The existence of the maximum likelihood estimate in a hierarchical log-linear model is crucial to the reliability of inference for this model. Determining whether the estimate exists is equivalent to finding whether the sufficient statistics vector $t$ belongs to the boundary of the marginal polytope of the model. The dimension of the smallest face $\mathbf{F}_{t}$ containing $t$ determines the dimension of the reduced model which should be considered for correct inference. For higher-dimensional problems, it is not possible to compute $\mathbf{F}_{t}$ exactly. Massam and Wang (2015) found an outer approximation to $\mathbf{F}_{t}$ using a collection of submodels of the original model. This paper refines the methodology to find an outer approximation and devises a new methodology to find an inner approximation. The inner approximation is given not in terms of a face of the marginal polytope, but in terms of a subset of the vertices of $\mathbf{F}_{t}$.

Knowing $\mathbf{F}_{t}$ exactly indicates which cell probabilities have maximum likelihood estimates equal to $0$. When $\mathbf{F}_{t}$ cannot be obtained exactly, we can use, first, the outer approximation $\mathbf{F}_{2}$ to reduce the dimension of the problem and then the inner approximation $\mathbf{F}_{1}$ to obtain correct estimates of cell probabilities corresponding to elements of $\mathbf{F}_{1}$ and improve the estimates of the remaining probabilities corresponding to elements in $\mathbf{F}_{2}\setminus\mathbf{F}_{1}$. Using both real-world and simulated data, we illustrate our results, and show that our methodology scales to high dimensions.

Unsupervised learning is an important problem in statistics and machine learning with a wide range of applications. In this paper, we study clustering of high-dimensional Gaussian mixtures and propose a procedure, called CHIME, that is based on the EM algorithm and a direct estimation method for the sparse discriminant vector. Both theoretical and numerical properties of CHIME are investigated. We establish the optimal rate of convergence for the excess misclustering error and show that CHIME is minimax rate optimal. In addition, the optimality of the proposed estimator of the discriminant vector is also established. Simulation studies show that CHIME outperforms the existing methods under a variety of settings. The proposed CHIME procedure is also illustrated in an analysis of a glioblastoma gene expression data set and shown to have superior performance.

Clustering of Gaussian mixtures in the conventional low-dimensional setting is also considered. The technical tools developed for the high-dimensional setting are used to establish the optimality of the clustering procedure that is based on the classical EM algorithm.

Nonreversible Markov chain Monte Carlo schemes based on piecewise deterministic Markov processes have been recently introduced in applied probability, automatic control, physics and statistics. Although these algorithms demonstrate experimentally good performance and are accordingly increasingly used in a wide range of applications, geometric ergodicity results for such schemes have only been established so far under very restrictive assumptions. We give here verifiable conditions on the target distribution under which the Bouncy Particle Sampler algorithm introduced in [Phys. Rev. E 85 (2012) 026703, 1671–1691] is geometrically ergodic and we provide a central limit theorem for the associated ergodic averages. This holds essentially whenever the target satisfies a curvature condition and the growth of the negative logarithm of the target is at least linear and at most quadratic. For target distributions with thinner tails, we propose an original modification of this scheme that is geometrically ergodic. For targets with thicker tails, we extend the idea pioneered in [Ann. Statist. 40 (2012) 3050–3076] in a random walk Metropolis context. We establish geometric ergodicity of the Bouncy Particle Sampler with respect to an appropriate transformation of the target. Mapping the resulting process back to the original parameterization, we obtain a geometrically ergodic piecewise deterministic Markov process.

Standard MCMC methods can scale poorly to big data settings due to the need to evaluate the likelihood at each iteration. There have been a number of approximate MCMC algorithms that use sub-sampling ideas to reduce this computational burden, but with the drawback that these algorithms no longer target the true posterior distribution. We introduce a new family of Monte Carlo methods based upon a multidimensional version of the Zig-Zag process of [Ann. Appl. Probab. 27 (2017) 846–882], a continuous-time piecewise deterministic Markov process. While traditional MCMC methods are reversible by construction (a property which is known to inhibit rapid convergence) the Zig-Zag process offers a flexible nonreversible alternative which we observe to often have favourable convergence properties. We show how the Zig-Zag process can be simulated without discretisation error, and give conditions for the process to be ergodic. Most importantly, we introduce a sub-sampling version of the Zig-Zag process that is an example of an exact approximate scheme, that is, the resulting approximate process still has the posterior as its stationary distribution. Furthermore, if we use a control-variate idea to reduce the variance of our unbiased estimator, then the Zig-Zag process can be super-efficient: after an initial preprocessing step, essentially independent samples from the posterior distribution are obtained at a computational cost which does not depend on the size of the data.

We consider the estimation of large covariance and precision matrices from high-dimensional sub-Gaussian or heavier-tailed observations with slowly decaying temporal dependence. The temporal dependence is allowed to be long-range so with longer memory than those considered in the current literature. We show that several commonly used methods for independent observations can be applied to the temporally dependent data. In particular, the rates of convergence are obtained for the generalized thresholding estimation of covariance and correlation matrices, and for the constrained $\ell_{1}$ minimization and the $\ell_{1}$ penalized likelihood estimation of precision matrix. Properties of sparsistency and sign-consistency are also established. A gap-block cross-validation method is proposed for the tuning parameter selection, which performs well in simulations. As a motivating example, we study the brain functional connectivity using resting-state fMRI time series data with long-range temporal dependence.

The paper focuses on the problem of model selection in linear Gaussian regression with unknown possibly inhomogeneous noise. For a given family of linear estimators $\{\widetilde{\boldsymbol{{\theta}}}_{m},m\in\mathscr{M}\}$, ordered by their variance, we offer a new “smallest accepted” approach motivated by Lepski’s device and the multiple testing idea. The procedure selects the smallest model which satisfies the acceptance rule based on comparison with all larger models. The method is completely data-driven and does not use any prior information about the variance structure of the noise: its parameters are adjusted to the underlying possibly heterogeneous noise by the so-called “propagation condition” using a wild bootstrap method. The validity of the bootstrap calibration is proved for finite samples with an explicit error bound. We provide a comprehensive theoretical study of the method, describe in details the set of possible values of the selected model $\widehat{m}\in\mathscr{M}$ and establish some oracle error bounds for the corresponding estimator $\widehat{\boldsymbol{{\theta}}}=\widetilde{\boldsymbol{{\theta}}}_{\widehat{m}}$.

We propose a new framework for the detection of change-points in online, sequential data analysis. The approach utilizes nearest neighbor information and can be applied to sequences of multivariate observations or non-Euclidean data objects, such as network data. Different stopping rules are explored, and one specific rule is recommended due to its desirable properties. An accurate analytic approximation of the average run length is derived for the recommended rule, making it an easy off-the-shelf approach for real multivariate/object sequential data monitoring applications. Simulations reveal that the new approach has better performance than likelihood-based approaches for high dimensional data. The new approach is illustrated through a real dataset in detecting global structural changes in social networks.

We study linear subset regression in the context of a high-dimensional linear model. Consider $y=\vartheta +\theta 'z+\epsilon $ with univariate response $y$ and a $d$-vector of random regressors $z$, and a submodel where $y$ is regressed on a set of $p$ explanatory variables that are given by $x=M'z$, for some $d\times p$ matrix $M$. Here, “high-dimensional” means that the number $d$ of available explanatory variables in the overall model is much larger than the number $p$ of variables in the submodel. In this paper, we present Pinsker-type results for prediction of $y$ given $x$. In particular, we show that the mean squared prediction error of the best linear predictor of $y$ given $x$ is close to the mean squared prediction error of the corresponding Bayes predictor $\mathbb{E}[y\|x]$, provided only that $p/\log d$ is small. We also show that the mean squared prediction error of the (feasible) least-squares predictor computed from $n$ independent observations of $(y,x)$ is close to that of the Bayes predictor, provided only that both $p/\log d$ and $p/n$ are small. Our results hold uniformly in the regression parameters and over large collections of distributions for the design variables $z$.

This paper considers testing the equality of two high dimensional means. Two approaches are utilized to formulate $L_{2}$-type tests for better power performance when the two high dimensional mean vectors differ only in sparsely populated coordinates and the differences are faint. One is to conduct thresholding to remove the nonsignal bearing dimensions for variance reduction of the test statistics. The other is to transform the data via the precision matrix for signal enhancement. It is shown that the thresholding and data transformation lead to attractive detection boundaries for the tests. Furthermore, we demonstrate explicitly the effects of precision matrix estimation on the detection boundary for the test with thresholding and data transformation. Extension to multi-sample ANOVA tests is also investigated. Numerical studies are performed to confirm the theoretical findings and demonstrate the practical implementations.

We consider inference post-model-selection in linear regression. In this setting, Berk et al. [Ann. Statist. 41 (2013a) 802–837] recently introduced a class of confidence sets, the so-called PoSI intervals, that cover a certain nonstandard quantity of interest with a user-specified minimal coverage probability, irrespective of the model selection procedure that is being used. In this paper, we generalize the PoSI intervals to confidence intervals for post-model-selection predictors.

A fundamental assumption used in causal inference with observational data is that treatment assignment is ignorable given measured confounding variables. This assumption of no missing confounders is plausible if a large number of baseline covariates are included in the analysis, as we often have no prior knowledge of which variables can be important confounders. Thus, estimation of treatment effects with a large number of covariates has received considerable attention in recent years. Most existing methods require specifying certain parametric models involving the outcome, treatment and confounding variables, and employ a variable selection procedure to identify confounders. However, selection of a proper set of confounders depends on correct specification of the working models. The bias due to model misspecification and incorrect selection of confounding variables can yield misleading results. We propose a robust and efficient approach for inference about the average treatment effect via a flexible modeling strategy incorporating penalized variable selection. Specifically, we consider an estimator constructed based on an efficient influence function that involves a propensity score and an outcome regression. We then propose a new sparse sufficient dimension reduction method to estimate these two functions without making restrictive parametric modeling assumptions. The proposed estimator of the average treatment effect is asymptotically normal and semiparametrically efficient without the need for variable selection consistency. The proposed methods are illustrated via simulation studies and a biomedical application.

Linear structural equation models postulate noisy linear relationships between variables of interest. Each model corresponds to a path diagram, which is a mixed graph with directed edges that encode the domains of the linear functions and bidirected edges that indicate possible correlations among noise terms. Using this graphical representation, we determine the maximum likelihood threshold, that is, the minimum sample size at which the likelihood function of a Gaussian structural equation model is almost surely bounded. Our result allows the model to have feedback loops and is based on decomposing the path diagram with respect to the connected components of its bidirected part. We also prove that if the sample size is below the threshold, then the likelihood function is almost surely unbounded. Our work clarifies, in particular, that standard likelihood inference is applicable to sparse high-dimensional models even if they feature feedback loops.

In this paper, we present a general convex optimization approach for solving high-dimensional multiple response tensor regression problems under low-dimensional structural assumptions. We consider using convex and weakly decomposable regularizers assuming that the underlying tensor lies in an unknown low-dimensional subspace. Within our framework, we derive general risk bounds of the resulting estimate under fairly general dependence structure among covariates. Our framework leads to upper bounds in terms of two very simple quantities, the Gaussian width of a convex set in tensor space and the intrinsic dimension of the low-dimensional tensor subspace. To the best of our knowledge, this is the first general framework that applies to multiple response problems. These general bounds provide useful upper bounds on rates of convergence for a number of fundamental statistical models of interest including multiresponse regression, vector autoregressive models, low-rank tensor models and pairwise interaction models. Moreover, in many of these settings we prove that the resulting estimates are minimax optimal. We also provide a numerical study that both validates our theoretical guarantees and demonstrates the breadth of our framework.

We develop large sample theory for merged data from multiple sources. Main statistical issues treated in this paper are (1) the same unit potentially appears in multiple datasets from overlapping data sources, (2) duplicated items are not identified and (3) a sample from the same data source is dependent due to sampling without replacement. We propose and study a new weighted empirical process and extend empirical process theory to a dependent and biased sample with duplication. Specifically, we establish the uniform law of large numbers and uniform central limit theorem over a class of functions along with several empirical process results under conditions identical to those in the i.i.d. setting. As applications, we study infinite-dimensional $M$-estimation and develop its consistency, rates of convergence and asymptotic normality. Our theoretical results are illustrated with simulation studies and a real data example.

Let $F_{n}$ denote the distribution function of the normalized sum of $n$ i.i.d. random variables. In this paper, polynomial rates of approximation of $F_{n}$ by the corrected normal laws are considered in the model where the underlying distribution has a convolution structure. As a basic tool, the convergence part of Khinchine’s theorem in metric theory of Diophantine approximations is extended to the class of product characteristic functions.

The increased availability of massive data sets provides a unique opportunity to discover subtle patterns in their distributions, but also imposes overwhelming computational challenges. To fully utilize the information contained in big data, we propose a two-step procedure: (i) estimate conditional quantile functions at different levels in a parallel computing environment; (ii) construct a conditional quantile regression process through projection based on these estimated quantile curves. Our general quantile regression framework covers both linear models with fixed or growing dimension and series approximation models. We prove that the proposed procedure does not sacrifice any statistical inferential accuracy provided that the number of distributed computing units and quantile levels are chosen properly. In particular, a sharp upper bound for the former and a sharp lower bound for the latter are derived to capture the minimal computational cost from a statistical perspective. As an important application, the statistical inference on conditional distribution functions is considered. Moreover, we propose computationally efficient approaches to conducting inference in the distributed estimation setting described above. Those approaches directly utilize the availability of estimators from subsamples and can be carried out at almost no additional computational cost. Simulations confirm our statistical inferential theory.

This paper establishes an upper bound for the Kolmogorov distance between the maximum of a high-dimensional vector of smooth Wiener functionals and the maximum of a Gaussian random vector. As a special case, we show that the maximum of multiple Wiener–Itô integrals with common orders is well approximated by its Gaussian analog in terms of the Kolmogorov distance if their covariance matrices are close to each other and the maximum of the fourth cumulants of the multiple Wiener–Itô integrals is close to zero. This may be viewed as a new kind of fourth moment phenomenon, which has attracted considerable attention in the recent studies of probability. This type of Gaussian approximation result has many potential applications to statistics. To illustrate this point, we present two statistical applications in high-frequency financial econometrics: One is the hypothesis testing problem for the absence of lead-lag effects and the other is the construction of uniform confidence bands for spot volatility.

Causal inference is known to be very challenging when only observational data are available. Randomized experiments are often costly and impractical and in instrumental variable regression the number of instruments has to exceed the number of causal predictors. It was recently shown in Peters, Bühlmann and Meinshausen (2016) (J. R. Stat. Soc. Ser. B. Stat. Methodol. 78 947–1012) that causal inference for the full model is possible when data from distinct observational environments are available, exploiting that the conditional distribution of a response variable is invariant under the correct causal model. Two shortcomings of such an approach are the high computational effort for large-scale data and the assumed absence of hidden confounders. Here, we show that these two shortcomings can be addressed if one is willing to make a more restrictive assumption on the type of interventions that generate different environments. Thereby, we look at a different notion of invariance, namely inner-product invariance. By avoiding a computationally cumbersome reverse-engineering approach such as in Peters, Bühlmann and Meinshausen (2016), it allows for large-scale causal inference in linear structural equation models. We discuss identifiability conditions for the causal parameter and derive asymptotic confidence intervals in the low-dimensional setting. In the case of nonidentifiability, we show that the solution set of causal Dantzig has predictive guarantees under certain interventions. We derive finite-sample bounds in the high-dimensional setting and investigate its performance on simulated datasets.

Standard penalized methods of variable selection and parameter estimation rely on the magnitude of coefficient estimates to decide which variables to include in the final model. However, coefficient estimates are unreliable when the design matrix is collinear. To overcome this challenge, an entirely new perspective on variable selection is presented within a generalized fiducial inference framework. This new procedure is able to effectively account for linear dependencies among subsets of covariates in a high-dimensional setting where $p$ can grow almost exponentially in $n$, as well as in the classical setting where $p\le n$. It is shown that the procedure very naturally assigns small probabilities to subsets of covariates which include redundancies by way of explicit $L_{0}$ minimization. Furthermore, with a typical sparsity assumption, it is shown that the proposed method is consistent in the sense that the probability of the true sparse subset of covariates converges in probability to 1 as $n\to\infty$, or as $n\to\infty$ and $p\to\infty$. Very reasonable conditions are needed, and little restriction is placed on the class of possible subsets of covariates to achieve this consistency result.

Exploiting the fact that most arrival processes exhibit cyclic behaviour, we propose a simple procedure for estimating the intensity of a nonhomogeneous Poisson process. The estimator is the super-resolution analogue to Shao (2010) and Shao and Lii [J. R. Stat. Soc. Ser. B. Stat. Methodol. 73 (2011) 99–122], which is a sum of $p$ sinusoids where $p$ and the amplitude and phase of each wave are not known and need to be estimated. This results in an interpretable yet flexible specification that is suitable for use in modelling as well as in high resolution simulations.

Our estimation procedure sits in between classic periodogram methods and atomic/total variation norm thresholding. Through a novel use of window functions in the point process domain, our approach attains super-resolution without semidefinite programming. Under suitable conditions, finite sample guarantees can be derived for our procedure. These resolve some open questions and expand existing results in spectral estimation literature.

The sequential multiple testing problem is considered under two generalized error metrics. Under the first one, the probability of at least $k$ mistakes, of any kind, is controlled. Under the second, the probabilities of at least $k_{1}$ false positives and at least $k_{2}$ false negatives are simultaneously controlled. For each formulation, the optimal expected sample size is characterized, to a first-order asymptotic approximation as the error probabilities go to 0, and a novel multiple testing procedure is proposed and shown to be asymptotically efficient under every signal configuration. These results are established when the data streams for the various hypotheses are independent and each local log-likelihood ratio statistic satisfies a certain strong law of large numbers. In the special case of i.i.d. observations in each stream, the gains of the proposed sequential procedures over fixed-sample size schemes are quantified.