We consider a clustering problem where we observe feature vectors $X_{i}\in R^{p}$, $i=1,2,\ldots,n$, from $K$ possible classes. The class labels are unknown and the main interest is to estimate them. We are primarily interested in the modern regime of $p\gg n$, where classical clustering methods face challenges.

We propose Influential Features PCA (IF-PCA) as a new clustering procedure. In IF-PCA, we select a small fraction of features with the largest Kolmogorov–Smirnov (KS) scores, obtain the first $(K-1)$ left singular vectors of the post-selection normalized data matrix, and then estimate the labels by applying the classical $k$-means procedure to these singular vectors. In this procedure, the only tuning parameter is the threshold in the feature selection step. We set the threshold in a data-driven fashion by adapting the recent notion of Higher Criticism. As a result, IF-PCA is a tuning-free clustering method.

We apply IF-PCA to $10$ gene microarray data sets. The method has competitive performance in clustering. Especially, in three of the data sets, the error rates of IF-PCA are only $29\%$ or less of the error rates by other methods. We have also rediscovered a phenomenon on empirical null by Efron [J. Amer. Statist. Assoc. 99 (2004) 96–104] on microarray data.

With delicate analysis, especially post-selection eigen-analysis, we derive tight probability bounds on the Kolmogorov–Smirnov statistics and show that IF-PCA yields clustering consistency in a broad context. The clustering problem is connected to the problems of sparse PCA and low-rank matrix recovery, but it is different in important ways. We reveal an interesting phase transition phenomenon associated with these problems and identify the range of interest for each.

We study a class of nonlinear nonparametric inverse problems. Specifically, we propose a nonparametric estimator of the dynamics of a monotonically increasing trajectory defined on a finite time interval. Under suitable regularity conditions, we show that in terms of $L^{2}$-loss, the optimal rate of convergence for the proposed estimator is the same as that for the estimation of the derivative of a function. We conduct simulation studies to examine the finite sample behavior of the proposed estimator and apply it to the Berkeley growth data.

Identifying causal parameters from observational data is fraught with subtleties due to the issues of selection bias and confounding. In addition, more complex questions of interest, such as effects of treatment on the treated and mediated effects may not always be identified even in data where treatment assignment is known and under investigator control, or may be identified under one causal model but not another.

Increasingly complex effects of interest, coupled with a diversity of causal models in use resulted in a fragmented view of identification. This fragmentation makes it unnecessarily difficult to determine if a given parameter is identified (and in what model), and what assumptions must hold for this to be the case. This, in turn, complicates the development of estimation theory and sensitivity analysis procedures.

In this paper, we give a unifying view of a large class of causal effects of interest, including novel effects not previously considered, in terms of a hierarchy of interventions, and show that identification theory for this large class reduces to an identification theory of random variables under interventions from this hierarchy. Moreover, we show that one type of intervention in the hierarchy is naturally associated with queries identified under the Finest Fully Randomized Causally Interpretable Structure Tree Graph (FFRCISTG) model of Robins (via the extended g-formula), and another is naturally associated with queries identified under the Non-Parametric Structural Equation Model with Independent Errors (NPSEM-IE) of Pearl, via a more general functional we call the edge g-formula.

Our results motivate the study of estimation theory for the edge g-formula, since we show it arises both in mediation analysis, and in settings where treatment assignment has unobserved causes, such as models associated with Pearl’s front-door criterion.

In this paper, we study asymptotic properties of model selection criteria for high-dimensional regression models where the number of covariates is much larger than the sample size. In particular, we consider a class of loss functions called the class of quadratically supported risks which is large enough to include the quadratic loss, Huber loss, quantile loss and logistic loss. We provide sufficient conditions for the model selection criteria, which are applicable to the class of quadratically supported risks. Our results extend most previous sufficient conditions for model selection consistency. In addition, sufficient conditions for pathconsistency of the Lasso and nonconvex penalized estimators are presented. Here, pathconsistency means that the probability of the solution path that includes the true model converges to 1. Pathconsistency makes it practically feasible to apply consistent model selection criteria to high-dimensional data. The data-adaptive model selection procedure is proposed which is selection consistent and performs well for finite samples. Results of simulation studies as well as real data analysis are presented to compare the finite sample performances of the proposed data-adaptive model selection criterion with other competitors.

We study the computational complexity of Markov chain Monte Carlo (MCMC) methods for high-dimensional Bayesian linear regression under sparsity constraints. We first show that a Bayesian approach can achieve variable-selection consistency under relatively mild conditions on the design matrix. We then demonstrate that the statistical criterion of posterior concentration need not imply the computational desideratum of rapid mixing of the MCMC algorithm. By introducing a truncated sparsity prior for variable selection, we provide a set of conditions that guarantee both variable-selection consistency and rapid mixing of a particular Metropolis–Hastings algorithm. The mixing time is linear in the number of covariates up to a logarithmic factor. Our proof controls the spectral gap of the Markov chain by constructing a canonical path ensemble that is inspired by the steps taken by greedy algorithms for variable selection.

Starting from a parallel between some minimax adaptive tests of a single null hypothesis, based on aggregation approaches, and some tests of multiple hypotheses, we propose a new second kind error-related evaluation criterion, as the core of an emergent minimax theory for multiple tests. Aggregation-based tests, proposed for instance by Baraud [Bernoulli 8 (2002) 577–606], Baraud, Huet and Laurent [Ann. Statist. 31 (2003) 225–251] or Fromont and Laurent [Ann. Statist. 34 (2006) 680–720], are justified through their first kind error rate, which is controlled by the prescribed level on the one hand, and through their separation rates over various classes of alternatives, rates which are minimax on the other hand. We show that some of these tests can be viewed as the first steps of classical step-down multiple testing procedures, and accordingly be evaluated from the multiple testing point of view also, through a control of their Family-Wise Error Rate (FWER). Conversely, many multiple testing procedures, from the historical ones of Bonferroni and Holm, to more recent ones like min-$p$ procedures or randomized procedures such as the ones proposed by Romano and Wolf [J. Amer. Statist. Assoc. 100 (2005) 94–108], can be investigated from the minimax adaptive testing point of view. To this end, we extend the notion of separation rate to the multiple testing field, by defining the weak Family-Wise Separation Rate and its stronger counterpart, the Family-Wise Separation Rate (FWSR). As for nonparametric tests of a single null hypothesis, we prove that these new concepts allow an accurate analysis of the second kind error of a multiple testing procedure, leading to clear definitions of minimax and minimax adaptive multiple tests. Some illustrations in classical Gaussian frameworks corroborate several expected results under particular conditions on the tested hypotheses, but also lead to new questions and perspectives.

We establish minimax optimal rates of convergence for estimation in a high dimensional additive model assuming that it is approximately sparse. Our results reveal a behavior universal to this class of high dimensional problems. In the sparse regime when the components are sufficiently smooth or the dimensionality is sufficiently large, the optimal rates are identical to those for high dimensional linear regression and, therefore, there is no additional cost to entertain a nonparametric model. Otherwise, in the so-called smooth regime, the rates coincide with the optimal rates for estimating a univariate function and, therefore, they are immune to the “curse of dimensionality.”

Model-free variable selection has been implemented under the sufficient dimension reduction framework since the seminal paper of Cook [Ann. Statist. 32 (2004) 1062–1092]. In this paper, we extend the marginal coordinate test for sliced inverse regression (SIR) in Cook (2004) and propose a novel marginal SIR utility for the purpose of ultrahigh dimensional feature selection. Two distinct procedures, Dantzig selector and sparse precision matrix estimation, are incorporated to get two versions of sample level marginal SIR utilities. Both procedures lead to model-free variable selection consistency with predictor dimensionality $p$ diverging at an exponential rate of the sample size $n$. As a special case of marginal SIR, we ignore the correlation among the predictors and propose marginal independence SIR. Marginal independence SIR is closely related to many existing independence screening procedures in the literature, and achieves model-free screening consistency in the ultrahigh dimensional setting. The finite sample performances of the proposed procedures are studied through synthetic examples and an application to the small round blue cell tumors data.

We study the problem of variable selection in convex nonparametric regression. Under the assumption that the true regression function is convex and sparse, we develop a screening procedure to select a subset of variables that contains the relevant variables. Our approach is a two-stage quadratic programming method that estimates a sum of one-dimensional convex functions, followed by one-dimensional concave regression fits on the residuals. In contrast to previous methods for sparse additive models, the optimization is finite dimensional and requires no tuning parameters for smoothness. Under appropriate assumptions, we prove that the procedure is faithful in the population setting, yielding no false negatives. We give a finite sample statistical analysis, and introduce algorithms for efficiently carrying out the required quadratic programs. The approach leads to computational and statistical advantages over fitting a full model, and provides an effective, practical approach to variable screening in convex regression.

Asymmetric least squares regression is an important method that has wide applications in statistics, econometrics and finance. The existing work on asymmetric least squares only considers the traditional low dimension and large sample setting. In this paper, we systematically study the Sparse Asymmetric LEast Squares (SALES) regression under high dimensions where the penalty functions include the Lasso and nonconvex penalties. We develop a unified efficient algorithm for fitting SALES and establish its theoretical properties. As an important application, SALES is used to detect heteroscedasticity in high-dimensional data. Another method for detecting heteroscedasticity is the sparse quantile regression. However, both SALES and the sparse quantile regression may fail to tell which variables are important for the conditional mean and which variables are important for the conditional scale/variance, especially when there are variables that are important for both the mean and the scale. To that end, we further propose a COupled Sparse Asymmetric LEast Squares (COSALES) regression which can be efficiently solved by an algorithm similar to that for solving SALES. We establish theoretical properties of COSALES. In particular, COSALES using the SCAD penalty or MCP is shown to consistently identify the two important subsets for the mean and scale simultaneously, even when the two subsets overlap. We demonstrate the empirical performance of SALES and COSALES by simulated and real data.

We discuss the possibilities and limitations of estimating the mean of a real-valued random variable from independent and identically distributed observations from a nonasymptotic point of view. In particular, we define estimators with a sub-Gaussian behavior even for certain heavy-tailed distributions. We also prove various impossibility results for mean estimators.

We establish minimax lower bounds and maximum likelihood convergence rates of parameter estimation for mean-covariance multivariate Gaussian mixtures, shape-rate Gamma mixtures and some variants of finite mixture models, including the setting where the number of mixing components is bounded but unknown. These models belong to what we call “weakly identifiable” classes, which exhibit specific interactions among mixing parameters driven by the algebraic structures of the class of kernel densities and their partial derivatives. Accordingly, both the minimax bounds and the maximum likelihood parameter estimation rates in these models, obtained under some compactness conditions on the parameter space, are shown to be typically much slower than the usual $n^{-1/2}$ or $n^{-1/4}$ rates of convergence.

The estimation of a log-concave density on $\mathbb{R}^{d}$ represents a central problem in the area of nonparametric inference under shape constraints. In this paper, we study the performance of log-concave density estimators with respect to global loss functions, and adopt a minimax approach. We first show that no statistical procedure based on a sample of size $n$ can estimate a log-concave density with respect to the squared Hellinger loss function with supremum risk smaller than order $n^{-4/5}$, when $d=1$, and order $n^{-2/(d+1)}$ when $d\geq2$. In particular, this reveals a sense in which, when $d\geq3$, log-concave density estimation is fundamentally more challenging than the estimation of a density with two bounded derivatives (a problem to which it has been compared). Second, we show that for $d\leq3$, the Hellinger $\varepsilon$-bracketing entropy of a class of log-concave densities with small mean and covariance matrix close to the identity grows like $\max\{\varepsilon^{-d/2},\varepsilon^{-(d-1)}\}$ (up to a logarithmic factor when $d=2$). This enables us to prove that when $d\leq3$ the log-concave maximum likelihood estimator achieves the minimax optimal rate (up to logarithmic factors when $d=2,3$) with respect to squared Hellinger loss.