In this paper, we study the problem of density deconvolution under general assumptions on the measurement error distribution. Typically, deconvolution estimators are constructed using Fourier transform techniques, and it is assumed that the characteristic function of the measurement errors does not have zeros on the real line. This assumption is rather strong and is not fulfilled in many cases of interest. In this paper, we develop a methodology for constructing optimal density deconvolution estimators in the general setting that covers vanishing and nonvanishing characteristic functions of the measurement errors. We derive upper bounds on the risk of the proposed estimators and provide sufficient conditions under which zeros of the corresponding characteristic function have no effect on estimation accuracy. Moreover, we show that the derived conditions are also necessary in some specific problem instances.

We tackle the problem of high-dimensional nonparametric density estimation by taking the class of log-concave densities on ${\mathbb{R}}^{\mathit{p}}$ and incorporating within it symmetry assumptions, which facilitate scalable estimation algorithms and can mitigate the curse of dimensionality. Our main symmetry assumption is that the super-level sets of the density are K-homothetic (i.e., scalar multiples of a convex body $\mathit{K}\subseteq {\mathbb{R}}^{\mathit{p}}$). When K is known, we prove that the K-homothetic log-concave maximum likelihood estimator based on n independent observations from such a density achieves the minimax optimal rate of convergence with respect to, for example, squared Hellinger loss, of order ${\mathit{n}}^{-4/5}$, independent of p. Moreover, we show that the estimator is adaptive in the sense that if the data generating density admits a special form, then a nearly parametric rate may be attained. We also provide worst case and adaptive risk bounds in cases where K is only known up to a positive definite transformation, and where it is completely unknown and must be estimated nonparametrically. Our estimation algorithms are fast even when n and p are on the order of hundreds of thousands, and we illustrate the strong finite-sample performance of our methods on simulated data.

We investigate large-sample properties of treatment effect estimators under unknown interference in randomized experiments. The inferential target is a generalization of the average treatment effect estimand that marginalizes over potential spillover effects. We show that estimators commonly used to estimate treatment effects under no interference are consistent for the generalized estimand for several common experimental designs under limited but otherwise arbitrary and unknown interference. The rates of convergence depend on the growth rate of the unit-average amount of interference and the degree to which the interference aligns with dependencies in treatment assignment. Importantly for practitioners, the results imply that even if one erroneously assumes that units do not interfere in a setting with moderate interference, standard estimators are nevertheless likely to be close to an average treatment effect if the sample is sufficiently large. Conventional confidence statements may, however, not be accurate.

For a nonlinear regression model, the information matrices of designs depend on the parameter of the model. The adaptive Wynn algorithm for D-optimal design estimates the parameter at each step on the basis of the observed responses and employed design points so far, and selects the next design point as in the classical Wynn algorithm for D-optimal design. The name “Wynn algorithm” is in honor of Henry P. Wynn who established the latter “classical” algorithm in his 1970 paper (Ann. Math. Stat. 41 (1970) 1655–1664). The asymptotics of the sequences of designs and maximum likelihood estimates generated by the adaptive algorithm is studied for an important class of nonlinear regression models: generalized linear models whose (univariate) response variables follow a distribution from a one-parameter exponential family. Under the assumptions of compactness of the experimental region and of the parameter space together with some natural continuity assumptions, it is shown that the adaptive ML-estimators are strongly consistent and the design sequence is asymptotically locally D-optimal at the true parameter point. If the true parameter point is an interior point of the parameter space, then under some smoothness assumptions the asymptotic normality of the adaptive ML-estimators is obtained.

Protection against disclosure is a legal and ethical obligation for agencies releasing microdata files for public use. Consider a microdata sample of size n from a finite population of size $\overline{\mathit{n}}=\mathit{n}+\mathit{\lambda }\mathit{n}$, with $\mathit{\lambda }>0$, such that each sample record contains two disjoint types of information: identifying categorical information and sensitive information. Any decision about releasing data is supported by the estimation of measures of disclosure risk, which are defined as discrete functionals of the number of sample records with a unique combination of values of identifying variables. The most common measure is arguably the number ${\mathit{\tau }}_1$ of sample unique records that are population uniques. In this paper, we first study nonparametric estimation of ${\mathit{\tau }}_1$ under the Poisson abundance model for sample records. We introduce a class of linear estimators of ${\mathit{\tau }}_1$ that are simple, computationally efficient and scalable to massive datasets, and we give uniform theoretical guarantees for them. In particular, we show that they provably estimate ${\mathit{\tau }}_1$ all of the way up to the sampling fraction ${(\mathit{\lambda }+1)}^{-1}\propto {(log\mathit{n})}^{-1}$, with vanishing normalized mean-square error (NMSE) for large n. We then establish a lower bound for the minimax NMSE for the estimation of ${\mathit{\tau }}_1$, which allows us to show that: (i) ${(\mathit{\lambda }+1)}^{-1}\propto {(log\mathit{n})}^{-1}$ is the smallest possible sampling fraction for consistently estimating ${\mathit{\tau }}_1$; (ii) estimators’ NMSE is near optimal, in the sense of matching the minimax lower bound, for large n. This is the main result of our paper, and it provides a rigorous answer to an open question about the feasibility of nonparametric estimation of ${\mathit{\tau }}_1$ under the Poisson abundance model and for a sampling fraction ${(\mathit{\lambda }+1)}^{-1}<1/2$.

Exchangeable random graphs serve as an important probabilistic framework for the statistical analysis of network data. In this work, we develop an alternative parameterization for a large class of exchangeable random graphs, where the nodes are independent random vectors in a linear space equipped with an indefinite inner product, and the edge probability between two nodes equals the inner product of the corresponding node vectors. Therefore, the distribution of exchangeable random graphs in this subclass can be represented by a node sampling distribution on this linear space, which we call the graph root distribution. We study existence and identifiability of such representations, the topological relationship between the graph root distribution and the exchangeable random graph sampling distribution and estimation of graph root distributions.

We consider the problem of nonparametric regression when the covariate is d dimensional, where $\mathit{d}\ge 1$. In this paper, we introduce and study two nonparametric least squares estimators (LSEs) in this setting—the entirely monotonic LSE and the constrained Hardy–Krause variation LSE. We show that these two LSEs are natural generalizations of univariate isotonic regression and univariate total variation denoising, respectively, to multiple dimensions. We discuss the characterization and computation of these two LSEs obtained from n data points. We provide a detailed study of their risk properties under the squared error loss and fixed uniform lattice design. We show that the finite sample risk of these LSEs is always bounded from above by ${\mathit{n}}^{-2/3}$ modulo logarithmic factors depending on d; thus these nonparametric LSEs avoid the curse of dimensionality to some extent. We also prove nearly matching minimax lower bounds. Further, we illustrate that these LSEs are particularly useful in fitting rectangular piecewise constant functions. Specifically, we show that the risk of the entirely monotonic LSE is almost parametric (at most $1/\mathit{n}$ up to logarithmic factors) when the true function is well approximable by a rectangular piecewise constant entirely monotone function with not too many constant pieces. A similar result is also shown to hold for the constrained Hardy–Krause variation LSE for a simple subclass of rectangular piecewise constant functions. We believe that the proposed LSEs yield a novel approach to estimating multivariate functions using convex optimization that avoid the curse of dimensionality to some extent.

In Learn-As-you-GO (LAGO) adaptive studies, the intervention is a complex multicomponent package, and is adapted in stages during the study based on past outcome data. This design formalizes standard practice in public health intervention studies. An effective intervention package is sought, while minimizing intervention package cost. In LAGO study data, the interventions in later stages depend upon the outcomes in the previous stages, violating standard statistical theory. We develop an estimator for the intervention effects, and prove consistency and asymptotic normality using a novel coupling argument, ensuring the validity of the test for the hypothesis of no overall intervention effect. We develop a confidence set for the optimal intervention package and confidence bands for the success probabilities under alternative package compositions. We illustrate our methods in the BetterBirth Study, which aimed to improve maternal and neonatal outcomes among 157,689 births in Uttar Pradesh, India through a multicomponent intervention package.

This paper investigates conditions for variable selection consistency of the LASSO in high dimensional regression models and gives necessary and sufficient conditions for the same, potentially allowing the model dimension p to grow arbitrarily fast as a function of the sample size n. These conditions require both upper and lower bounds on the growth rate of the penalty parameter. It turns out that a variant of the irrepresentable Condition (IRC) of (J. Mach. Learn. Res. 7 (2006) 2541–2563), herein called the lower irrepresentable Condition (or LIRC), is determined by the lower bound considerations while the upper bound considerations lead to a new condition, called the upper irrepresentable Condition (or UIRC) in this paper. It is shown that the LIRC together with the UIRC is necessary and sufficient for the variable selection consistency of the LASSO, thereby settling a conjecture of (J. Mach. Learn. Res. 7 (2006) 2541–2563). Further, it is shown that under some mild regularity conditions, the penalty parameter must necessarily tend to infinity at a certain minimal rate to ensure variable selection consistency of the LASSO and that the corresponding LASSO estimators of the nonzero regression parameters cannot be $\sqrt{\mathit{n}}$-consistent (even for individual parameters). Thus, under fairly general conditions, the LASSO with a single choice of the penalty parameter cannot achieve both variable selection consistency and $\sqrt{\mathit{n}}$-consistency simultaneously.

Exchangeable arrays are natural tools to model common forms of dependence between units of a sample. Jointly exchangeable arrays are well suited to dyadic data, where observed random variables are indexed by two units from the same population. Examples include trade flows between countries or relationships in a network. Separately exchangeable arrays are well suited to multiway clustering, where units sharing the same cluster (e.g., geographical areas or sectors of activity when considering individual wages) may be dependent in an unrestricted way. We prove uniform laws of large numbers and central limit theorems for such exchangeable arrays. We obtain these results under the same moment restrictions and conditions on the class of functions as those typically assumed with i.i.d. data. We also show the convergence of bootstrap processes adapted to such arrays.

Hierarchical nonparametric processes are popular tools for defining priors on collections of probability distributions, which induce dependence across multiple samples. In survival analysis problems, one is typically interested in modeling the hazard rates, rather than the probability distributions themselves, and the currently available methodologies are not applicable. Here, we fill this gap by introducing a novel, and analytically tractable, class of multivariate mixtures whose distribution acts as a prior for the vector of sample-specific baseline hazard rates. The dependence is induced through a hierarchical specification of the mixing random measures that ultimately corresponds to a composition of random discrete combinatorial structures. Our theoretical results allow to develop a full Bayesian analysis for this class of models, which can also account for right-censored survival data and covariates, and we also show posterior consistency. In particular, we emphasize that the posterior characterization we achieve is the key for devising both marginal and conditional algorithms for evaluating Bayesian inferences of interest. The effectiveness of our proposal is illustrated through some synthetic and real data examples.

This paper investigates adaptive importance sampling algorithms for which the policy, the sequence of distributions used to generate the particles, is a mixture distribution between a flexible kernel density estimate (based on the previous particles), and a “safe” heavy-tailed density. When the share of samples generated according to the safe density goes to zero but not too quickly, two results are established: (i) uniform convergence rates are derived for the policy toward the target density; (ii) a central limit theorem is obtained for the resulting integral estimates. The fact that the asymptotic variance is the same as the variance of an “oracle” procedure with variance-optimal policy, illustrates the benefits of the approach. In addition, a subsampling step (among the particles) can be conducted before constructing the kernel estimate in order to decrease the computational effort without altering the performance of the method. The practical behavior of the algorithms is illustrated in a simulation study.

Distributed statistical learning problems arise commonly when dealing with large datasets. In this setup, datasets are partitioned over machines, which compute locally, and communicate short messages. Communication is often the bottleneck. In this paper, we study one-step and iterative weighted parameter averaging in statistical linear models under data parallelism. We do linear regression on each machine, send the results to a central server and take a weighted average of the parameters. Optionally, we iterate, sending back the weighted average and doing local ridge regressions centered at it. How does this work compared to doing linear regression on the full data? Here, we study the performance loss in estimation and test error, and confidence interval length in high dimensions, where the number of parameters is comparable to the training data size.

We find the performance loss in one-step weighted averaging, and also give results for iterative averaging. We also find that different problems are affected differently by the distributed framework. Estimation error and confidence interval length increases a lot, while prediction error increases much less. We rely on recent results from random matrix theory, where we develop a new calculus of deterministic equivalents as a tool of broader interest.

This paper is concerned with estimating the column space of an unknown low-rank matrix ${\mathit{A}}^{\star }\in {\mathbb{R}}^{{\mathit{d}}_1\times {\mathit{d}}_2}$, given noisy and partial observations of its entries. There is no shortage of scenarios where the observations—while being too noisy to support faithful recovery of the entire matrix—still convey sufficient information to enable reliable estimation of the column space of interest. This is particularly evident and crucial for the highly unbalanced case where the column dimension ${\mathit{d}}_2$ far exceeds the row dimension ${\mathit{d}}_1$, which is the focal point of the current paper.

We investigate an efficient spectral method, which operates upon the sample Gram matrix with diagonal deletion. While this algorithmic idea has been studied before, we establish new statistical guarantees for this method in terms of both ${\ell }_2$ and ${\ell }_{2,\infty }$ estimation accuracy, which improve upon prior results if ${\mathit{d}}_2$ is substantially larger than ${\mathit{d}}_1$. To illustrate the effectiveness of our findings, we derive matching minimax lower bounds with respect to the noise levels, and develop consequences of our general theory for three applications of practical importance: (1) tensor completion from noisy data, (2) covariance estimation/principal component analysis with missing data and (3) community recovery in bipartite graphs. Our theory leads to improved performance guarantees for all three cases.

This paper is devoted to the stochastic approximation of entropically regularized Wasserstein distances between two probability measures, also known as Sinkhorn divergences. The semi-dual formulation of such regularized optimal transportation problems can be rewritten as a nonstrongly concave optimisation problem. It allows to implement a Robbins–Monro stochastic algorithm to estimate the Sinkhorn divergence using a sequence of data sampled from one of the two distributions. Our main contribution is to establish the almost sure convergence and the asymptotic normality of a new recursive estimator of the Sinkhorn divergence between two probability measures in the discrete and semi-discrete settings. We also study the rate of convergence of the expected excess risk of this estimator in the absence of strong concavity of the objective function. Numerical experiments on synthetic and real datasets are also provided to illustrate the usefulness of our approach for data analysis.

A method for extracting multiscale geometric features from a data cloud is proposed and analyzed. Based on geometric considerations, we map each pair of data points into a real-valued feature function defined on the unit interval. Further statistical analysis is then based on the collection of feature functions. The potential of the method is illustrated by different applications, including classification and anomaly detection. Connections to other concepts, such as random set theory, localized depth measures and nonlinear dimension reduction, are also explored.

For nonparametric univariate regression under a monotonicity constraint on the regression function f, we study the coverage of a Bayesian credible interval for $\mathit{f}({\mathit{x}}_0)$, where ${\mathit{x}}_0$ is an interior point. Analysis of the posterior becomes a lot more tractable by considering a “projection-posterior” distribution based on a finite random series of step functions with normal basis coefficients as a prior for f. A sample f from the resulting conjugate posterior distribution is projected on the space of monotone increasing functions to obtain a monotone function ${\mathit{f}}^{\ast }$ closest to f, inducing the “projection-posterior.” We use projection-posterior samples to obtain credible intervals for $\mathit{f}({\mathit{x}}_0)$. We obtain the asymptotic coverage of the credible interval thus constructed and observe that it is free of nuisance parameters involving the true function. We observe a very interesting phenomenon that the coverage is typically higher than the nominal credibility level, the opposite of a phenomenon observed by Cox (Ann. Statist. 21 (1993) 903–923) in the Gaussian sequence model. We further show that a recalibration gives the right asymptotic coverage by starting from a lower credibility level that can be explicitly calculated.

We consider the problem of learning an unknown function ${\mathit{f}}_{\star }$ on the d-dimensional sphere with respect to the square loss, given i.i.d. samples ${\{({\mathit{y}}_{\mathit{i}},{\mathit{x}}_{\mathit{i}})\}}_{\mathit{i}\le \mathit{n}}$ where ${\mathit{x}}_{\mathit{i}}$ is a feature vector uniformly distributed on the sphere and ${\mathit{y}}_{\mathit{i}}={\mathit{f}}_{\star }({\mathit{x}}_{\mathit{i}})+{\mathit{\epsilon }}_{\mathit{i}}$. We study two popular classes of models that can be regarded as linearizations of two-layers neural networks around a random initialization: the random features model of Rahimi–Recht (RF); the neural tangent model of Jacot–Gabriel–Hongler (NT). Both these models can also be regarded as randomized approximations of kernel ridge regression (with respect to different kernels), and enjoy universal approximation properties when the number of neurons N diverges, for a fixed dimension d.

We consider two specific regimes: the infinite-sample finite-width regime, in which $\mathit{n}=\infty$ while d and N are large but finite, and the infinite-width finite-sample regime in which $\mathit{N}=\infty$ while d and n are large but finite. In the first regime, we prove that if ${\mathit{d}}^{\ell +\mathit{\delta }}\le \mathit{N}\le {\mathit{d}}^{\ell +1-\mathit{\delta }}$ for small $\mathit{\delta }>0$, then RF effectively fits a degree-ℓ polynomial in the raw features, and NT fits a degree-$(\ell +1)$ polynomial. In the second regime, both RF and NT reduce to kernel methods with rotationally invariant kernels. We prove that, if the sample size satisfies ${\mathit{d}}^{\ell +\mathit{\delta }}\le \mathit{n}\le {\mathit{d}}^{\ell +1-\mathit{\delta }}$, then kernel methods can fit at most a degree-ℓ polynomial in the raw features. This lower bound is achieved by kernel ridge regression, and near-optimal prediction error is achieved for vanishing ridge regularization.

A confidence sequence is a sequence of confidence intervals that is uniformly valid over an unbounded time horizon. Our work develops confidence sequences whose widths go to zero, with nonasymptotic coverage guarantees under nonparametric conditions. We draw connections between the Cramér–Chernoff method for exponential concentration, the law of the iterated logarithm (LIL) and the sequential probability ratio test—our confidence sequences are time-uniform extensions of the first; provide tight, nonasymptotic characterizations of the second; and generalize the third to nonparametric settings, including sub-Gaussian and Bernstein conditions, self-normalized processes and matrix martingales. We illustrate the generality of our proof techniques by deriving an empirical-Bernstein bound growing at a LIL rate, as well as a novel upper LIL for the maximum eigenvalue of a sum of random matrices. Finally, we apply our methods to covariance matrix estimation and to estimation of sample average treatment effect under the Neyman–Rubin potential outcomes model.

We study the detection of a sparse change in a high-dimensional mean vector as a minimax testing problem. Our first main contribution is to derive the exact minimax testing rate across all parameter regimes for n independent, p-variate Gaussian observations. This rate exhibits a phase transition when the sparsity level is of order $\sqrt{\mathit{p}loglog(8\mathit{n})}$ and has a very delicate dependence on the sample size: in a certain sparsity regime, it involves a triple iterated logarithmic factor in n. Further, in a dense asymptotic regime, we identify the sharp leading constant, while in the corresponding sparse asymptotic regime, this constant is determined to within a factor of $\sqrt{2}$. Extensions that cover spatial and temporal dependence, primarily in the dense case, are also provided.

We study a class of separable sample covariance matrices of the form ${\tilde{\mathcal{Q}}}_1:={\tilde{\mathit{A}}}^{1/2}\mathit{X}\tilde{\mathit{B}}{\mathit{X}}^{\ast }{\tilde{\mathit{A}}}^{1/2}$. Here, $\tilde{\mathit{A}}$ and $\tilde{\mathit{B}}$ are positive definite matrices whose spectrums consist of bulk spectrums plus several spikes, that is, larger eigenvalues that are separated from the bulks. Conceptually, we call ${\tilde{\mathcal{Q}}}_1$ a spiked separable covariance matrix model. On the one hand, this model includes the spiked covariance matrix as a special case with $\tilde{\mathit{B}}=\mathit{I}$. On the other hand, it allows for more general correlations of datasets. In particular, for spatio-temporal dataset, $\tilde{\mathit{A}}$ and $\tilde{\mathit{B}}$ represent the spatial and temporal correlations, respectively.

In this paper, we study the outlier eigenvalues and eigenvectors, that is, the principal components, of the spiked separable covariance model ${\tilde{\mathcal{Q}}}_1$. We prove the convergence of the outlier eigenvalues ${\tilde{\mathit{\lambda }}}_{\mathit{i}}$ and the generalized components (i.e., $⟨\mathbf{v},{\tilde{\mathit{\xi }}}_{\mathit{i}}⟩$ for any deterministic vector v) of the outlier eigenvectors ${\tilde{\mathit{\xi }}}_{\mathit{i}}$ with optimal convergence rates. Moreover, we also prove the delocalization of the nonoutlier eigenvectors. We state our results in full generality, in the sense that they also hold near the so-called BBP transition and for degenerate outliers. Our results highlight both the similarity and difference between the spiked separable covariance matrix model and the spiked covariance matrix model in (Probab. Theory Related Fields 164 (2016) 459–552). In particular, we show that the spikes of both $\tilde{\mathit{A}}$ and $\tilde{\mathit{B}}$ will cause outliers of the eigenvalue spectrum, and the eigenvectors can help to select the outliers that correspond to the spikes of $\tilde{\mathit{A}}$ (or $\tilde{\mathit{B}}$).

Unlike the real line, the real space ${\mathbb{R}}^{\mathit{d}}$, for $\mathit{d}\ge 2$, is not canonically ordered. As a consequence, such fundamental univariate concepts as quantile and distribution functions and their empirical counterparts, involving ranks and signs, do not canonically extend to the multivariate context. Palliating that lack of a canonical ordering has been an open problem for more than half a century, generating an abundant literature and motivating, among others, the development of statistical depth and copula-based methods. We show that, unlike the many definitions proposed in the literature, the measure transportation-based ranks and signs introduced in Chernozhukov, Galichon, Hallin and Henry (Ann. Statist. 45 (2017) 223–256) enjoy all the properties that make univariate ranks a successful tool for semiparametric inference. Related with those ranks, we propose a new center-outward definition of multivariate distribution and quantile functions, along with their empirical counterparts, for which we establish a Glivenko–Cantelli result. Our approach is based on McCann (Duke Math. J. 80 (1995) 309–323) and our results do not require any moment assumptions. The resulting ranks and signs are shown to be strictly distribution-free and essentially maximal ancillary in the sense of Basu (Sankhyā 21 (1959) 247–256) which, in semiparametric models involving noise with unspecified density, can be interpreted as a finite-sample form of semiparametric efficiency. Although constituting a sufficient summary of the sample, empirical center-outward distribution functions are defined at observed values only. A continuous extension to the entire d-dimensional space, yielding smooth empirical quantile contours and sign curves while preserving the essential monotonicity and Glivenko–Cantelli features of the concept, is provided. A numerical study of the resulting empirical quantile contours is conducted.

Brenier’s theorem is a cornerstone of optimal transport that guarantees the existence of an optimal transport map T between two probability distributions P and Q over ${\mathbb{R}}^{\mathit{d}}$ under certain regularity conditions. The main goal of this work is to establish the minimax estimation rates for such a transport map from data sampled from P and Q under additional smoothness assumptions on T. To achieve this goal, we develop an estimator based on the minimization of an empirical version of the semidual optimal transport problem, restricted to truncated wavelet expansions. This estimator is shown to achieve near minimax optimality using new stability arguments for the semidual and a complementary minimax lower bound. Furthermore, we provide numerical experiments on synthetic data supporting our theoretical findings and highlighting the practical benefits of smoothness regularization. These are the first minimax estimation rates for transport maps in general dimension.

We provide new limit theory for functionals of a general class of processes lying at the boundary between stationarity and nonstationarity—what we term weakly nonstationary processes (WNPs). This includes, as leading examples, fractional processes with $\mathit{d}=1/2$, and arrays of autoregressive processes with roots drifting slowly towards unity. We first apply the theory to study inference in parametric and nonparametric regression models involving WNPs as covariates. We then use these results to develop a new specification test for parametric regression models. By construction, our specification test statistic has a ${\mathit{\chi }}^2$ limiting distribution regardless of the form and extent of persistence of the regressor, implying that a practitioner can validly perform the test using a fixed critical value, while remaining agnostic about the mechanism generating the regressor. Simulation exercises confirm that the test controls size across a wide range of data generating processes, and outperforms a comparable test due to Wang and Phillips (Ann. Statist. 40 (2012) 727–758) against many alternatives.

We consider the class of all multiple testing methods controlling tail probabilities of the false discovery proportion, either for one random set or simultaneously for many such sets. This class encompasses methods controlling familywise error rate, generalized familywise error rate, false discovery exceedance, joint error rate, simultaneous control of all false discovery proportions, and others, as well as gene set testing in genomics and cluster inference in neuroimaging. We show that all such methods are either equivalent to a closed testing procedure, or are uniformly improved by one. Moreover, we show that a closed testing method is admissible if and only if all its local tests are admissible. This implies that, when designing methods, it is sufficient to restrict attention to closed testing. We demonstrate the practical usefulness of this design principle by obtaining more informative inferences from the method of higher criticism, and by constructing a uniform improvement of a recently proposed method.