In observational studies, treatment may be adapted to covariates at several times without a fixed protocol, in continuous time. Treatment influences covariates, which influence treatment, which influences covariates and so on. Then even time-dependent Cox-models cannot be used to estimate the net treatment effect. Structural nested models have been applied in this setting. Structural nested models are based on counterfactuals: the outcome a person would have had had treatment been withheld after a certain time. Previous work on continuous-time structural nested models assumes that counterfactuals depend deterministically on observed data, while conjecturing that this assumption can be relaxed. This article proves that one can mimic counterfactuals by constructing random variables, solutions to a differential equation, that have the same distribution as the counterfactuals, even given past observed data. These “mimicking” variables can be used to estimate the parameters of structural nested models without assuming the treatment effect to be deterministic.

The stochastic block model (SBM) provides a popular framework for modeling community structures in networks. However, more attention has been devoted to problems concerning estimating the latent node labels and the model parameters than the issue of choosing the number of blocks. We consider an approach based on the log likelihood ratio statistic and analyze its asymptotic properties under model misspecification. We show the limiting distribution of the statistic in the case of underfitting is normal and obtain its convergence rate in the case of overfitting. These conclusions remain valid when the average degree grows at a polylog rate. The results enable us to derive the correct order of the penalty term for model complexity and arrive at a likelihood-based model selection criterion that is asymptotically consistent. Our analysis can also be extended to a degree-corrected block model (DCSBM). In practice, the likelihood function can be estimated using more computationally efficient variational methods or consistent label estimation algorithms, allowing the criterion to be applied to large networks.

A topological multiple testing scheme is presented for detecting peaks in images under stationary ergodic Gaussian noise, where tests are performed at local maxima of the smoothed observed signals. The procedure generalizes the one-dimensional scheme of Schwartzman, Gavrilov and Adler [Ann. Statist. 39 (2011) 3290–3319] to Euclidean domains of arbitrary dimension. Two methods are developed according to two different ways of computing p-values: (i) using the exact distribution of the height of local maxima, available explicitly when the noise field is isotropic [Extremes 18 (2015) 213–240; Expected number and height distribution of critical points of smooth isotropic Gaussian random fields (2015) Preprint]; (ii) using an approximation to the overshoot distribution of local maxima above a pre-threshold, applicable when the exact distribution is unknown, such as when the stationary noise field is nonisotropic [Extremes 18 (2015) 213–240]. The algorithms, combined with the Benjamini–Hochberg procedure for thresholding p-values, provide asymptotic strong control of the False Discovery Rate (FDR) and power consistency, with specific rates, as the search space and signal strength get large. The optimal smoothing bandwidth and optimal pre-threshold are obtained to achieve maximum power. Simulations show that FDR levels are maintained in nonasymptotic conditions. The methods are illustrated in the analysis of functional magnetic resonance images of the brain.

The paper considers the problem of recovering the sparse different components between two high-dimensional means of column-wise dependent random vectors. We show that dependence can be utilized to lower the identification boundary for signal recovery. Moreover, an optimal convergence rate for the marginal false nondiscovery rate (mFNR) is established under dependence. The convergence rate is faster than the optimal rate without dependence. To recover the sparse signal bearing dimensions, we propose a Dependence-Assisted Thresholding and Excising (DATE) procedure, which is shown to be rate optimal for the mFNR with the marginal false discovery rate (mFDR) controlled at a pre-specified level. Extensions of the DATE to recover the differences in contrasts among multiple population means and differences between two covariance matrices are also provided. Simulation studies and case study are given to demonstrate the performance of the proposed signal identification procedure.

Estimation procedures based on recursive algorithms are interesting and powerful techniques that are able to deal rapidly with very large samples of high dimensional data. The collected data may be contaminated by noise so that robust location indicators, such as the geometric median, may be preferred to the mean. In this context, an estimator of the geometric median based on a fast and efficient averaged nonlinear stochastic gradient algorithm has been developed by [Bernoulli 19 (2013) 18–43]. This work aims at studying more precisely the nonasymptotic behavior of this nonlinear algorithm by giving nonasymptotic confidence balls in general separable Hilbert spaces. This new result is based on the derivation of improved $L^{2}$ rates of convergence as well as an exponential inequality for the nearly martingale terms of the recursive nonlinear Robbins–Monro algorithm.

Confidence sets play a fundamental role in statistical inference. In this paper, we consider confidence intervals for high-dimensional linear regression with random design. We first establish the convergence rates of the minimax expected length for confidence intervals in the oracle setting where the sparsity parameter is given. The focus is then on the problem of adaptation to sparsity for the construction of confidence intervals. Ideally, an adaptive confidence interval should have its length automatically adjusted to the sparsity of the unknown regression vector, while maintaining a pre-specified coverage probability. It is shown that such a goal is in general not attainable, except when the sparsity parameter is restricted to a small region over which the confidence intervals have the optimal length of the usual parametric rate. It is further demonstrated that the lack of adaptivity is not due to the conservativeness of the minimax framework, but is fundamentally caused by the difficulty of learning the bias accurately.

We consider the estimation of joint causal effects from observational data. In particular, we propose new methods to estimate the effect of multiple simultaneous interventions (e.g., multiple gene knockouts), under the assumption that the observational data come from an unknown linear structural equation model with independent errors. We derive asymptotic variances of our estimators when the underlying causal structure is partly known, as well as high-dimensional consistency when the causal structure is fully unknown and the joint distribution is multivariate Gaussian. We also propose a generalization of our methodology to the class of nonparanormal distributions. We evaluate the estimators in simulation studies and also illustrate them on data from the DREAM4 challenge.

Statistical latent class models are widely used in social and psychological researches, yet it is often difficult to establish the identifiability of the model parameters. In this paper, we consider the identifiability issue of a family of restricted latent class models, where the restriction structures are needed to reflect pre-specified assumptions on the related assessment. We establish the identifiability results in the strict sense and specify which types of restriction structure would give the identifiability of the model parameters. The results not only guarantee the validity of many of the popularly used models, but also provide a guideline for the related experimental design, where in the current applications the design is usually experience based and identifiability is not guaranteed. Theoretically, we develop a new technique to establish the identifiability result, which may be extended to other restricted latent class models.

We establish a Bernstein-type inequality for a class of stochastic processes that includes the classical geometrically $\phi$-mixing processes, Rio’s generalization of these processes and many time-discrete dynamical systems. Modulo a logarithmic factor and some constants, our Bernstein-type inequality coincides with the classical Bernstein inequality for i.i.d. data. We further use this new Bernstein-type inequality to derive an oracle inequality for generic regularized empirical risk minimization algorithms and data generated by such processes. Applying this oracle inequality to support vector machines using the Gaussian kernels for binary classification, we obtain essentially the same rate as for i.i.d. processes, and for least squares and quantile regression; it turns out that the resulting learning rates match, up to some arbitrarily small extra term in the exponent, the optimal rates for i.i.d. processes.

Strong consistency of the maximum likelihood estimator (MLE) for parametric Gibbs point process models is established. The setting is very general. It includes pairwise pair potentials, finite and infinite multibody interactions and geometrical interactions, where the range can be finite or infinite. The Gibbs interaction may depend linearly or nonlinearly on the parameters, a particular case being hardcore parameters and interaction range parameters. As important examples, we deduce the consistency of the MLE for all parameters of the Strauss model, the hardcore Strauss model, the Lennard–Jones model and the area-interaction model.

Tests based on mean vectors and spatial signs and ranks for a zero mean in one-sample problems and for the equality of means in two-sample problems have been studied in the recent literature for high-dimensional data with the dimension larger than the sample size. For the above testing problems, we show that under suitable sequences of alternatives, the powers of the mean-based tests and the tests based on spatial signs and ranks tend to be same as the data dimension tends to infinity for any sample size when the coordinate variables satisfy appropriate mixing conditions. Further, their limiting powers do not depend on the heaviness of the tails of the distributions. This is in striking contrast to the asymptotic results obtained in the classical multivariate setting. On the other hand, we show that in the presence of stronger dependence among the coordinate variables, the spatial-sign- and rank-based tests for high-dimensional data can be asymptotically more powerful than the mean-based tests if, in addition to the data dimension, the sample size also tends to infinity. The sizes of some mean-based tests for high-dimensional data studied in the recent literature are observed to be significantly different from their nominal levels. This is due to the inadequacy of the asymptotic approximations used for the distributions of those test statistics. However, our asymptotic approximations for the tests based on spatial signs and ranks are observed to work well when the tests are applied on a variety of simulated and real datasets.

We revisit, in an original and challenging perspective, the problem of testing the null hypothesis that the mode of a directional signal is equal to a given value. Motivated by a real data example where the signal is weak, we consider this problem under asymptotic scenarios for which the signal strength goes to zero at an arbitrary rate $\eta_{n}$. Both under the null and the alternative, we focus on rotationally symmetric distributions. We show that, while they are asymptotically equivalent under fixed signal strength, the classical Wald and Watson tests exhibit very different (null and nonnull) behaviours when the signal becomes arbitrarily weak. To fully characterize how challenging the problem is as a function of $\eta_{n}$, we adopt a Le Cam, convergence-of-statistical-experiments, point of view and show that the resulting limiting experiments crucially depend on $\eta_{n}$. In the light of these results, the Watson test is shown to be adaptively rate-consistent and essentially adaptively Le Cam optimal. Throughout, our theoretical findings are illustrated via Monte-Carlo simulations. The practical relevance of our results is also shown on the real data example that motivated the present work.

We consider the asymptotic behaviour of the marginal maximum likelihood empirical Bayes posterior distribution in general setting. First, we characterize the set where the maximum marginal likelihood estimator is located with high probability. Then we provide oracle type of upper and lower bounds for the contraction rates of the empirical Bayes posterior. We also show that the hierarchical Bayes posterior achieves the same contraction rate as the maximum marginal likelihood empirical Bayes posterior. We demonstrate the applicability of our general results for various models and prior distributions by deriving upper and lower bounds for the contraction rates of the corresponding empirical and hierarchical Bayes posterior distributions.

We study theoretical properties of regularized robust $M$-estimators, applicable when data are drawn from a sparse high-dimensional linear model and contaminated by heavy-tailed distributions and/or outliers in the additive errors and covariates. We first establish a form of local statistical consistency for the penalized regression estimators under fairly mild conditions on the error distribution: When the derivative of the loss function is bounded and satisfies a local restricted curvature condition, all stationary points within a constant radius of the true regression vector converge at the minimax rate enjoyed by the Lasso with sub-Gaussian errors. When an appropriate nonconvex regularizer is used in place of an $\ell_{1}$-penalty, we show that such stationary points are in fact unique and equal to the local oracle solution with the correct support; hence, results on asymptotic normality in the low-dimensional case carry over immediately to the high-dimensional setting. This has important implications for the efficiency of regularized nonconvex $M$-estimators when the errors are heavy-tailed. Our analysis of the local curvature of the loss function also has useful consequences for optimization when the robust regression function and/or regularizer is nonconvex and the objective function possesses stationary points outside the local region. We show that as long as a composite gradient descent algorithm is initialized within a constant radius of the true regression vector, successive iterates will converge at a linear rate to a stationary point within the local region. Furthermore, the global optimum of a convex regularized robust regression function may be used to obtain a suitable initialization. The result is a novel two-step procedure that uses a convex $M$-estimator to achieve consistency and a nonconvex $M$-estimator to increase efficiency. We conclude with simulation results that corroborate our theoretical findings.

Feature interactions can contribute to a large proportion of variation in many prediction models. In the era of big data, the coexistence of high dimensionality in both responses and covariates poses unprecedented challenges in identifying important interactions. In this paper, we suggest a two-stage interaction identification method, called the interaction pursuit via distance correlation (IPDC), in the setting of high-dimensional multi-response interaction models that exploits feature screening applied to transformed variables with distance correlation followed by feature selection. Such a procedure is computationally efficient, generally applicable beyond the heredity assumption, and effective even when the number of responses diverges with the sample size. Under mild regularity conditions, we show that this method enjoys nice theoretical properties including the sure screening property, support union recovery and oracle inequalities in prediction and estimation for both interactions and main effects. The advantages of our method are supported by several simulation studies and real data analysis.