 shows that such a solution enjoys the sure screen property.
Acknowledgments Yang’s research was supported by the National Nature Science Foundation of China (NNSFC) grants 11471086 and 11871173, the National Social Science Foundation of China (NSSFC) grant 16BTJ032, the National Statistical Scientific Center grant 2015LD02, China; and the Fundamental Research Funds for the Central Universities of Jinan University Qimingxing Plan 15JNQM019, China. Zhang and Li’s research was supported by National Institute on Drug Abuse (NIDA) grant P50 DA039838, USA; National Science Foundation (NSF) grant DMS 1820702, USA; and NNSFC grants 11690014 and 11690015. The content is solely the responsibility of the authors and does not necessarily represent the official views of the NIDA, the NNSFC, the NSF, or the NSSFC.
Introduction Cyclooxygense (COX) is a key enzyme catalyzing the rate-limiting step that converts free arachidonic ALW-II-41-27 to prostaglandin (PG) H2 on the arachidonic cascade . COX exists in two distinct isozymes (COX-1 and COX-2), one of which, COX-2, is primarily responsible for inflammation . COX-2 is not normally present under the basal conditions or is present in very low amounts; however, it is rapidly induced in response to a wide variety of cytokines, growth factors, and ligands of G protein-coupled receptors. COX-2 is responsible for high levels of PG production during inflammation and immune responses and mediates a variety of biological actions involved in vascular pathophysiology. The induction of the COX-2 gene expression is regulated at both transcriptional (promoter-based) and post-transcriptional levels , , . Both mitogen-activated protein kinase (MAPK) and nuclear factor-kB (NF-kB) signaling pathways have been shown to mediate the COX-2 gene expression .
Identification of HNE as an inducer of COX-2 In view of the observation that liver injury associated with oxidative stress is accompanied by increased PG synthesis, it is hypothesized that lipid peroxidation products may be involved in the up-regulation of the PG biosynthesis. Indeed, in an alcohol-fed rat, a model of alcoholic liver disease, alcohol over-intake increases the formation of HNE-modified proteins and is associated with the COX-2 and proinflammatory cytokine TNF-α expression , . In addition, the HNE-specific epitopes have been detected in foamy macrophages within human atheromatous lesions  where the pro-inflammatory responses, including COX-2 expression, are being accelerated. To determine if lipid peroxidation could be involved in the COX-2 expression, Kumagai et al.  conducted a screen of oxidized fatty acids on COX-2 induction in rat liver epithelial RL34 and mouse macrophage RAW264.7 cell lines and demonstrated that HNE could specifically stimulate the COX-2 expression (Fig. 1). They have also shown that the depletion of the GSH pools in the cells with L-buthionine-S,R-sulfoximine significantly reduced the HNE-induced expression of COX-2 whereas the N-acetylcysteine pretreatment reversely led to a dose-dependent enhancement of the COX-2 expression . These findings suggest the intracellular GSH status may be strictly related to the HNE-induced COX-2 expression. Of interest, they also observed that the α,β-unsaturated aldehydes, such as acrolein, crotonaldehyde, and 2-nonenal, possessing an analogous functionality to HNE, were all inactive on the COX-2 induction. These studies represent a first demonstration of a link between COX-2 and HNE.