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.

Factor models are a class of powerful statistical models that have been widely used to deal with dependent measurements that arise frequently from various applications from genomics and neuroscience to economics and finance. As data are collected at an ever-growing scale, statistical machine learning faces some new challenges: high dimensionality, strong dependence among observed variables, heavy-tailed variables and heterogeneity. High-dimensional robust factor analysis serves as a powerful toolkit to conquer these challenges.

Most papers on high-dimensional statistics are based on the assumption that none of the regressors are correlated with the regression error, namely, they are exogeneous. Yet, endogeneity arises easily in high-dimensional regression due to a large pool of regressors and this causes the inconsistency of the penalized least-squares methods and possible false scientific discoveries. A necessary condition for model selection of a very general class of penalized regression methods is given, which allows us to prove formally the inconsistency claim. To cope with the possible endogeneity, we construct a novel penalized focussed generalized method of moments (FGMM) criterion function and offer a new optimization algorithm. The FGMM is not a smooth function. To establish its asymptotic properties, we first study the model selection consistency and an oracle property for a general class of penalized regression methods. These results are then used to show that the FGMM possesses an oracle property even in the presence of endogenous predictors, and that the solution is also near global minimum under the over-identification assumption. Finally, we also show how the semi-parametric efficiency of estimation can be achieved via a two-step approach.

Measuring conditional dependence is an important topic in statistics with broad applications including graphical models. Under a factor model setting, a new conditional dependence measure based on projection is proposed. The corresponding conditional independence test is developed with the asymptotic null distribution unveiled where the number of factors could be high-dimensional. It is also shown that the new test has control over the asymptotic significance level and can be calculated efficiently. A generic method for building dependency graphs without Gaussian assumption using the new test is elaborated. Numerical results and real data analysis show the superiority of the new method.

Many data mining and statistical machine learning algorithms have been developed to select a subset of covariates to associate with a response variable. Spurious discoveries can easily arise in high-dimensional data analysis due to enormous possibilities of such selections. How can we know statistically our discoveries better than those by chance? In this paper, we define a measure of goodness of spurious fit, which shows how good a response variable can be fitted by an optimally selected subset of covariates under the null model, and propose a simple and effective LAMM algorithm to compute it. It coincides with the maximum spurious correlation for linear models and can be regarded as a generalized maximum spurious correlation. We derive the asymptotic distribution of such goodness of spurious fit for generalized linear models and L1 regression. Such an asymptotic distribution depends on the sample size, ambient dimension, the number of variables used in the fit, and the covariance information. It can be consistently estimated by multiplier bootstrapping and used as a benchmark to guard against spurious discoveries. It can also be applied to model selection, which considers only candidate models with goodness of fits better than those by spurious fits. The theory and method are convincingly illustrated by simulated examples and an application to the binary outcomes from German Neuroblastoma Trials.