We derive nonasymptotic bounds for the minimax risk of variable selection under expected Hamming loss in the Gaussian mean model in $\mathbb{R}^{d}$ for classes of at most $s$-sparse vectors separated from 0 by a constant $a>0$. In some cases, we get exact expressions for the nonasymptotic minimax risk as a function of $d,s,a$ and find explicitly the minimax selectors. These results are extended to dependent or non-Gaussian observations and to the problem of crowdsourcing. Analogous conclusions are obtained for the probability of wrong recovery of the sparsity pattern. As corollaries, we derive necessary and sufficient conditions for such asymptotic properties as almost full recovery and exact recovery. Moreover, we propose data-driven selectors that provide almost full and exact recovery adaptively to the parameters of the classes.

Under the potential outcomes framework, we propose a randomization based estimation procedure for causal inference from split-plot designs, with special emphasis on $2^{2}$ designs that naturally arise in many social, behavioral and biomedical experiments. Point estimators of factorial effects are obtained and their sampling variances are derived in closed form as linear combinations of the between- and within-group covariances of the potential outcomes. Results are compared to those under complete randomization as measures of design efficiency. Conservative estimators of these sampling variances are proposed. Connection of the randomization-based approach to inference based on the linear mixed effects model is explored. Results on sampling variances of point estimators and their estimators are extended to general split-plot designs. The superiority over existing model-based alternatives in frequency coverage properties is reported under a variety of simulation settings for both binary and continuous outcomes.

Heavy-tailed errors impair the accuracy of the least squares estimate, which can be spoiled by a single grossly outlying observation. As argued in the seminal work of Peter Huber in 1973 [Ann. Statist. 1 (1973) 799–821], robust alternatives to the method of least squares are sorely needed. To achieve robustness against heavy-tailed sampling distributions, we revisit the Huber estimator from a new perspective by letting the tuning parameter involved diverge with the sample size. In this paper, we develop nonasymptotic concentration results for such an adaptive Huber estimator, namely, the Huber estimator with the tuning parameter adapted to sample size, dimension and the variance of the noise. Specifically, we obtain a sub-Gaussian-type deviation inequality and a nonasymptotic Bahadur representation when noise variables only have finite second moments. The nonasymptotic results further yield two conventional normal approximation results that are of independent interest, the Berry–Esseen inequality and Cramér-type moderate deviation. As an important application to large-scale simultaneous inference, we apply these robust normal approximation results to analyze a dependence-adjusted multiple testing procedure for moderately heavy-tailed data. It is shown that the robust dependence-adjusted procedure asymptotically controls the overall false discovery proportion at the nominal level under mild moment conditions. Thorough numerical results on both simulated and real datasets are also provided to back up our theory.

Covariance matrix estimation is one of the most important problems in statistics. To accommodate the complexity of modern datasets, it is desired to have estimation procedures that not only can incorporate the structural assumptions of covariance matrices, but are also robust to outliers from arbitrary sources. In this paper, we define a new concept called matrix depth and then propose a robust covariance matrix estimator by maximizing the empirical depth function. The proposed estimator is shown to achieve minimax optimal rate under Huber’s $\varepsilon$-contamination model for estimating covariance/scatter matrices with various structures including bandedness and sparsity.

In regression models involving economic variables such as income, log transformation is typically taken to achieve approximate normality and stabilize the variance. However, often the interest is predicting individual values or means of the variable in the original scale. Under a nested error model for the log transformation of the target variable, we show that the usual approach of back transforming the predicted values may introduce a substantial bias. We obtain the optimal (or “best”) predictors of individual values of the original variable and of small area means under that model. Empirical best predictors are defined by estimating the unknown model parameters in the best predictors. When estimation is desired for subpopulations with small sample sizes (small areas), nested error models are widely used to “borrow strength” from the other areas and obtain estimators with greater efficiency than direct estimators based on the scarce area-specific data. We show that naive predictors of small area means obtained by back-transformation under the mentioned model may even underperform direct estimators. Moreover, assessing the uncertainty of the considered predictor is not straightforward. Exact mean squared errors of the best predictors and second-order approximations to the mean squared errors of the empirical best predictors are derived. Estimators of the mean squared errors that are second-order correct are also obtained. Simulation studies and an example with Mexican data on living conditions illustrate the procedures.

For sequences of random backward nested subspaces as occur, say, in dimension reduction for manifold or stratified space valued data, asymptotic results are derived. In fact, we formulate our results more generally for backward nested families of descriptors (BNFD). Under rather general conditions, asymptotic strong consistency holds. Under additional, still rather general hypotheses, among them existence of a.s. local twice differentiable charts, asymptotic joint normality of a BNFD can be shown. If charts factor suitably, this leads to individual asymptotic normality for the last element, a principal nested mean or a principal nested geodesic, say. It turns out that these results pertain to principal nested spheres (PNS) and principal nested great subsphere (PNGS) analysis by Jung, Dryden and Marron [Biometrika 99 (2012) 551–568] as well as to the intrinsic mean on a first geodesic principal component (IMo1GPC) for manifolds and Kendall’s shape spaces. A nested bootstrap two-sample test is derived and illustrated with simulations. In a study on real data, PNGS is applied to track early human mesenchymal stem cell differentiation over a coarse time grid and, among others, to locate a change point with direct consequences for the design of further studies.

We consider a sequence of multinomial data for which the probabilities associated with the categories are subject to abrupt changes of unknown magnitudes at unknown locations. When the number of categories is comparable to or even larger than the number of subjects allocated to these categories, conventional methods such as the classical Pearson’s chi-squared test and the deviance test may not work well. Motivated by high-dimensional homogeneity tests, we propose a novel change-point detection procedure that allows the number of categories to tend to infinity. The null distribution of our test statistic is asymptotically normal and the test performs well with finite samples. The number of change-points is determined by minimizing a penalized objective function based on segmentation, and the locations of the change-points are estimated by minimizing the objective function with the dynamic programming algorithm. Under some mild conditions, the consistency of the estimators of multiple change-points is established. Simulation studies show that the proposed method performs satisfactorily for identifying change-points in terms of power and estimation accuracy, and it is illustrated with an analysis of a real data set.

Local Asymptotic Normality (LAN) property for fractional Gaussian noise under high-frequency observations is proved with nondiagonal rate matrices depending on the parameter to be estimated. In contrast to the LAN families in the literature, nondiagonal rate matrices are inevitable. As consequences of the LAN property, a maximum likelihood sequence of estimators is shown to be asymptotically efficient and the likelihood ratio test on the Hurst parameter is shown to be an asymptotically uniformly most powerful unbiased test for two-sided hypotheses.

In this paper, we study the effect of dependence on detecting sparse signals. In particular, we focus on global testing against sparse alternatives for the means of binary outcomes following an Ising model, and establish how the interplay between the strength and sparsity of a signal determines its detectability under various notions of dependence. The profound impact of dependence is best illustrated under the Curie–Weiss model where we observe the effect of a “thermodynamic” phase transition. In particular, the critical state exhibits a subtle “blessing of dependence” phenomenon in that one can detect much weaker signals at criticality than otherwise. Furthermore, we develop a testing procedure that is broadly applicable to account for dependence and show that it is asymptotically minimax optimal under fairly general regularity conditions.

We extend the principal component analysis (PCA) to second-order stationary vector time series in the sense that we seek for a contemporaneous linear transformation for a $p$-variate time series such that the transformed series is segmented into several lower-dimensional subseries, and those subseries are uncorrelated with each other both contemporaneously and serially. Therefore, those lower-dimensional series can be analyzed separately as far as the linear dynamic structure is concerned. Technically, it boils down to an eigenanalysis for a positive definite matrix. When $p$ is large, an additional step is required to perform a permutation in terms of either maximum cross-correlations or FDR based on multiple tests. The asymptotic theory is established for both fixed $p$ and diverging $p$ when the sample size $n$ tends to infinity. Numerical experiments with both simulated and real data sets indicate that the proposed method is an effective initial step in analyzing multiple time series data, which leads to substantial dimension reduction in modelling and forecasting high-dimensional linear dynamical structures. Unlike PCA for independent data, there is no guarantee that the required linear transformation exists. When it does not, the proposed method provides an approximate segmentation which leads to the advantages in, for example, forecasting for future values. The method can also be adapted to segment multiple volatility processes.

We study the nonparametric estimation of a decreasing density function $g_{0}$ in a general $s$-sample biased sampling model with weight (or bias) functions $w_{i}$ for $i=1,\ldots,s$. The determination of the monotone maximum likelihood estimator $\hat{g}_{n}$ and its asymptotic distribution, except for the case when $s=1$, has been long missing in the literature due to certain nonstandard structures of the likelihood function, such as nonseparability and a lack of strictly positive second order derivatives of the negative of the log-likelihood function. The existence, uniqueness, self-characterization, consistency of $\hat{g}_{n}$ and its asymptotic distribution at a fixed point are established in this article. To overcome the barriers caused by nonstandard likelihood structures, for instance, we show the tightness of $\hat{g}_{n}$ via a purely analytic argument instead of an intrinsic geometric one and propose an indirect approach to attain the $\sqrt{n}$-rate of convergence of the linear functional $\int w_{i}\hat{g}_{n}$.

Community detection is a central problem of network data analysis. Given a network, the goal of community detection is to partition the network nodes into a small number of clusters, which could often help reveal interesting structures. The present paper studies community detection in Degree-Corrected Block Models (DCBMs). We first derive asymptotic minimax risks of the problem for a misclassification proportion loss under appropriate conditions. The minimax risks are shown to depend on degree-correction parameters, community sizes and average within and between community connectivities in an intuitive and interpretable way. In addition, we propose a polynomial time algorithm to adaptively perform consistent and even asymptotically optimal community detection in DCBMs.

Let $\{Z_{ij}\}$ be independent and identically distributed (i.i.d.) random variables with $EZ_{ij}=0$, $E\vert Z_{ij}\vert^{2}=1$ and $E\vert Z_{ij}\vert^{4}<\infty$. Define linear processes $Y_{tj}=\sum_{k=0}^{\infty}b_{k}Z_{t-k,j}$ with $\sum_{i=0}^{\infty}\vert b_{i}\vert <\infty$. Consider a $p$-dimensional time series model of the form $\mathbf{x}_{t}=\boldsymbol{\Pi} \mathbf{x}_{t-1}+\Sigma^{1/2}\mathbf{y}_{t},\ 1\leq t\leq T$ with $\mathbf{y}_{t}=(Y_{t1},\ldots,Y_{tp})'$ and $\Sigma^{1/2}$ be the square root of a symmetric positive definite matrix. Let $\mathbf{B}=(1/p)\mathbf{XX}^{*}$ with $\mathbf{X}=(\mathbf{x_{1}},\ldots,\mathbf{x_{T}})'$ and $X^{*}$ be the conjugate transpose. This paper establishes both the convergence in probability and the asymptotic joint distribution of the first $k$ largest eigenvalues of $\mathbf{B}$ when $\mathbf{x}_{t}$ is nonstationary. As an application, two new unit root tests for possible nonstationarity of high-dimensional time series are proposed and then studied both theoretically and numerically.

In this work, we develop a new smooth backfitting method and theory for estimating additive nonparametric regression models when the covariates are contaminated by measurement errors. For this, we devise a new kernel function that suitably deconvolutes the bias due to measurement errors as well as renders a projection interpretation to the resulting estimator in the space of additive functions. The deconvolution property and the projection interpretation are essential for a successful solution of the problem. We prove that the method based on the new kernel weighting scheme achieves the optimal rate of convergence in one-dimensional deconvolution problems when the smoothness of measurement error distribution is less than a threshold value. We find that the speed of convergence is slower than the univariate rate when the smoothness of measurement error distribution is above the threshold, but it is still much faster than the optimal rate in multivariate deconvolution problems. The theory requires a deliberate analysis of the nonnegligible effects of measurement errors being propagated to other additive components through backfitting operation. We present the finite sample performance of the deconvolution smooth backfitting estimators that confirms our theoretical findings.

Several types of graphs with different conditional independence interpretations—also known as Markov properties—have been proposed and used in graphical models. In this paper, we unify these Markov properties by introducing a class of graphs with four types of edges—lines, arrows, arcs and dotted lines—and a single separation criterion. We show that independence structures defined by this class specialize to each of the previously defined cases, when suitable subclasses of graphs are considered. In addition, we define a pairwise Markov property for the subclass of chain mixed graphs, which includes chain graphs with the LWF interpretation, as well as summary graphs (and consequently ancestral graphs). We prove the equivalence of this pairwise Markov property to the global Markov property for compositional graphoid independence models.

The log-concave maximum likelihood estimator of a density on the real line based on a sample of size $n$ is known to attain the minimax optimal rate of convergence of $O(n^{-4/5})$ with respect to, for example, squared Hellinger distance. In this paper, we show that it also enjoys attractive adaptation properties, in the sense that it achieves a faster rate of convergence when the logarithm of the true density is $k$-affine (i.e., made up of $k$ affine pieces), or close to $k$-affine, provided in each case that $k$ is not too large. Our results use two different techniques: the first relies on a new Marshall’s inequality for log-concave density estimation, and reveals that when the true density is close to log-linear on its support, the log-concave maximum likelihood estimator can achieve the parametric rate of convergence in total variation distance. Our second approach depends on local bracketing entropy methods, and allows us to prove a sharp oracle inequality, which implies in particular a risk bound with respect to various global loss functions, including Kullback–Leibler divergence, of $O(\frac{k}{n}\log^{5/4}(en/k))$ when the true density is log-concave and its logarithm is close to $k$-affine.

The extremes of a stationary time series typically occur in clusters. A primary measure for this phenomenon is the extremal index, representing the reciprocal of the expected cluster size. Both disjoint and sliding blocks estimator for the extremal index are analyzed in detail. In contrast to many competitors, the estimators only depend on the choice of one parameter sequence. We derive an asymptotic expansion, prove asymptotic normality and show consistency of an estimator for the asymptotic variance. Explicit calculations in certain models and a finite-sample Monte Carlo simulation study reveal that the sliding blocks estimator outperforms other blocks estimators, and that it is competitive to runs- and inter-exceedance estimators in various models. The methods are applied to a variety of financial time series.

Asymptotic lower bounds for estimation play a fundamental role in assessing the quality of statistical procedures. In this paper, we propose a framework for obtaining semiparametric efficiency bounds for sparse high-dimensional models, where the dimension of the parameter is larger than the sample size. We adopt a semiparametric point of view: we concentrate on one-dimensional functions of a high-dimensional parameter. We follow two different approaches to reach the lower bounds: asymptotic Cramér–Rao bounds and Le Cam’s type of analysis. Both of these approaches allow us to define a class of asymptotically unbiased or “regular” estimators for which a lower bound is derived. Consequently, we show that certain estimators obtained by de-sparsifying (or de-biasing) an $\ell_{1}$-penalized M-estimator are asymptotically unbiased and achieve the lower bound on the variance: thus in this sense they are asymptotically efficient. The paper discusses in detail the linear regression model and the Gaussian graphical model.

We prove a central limit theorem for the components of the eigenvectors corresponding to the $d$ largest eigenvalues of the normalized Laplacian matrix of a finite dimensional random dot product graph. As a corollary, we show that for stochastic blockmodel graphs, the rows of the spectral embedding of the normalized Laplacian converge to multivariate normals and, furthermore, the mean and the covariance matrix of each row are functions of the associated vertex’s block membership. Together with prior results for the eigenvectors of the adjacency matrix, we then compare, via the Chernoff information between multivariate normal distributions, how the choice of embedding method impacts subsequent inference. We demonstrate that neither embedding method dominates with respect to the inference task of recovering the latent block assignments.

A central problem of random matrix theory is to understand the eigenvalues of spiked random matrix models, introduced by Johnstone, in which a prominent eigenvector (or “spike”) is planted into a random matrix. These distributions form natural statistical models for principal component analysis (PCA) problems throughout the sciences. Baik, Ben Arous and Péché showed that the spiked Wishart ensemble exhibits a sharp phase transition asymptotically: when the spike strength is above a critical threshold, it is possible to detect the presence of a spike based on the top eigenvalue, and below the threshold the top eigenvalue provides no information. Such results form the basis of our understanding of when PCA can detect a low-rank signal in the presence of noise. However, under structural assumptions on the spike, not all information is necessarily contained in the spectrum. We study the statistical limits of tests for the presence of a spike, including nonspectral tests. Our results leverage Le Cam’s notion of contiguity and include:

(i) For the Gaussian Wigner ensemble, we show that PCA achieves the optimal detection threshold for certain natural priors for the spike.

(ii) For any non-Gaussian Wigner ensemble, PCA is sub-optimal for detection. However, an efficient variant of PCA achieves the optimal threshold (for natural priors) by pre-transforming the matrix entries.

(iii) For the Gaussian Wishart ensemble, the PCA threshold is optimal for positive spikes (for natural priors) but this is not always the case for negative spikes.

In this paper, we study the statistical behaviour of the Exponentially Weighted Aggregate (EWA) in the problem of high-dimensional regression with fixed design. Under the assumption that the underlying regression vector is sparse, it is reasonable to use the Laplace distribution as a prior. The resulting estimator and, specifically, a particular instance of it referred to as the Bayesian lasso, was already used in the statistical literature because of its computational convenience, even though no thorough mathematical analysis of its statistical properties was carried out. The present work fills this gap by establishing sharp oracle inequalities for the EWA with the Laplace prior. These inequalities show that if the temperature parameter is small, the EWA with the Laplace prior satisfies the same type of oracle inequality as the lasso estimator does, as long as the quality of estimation is measured by the prediction loss. Extensions of the proposed methodology to the problem of prediction with low-rank matrices are considered.

This paper investigates a class of goodness-of-fit tests for fitting an error density in linear regression models with measurement error in covariates. Each test statistic is the integrated square difference between the deconvolution kernel density estimator of the regression model error density and a smoothed version of the null error density, an analog of the so-called Bickel and Rosenblatt test statistic. The asymptotic null distributions of the proposed test statistics are derived for both the ordinary smooth and super smooth cases. The asymptotic power behavior of the proposed tests against a fixed alternative and a class of local nonparametric alternatives for both cases is also described. The finite sample performance of the proposed test is evaluated by a simulation study. The simulation study shows some superiority of the proposed test over some other tests. Finally, a real data is used to illustrate the proposed test.