Using high-frequency data, we estimate the risk of a large portfolio with weights being the solution of an optimization problem subject to some linear inequality constraints. We propose a fully nonparametric approach as a benchmark, as well as a factor-based semiparametric approach with observable factors to attack the curse of dimensionality. We provide in-fill asymptotic distributions of the realized volatility estimators of the optimal portfolio, while taking into account the estimation error in the optimal portfolio weights as a result of the covariance matrix estimation. Our theoretical findings suggest that ignoring such an error leads to a first-order asymptotic bias which undermines the statistical inference. Such a bias is related to in-sample optimism in portfolio allocation. Our simulation results suggest satisfactory finite sample performance after bias correction, and that the factor-based approach becomes increasingly superior with a growing cross-sectional dimension. Empirically, using a large cross-section of high-frequency stock returns, we find our estimator successfully addresses the issue of in-sample optimism.

Large-scale multiple testing with correlated and heavy-tailed data arises in a wide range of research areas from genomics, medical imaging to finance. Conventional methods for estimating the false discovery proportion (FDP) often ignore the effect of heavy-tailedness and the dependence structure among test statistics, and thus may lead to inefficient or even inconsistent estimation. Also, the assumption of joint normality is often imposed, which is too stringent for many applications. To address these challenges, in this paper we propose a factoradjusted robust procedure for large-scale simultaneous inference with control of the false discovery proportion. We demonstrate that robust factor adjustments are extremely important in both improving the power of the tests and controlling FDP. We identify general conditions under which the proposed method produces consistent estimate of the FDP. As a byproduct that is of independent interest, we establish an exponential-type deviation inequality for a robust U-type covariance estimator under the spectral norm. Extensive numerical experiments demonstrate the advantage of the proposed method over several state-of-the-art methods especially when the data are generated from heavy-tailed distributions. Our proposed procedures are implemented in the R-package FarmTest. Supported by NSF Grants DMS-1662139, DMS-1712591, and NIH Grant R01-GM072611-12.

A class of variable selection procedures for parametric models via nonconcave penalized likelihood was proposed by Fan and Li to simultaneously estimate parameters and select important variables. They demonstrated that this class of procedures has an oracle property when the number of parameters is finite. However, in most model selection problems the number of parameters should be large and grow with the sample size. In this paper some asymptotic properties of the nonconcave penalized likelihood are established for situations in which the number of parameters tends to as the sample size increases. Under regularity conditions we have established an oracle property and the asymptotic normality of the penalized likelihood estimators. Furthermore, the consistency of the sandwich formula of the covariance matrix is demonstrated. Nonconcave penalized likelihood ratio statistics are discussed, and their

Ultrahigh-dimensional variable selection plays an increasingly important role in contemporary scientific discoveries and statistical research. Among others, Fan and Lv [JR Stat. Soc. Ser. B Stat. Methodol. 70 (2008) 849911] propose an independent screening framework by ranking the marginal correlations. They showed that the correlation ranking procedure possesses a sure independence screening property within the context of the linear model with Gaussian covariates and responses. In this paper, we propose a more general version of the independent learning with ranking the maximum marginal likelihood estimates or the maximum marginal likelihood itself in generalized linear models. We show that the proposed methods, with Fan and Lv [JR Stat. Soc. Ser. B Stat. Methodol. 70 (2008) 849911] as a very special case, also possess the sure screening property with vanishing false selection rate. The

Big data is transforming our world, revolutionizing operations and analytics everywhere, from financial engineering to biomedical sciences. The complexity of big data often makes dimension reduction techniques necessary before conducting statistical inference. Principal component analysis, commonly referred to as PCA, has become an essential tool for multivariate data analysis and unsupervised dimension reduction, the goal of which is to find a lower dimensional subspace that captures most of the variation in the dataset. This article provides an overview of methodological and theoretical developments of PCA over the past decade, with focus on its applications to big data analytics. We first review the mathematical formulation of PCA and its theoretical development from the view point of perturbation analysis. We then briefly discuss the relationship between PCA and factor analysis as well as its applications to