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With the recent explosion of scientific data of unprecedented size and complexity, feature ranking and screening are playing an increasingly important role in many scientific studies. In this article, we propose a novel feature screening procedure under a unified model framework, which covers a wide variety of commonly used parametric and semiparametric models. The new method does not require imposing a specific model structure on regression functions, and thus is particularly appealing to ultrahigh-dimensional regressions, where there are a huge number of candidate predictors but little information about the actual model forms. We demonstrate that, with the number of predictors growing at an exponential rate of the sample size, the proposed procedure possesses consistency in ranking, which is both useful in its own right and can lead to consistency in selection. The new procedure is computationally efficient and simple, and exhibits a competent empirical performance in our intensive simulations and real data analysis.

Searching for an effective dimension reduction space is an important problem in regression, especially for high dimensional data. We propose an adaptive approach based on semiparametric models, which we call the (conditional) minimum average variance estimation (MAVE) method, within quite a general setting. The MAVE method has the following advantages. Most existing methods must undersmooth the nonparametric link function estimator to achieve a faster rate of consistency for the estimator of the parameters (than for that of the nonparametric function). In contrast, a faster consistency rate can be achieved by the MAVE method even without undersmoothing the nonparametric link function estimator. The MAVE method is applicable to a wide range of models, with fewer restrictions on the distribution of the covariates, to the extent that even time series can be included. Because of the faster rate of consistency for the parameter estimators, it is possible for us to estimate the dimension of the space consistently. The relationship of the MAVE method with other methods is also investigated. In particular, a simple outer product gradient estimator is proposed as an initial estimator. In addition to theoretical results, we demonstrate the efficacy of the MAVE method for high dimensional data sets through simulation. Two real data sets are analysed by using the MAVE approach.

In this paper, we systematically study the consistency of sliced average variance estimation (SAVE). The findings reveal that when the response is continuous, the asymptotic behavior of SAVE is rather different from that of sliced inverse regression (SIR). SIR can achieve √n consistency even when each slice contains only two data points. However, SAVE cannot be √n consistent and it even turns out to be not consistent when each slice contains a fixed number of data points that do not depend on n, where n is the sample size. These results theoretically confirm the notion that SAVE is more sensitive to the number of slices than SIR. Taking this into account, a bias correction is recommended in order to allow SAVE to be √n consistent. In contrast, when the response is discrete and takes finite values, √n consistency can be achieved. Therefore, an approximation through discretization, which is commonly used in practice, is studied. A simulation study is carried out for the purposes of illustration.

Styrax japonicus is a medicinal and ornamental shrub belonging to the Styracaceae family. To explore the diversity and characteristics of the chloroplast genome of S. japonicus, we conducted sequencing and comparison of the chloroplast genomes of four naturally distributed S. japonicus. The results demonstrated that the four chloroplast genomes (157,914–157,962 bp) exhibited a typical quadripartite structure consisting of a large single copy (LSC) region, a small single copy (SSC) region, and a pair of reverse repeats (IRa and IRb), and the structure was highly conserved. DNA polymorphism analysis revealed that three coding genes (infA, psbK, and rpl33) and five intergene regions (petA-psbJ, trnC-petN, trnD-trnY, trnE-trnT, and trnY-trnE) were identified as mutation hotspots. These genetic fragments have the potential to be utilized as DNA barcodes for future identification purposes. When comparing the boundary genes, a small contraction was observed in the IR region of four S. japonicus. Selection pressure analysis indicated positive selection for ycf1 and ndhD. These findings collectively suggest the adaptive evolution of S. japonicus. The phylogenetic structure revealed conflicting relationships among several S. japonicus, indicating divergent evolutionary paths within this species. Our study concludes by uncovering the genetic traits of the chloroplast genome in the differentiation of S. japonicus variety, offering fresh perspectives on the evolutionary lineage of this species.

Functional constipation (FC) is a common clinical complaint. Despite a lack of consolidated evidence, Chinese herbal medicine (CHM) has become a popular alternative treatment for this condition. The aim of this study was to assess, with a rigidly designed study, the efficacy and safety of a CHM proprietary medicine, Hemp Seed Pill (HSP), in optimal dosage for treating FC. This study comprised two parts: trial I, a dose determination study, and trial II, a placebo-controlled clinical study. In trial I, the optimal dosage of HSP was first determined from among three doses (2.5, 5.0, and 7.5 g b.i.d.). In trial II, a randomized double-blind study, the efficacy and safety of HSP for FC patients (Rome III criteria) in excessive syndrome as defined by traditional Chinese medicine (TCM) theory were compared with placebo. All participants in trials underwent a 2-week run-in, an 8-week treatment, and an 8-week follow-up. The primary end point was the responder rate for complete spontaneous bowel movement (CSBM) during treatment. Participants with a mean increase of CSBM ≧ 1/week compared with their baselines were defined as responders. Secondary outcome measures included responder rate during follow-up, individual and global symptom assessments, and reported adverse effects (AEs). The dose of 7.5 g b.i.d. showed better therapeutic effect than that of 2.5 and 5.0 g b.i.d. among 96 subjects (32 per arm) in trial I and was therefore selected for comparison with placebo in trial II. In trial II, 120 subjects were randomized into two arms (60 per arm). Responder rates for the HSP and placebo groups were 43.3 and 8.3% during treatment and 30.0 and 15.0% in the follow-up period, respectively (P<0.05). Those in the HSP group showed benefit in terms of increased CSBM, relief in the severity of constipation and straining of evacuation, and effective reduction in the use of rescue therapy when compared with placebo. No serious AE was reported. HSP (7.5 g b.i.d.) is safe and effective for alleviating FC for subjects in excessive syndrome. Optimal dose determination may be crucial for all CHM studies.

We propose trace pursuit for model-free variable selection under the sufficient dimension-reduction paradigm. Two distinct algorithms are proposed: stepwise trace pursuit and forward trace pursuit. Stepwise trace pursuit achieves selection consistency with fixed p. Forward trace pursuit can serve as an initial screening step to speed up the computation in the case of ultrahigh dimensionality. The screening consistency property of forward trace pursuit based on sliced inverse regression is established. Finite sample performances of trace pursuit and other model-free variable selection methods are compared through numerical studies. Supplementary materials for this article are available online.

In many regression applications, the predictors fall naturally into a number of groups or domains, and it is often desirable to establish a domain-specific relation between the predictors and the response. In this article, we consider dimension reduction that incorporates such domain knowledge. The proposed method is based on the derivative of the conditional mean, where the differential operator is constrained to the form of a direct sum. This formulation also accommodates the situations where dimension reduction is focused only on part of the predictors; as such it extends Partial Dimension Reduction to cases where the blocked predictors are continuous. Through simulation and real data analyses, we show that the proposed method achieves greater accuracy and interpretability than the dimension reduction methods that ignore group information. Furthermore, the new method does not require the stringent conditions on the predictor distribution that are required by existing methods.

We propose an omnibus lack-of-fit test for linear or nonlinear quantile regression based on a cusum process of the gradient vector. The test does not involve nonparametric smoothing but is consistent for all nonparametric alternatives without any moment conditions on the regression error. In addition, the test is suitable for detecting the local alternatives of any order arbitrarily close to n−1/2 from the null hypothesis. The limiting distribution of the proposed test statistic is non-Gaussian but can be characterized by a Gaussian process. We propose a simple sequential resampling scheme to carry out the test whose nominal levels are well approximated in our empirical study for

In this paper we offer a complete methodology of cumulative slicing estimation to sufficient dimension reduction. In parallel to the classical slicing estimation, we develop three methods that are termed, respectively, as cumulative mean estimation, cumulative variance estimation, and cumulative directional regression. The strong consistency for p=O(n 1 / 2  / log n) and the asymptotic normality for p=o(n 1 / 2 ) are established, where p is the dimension of the predictors and n is sample size. Such asymptotic results improve the rate p=o(n 1 / 3 ) in many existing contexts of semiparametric modeling. In addition, we propose a modified BIC-type criterion to estimate the structural dimension of the central subspace. Its consistency is established when p=o(n 1 / 2 ). Extensive simulations are carried out for comparison with existing methods and a real data example is presented for illustration.

Because highly correlated data arise from many scientific fields, we investigate parameter estimation in a semiparametric regression model with diverging number of predictors that are highly correlated. For this, we first develop a distribution-weighted least squares estimator that can recover directions in the central subspace, then use the distribution-weighted least squares estimator as a seed vector and project it onto a Krylov space by partial least squares to avoid computing the inverse of the covariance of predictors. Thus, distrbution-weighted partial least squares can handle the cases with high dimensional and highly correlated predictors. Furthermore, we also suggest an iterative algorithm for obtaining a better initial value before implementing partial least squares. For theoretical investigation, we obtain strong consistency and asymptotic normality when the dimension p of predictors is of convergence rate O{n¹/²/ log (n)} and o(n¹/³) respectively where n is the sample size. When there are no other constraints on the covariance of predictors, the rates n¹/² and n¹/³ are optimal. We also propose a Bayesian information criterion type of criterion to estimate the dimension of the Krylov space in the partial least squares procedure. Illustrative examples with a real data set and comprehensive simulations demonstrate that the method is robust to non-ellipticity and works well even in 'small n-large p' problems.