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Søren Wengel Mogensen, Niels Richard Hansen.

Source: The Annals of Statistics, Volume 48, Number 1, 539--559.

Abstract:
Symmetric independence relations are often studied using graphical representations. Ancestral graphs or acyclic directed mixed graphs with $m$-separation provide classes of symmetric graphical independence models that are closed under marginalization. Asymmetric independence relations appear naturally for multivariate stochastic processes, for instance, in terms of local independence. However, no class of graphs representing such asymmetric independence relations, which is also closed under marginalization, has been developed. We develop the theory of directed mixed graphs with $mu $-separation and show that this provides a graphical independence model class which is closed under marginalization and which generalizes previously considered graphical representations of local independence. Several graphs may encode the same set of independence relations and this means that in many cases only an equivalence class of graphs can be identified from observational data. For statistical applications, it is therefore pivotal to characterize graphs that induce the same independence relations. Our main result is that for directed mixed graphs with $mu $-separation each equivalence class contains a maximal element which can be constructed from the independence relations alone. Moreover, we introduce the directed mixed equivalence graph as the maximal graph with dashed and solid edges. This graph encodes all information about the edges that is identifiable from the independence relations, and furthermore it can be computed efficiently from the maximal graph.

Edgar Dobriban, William Leeb, Amit Singer.

Source: The Annals of Statistics, Volume 48, Number 1, 491--513.

Abstract:
We consider the linearly transformed spiked model , where the observations $Y_{i}$ are noisy linear transforms of unobserved signals of interest $X_{i}$: egin{equation*}Y_{i}=A_{i}X_{i}+varepsilon_{i},end{equation*} for $i=1,ldots ,n$. The transform matrices $A_{i}$ are also observed. We model the unobserved signals (or regression coefficients) $X_{i}$ as vectors lying on an unknown low-dimensional space. Given only $Y_{i}$ and $A_{i}$ how should we predict or recover their values? The naive approach of performing regression for each observation separately is inaccurate due to the large noise level. Instead, we develop optimal methods for predicting $X_{i}$ by “borrowing strength” across the different samples. Our linear empirical Bayes methods scale to large datasets and rely on weak moment assumptions. We show that this model has wide-ranging applications in signal processing, deconvolution, cryo-electron microscopy, and missing data with noise. For missing data, we show in simulations that our methods are more robust to noise and to unequal sampling than well-known matrix completion methods.

François Bachoc, David Preinerstorfer, Lukas Steinberger.

Source: The Annals of Statistics, Volume 48, Number 1, 440--463.

Abstract:
We suggest general methods to construct asymptotically uniformly valid confidence intervals post-model-selection. The constructions are based on principles recently proposed by Berk et al. ( Ann. Statist. 41 (2013) 802–837). In particular, the candidate models used can be misspecified, the target of inference is model-specific, and coverage is guaranteed for any data-driven model selection procedure. After developing a general theory, we apply our methods to practically important situations where the candidate set of models, from which a working model is selected, consists of fixed design homoskedastic or heteroskedastic linear models, or of binary regression models with general link functions. In an extensive simulation study, we find that the proposed confidence intervals perform remarkably well, even when compared to existing methods that are tailored only for specific model selection procedures.

Michael Schweinberger, Jonathan Stewart.

Source: The Annals of Statistics, Volume 48, Number 1, 374--396.

Abstract:
Statistical inference for exponential-family models of random graphs with dependent edges is challenging. We stress the importance of additional structure and show that additional structure facilitates statistical inference. A simple example of a random graph with additional structure is a random graph with neighborhoods and local dependence within neighborhoods. We develop the first concentration and consistency results for maximum likelihood and $M$-estimators of a wide range of canonical and curved exponential-family models of random graphs with local dependence. All results are nonasymptotic and applicable to random graphs with finite populations of nodes, although asymptotic consistency results can be obtained as well. In addition, we show that additional structure can facilitate subgraph-to-graph estimation, and present concentration results for subgraph-to-graph estimators. As an application, we consider popular curved exponential-family models of random graphs, with local dependence induced by transitivity and parameter vectors whose dimensions depend on the number of nodes.

Davy Paindaveine, Julien Remy, Thomas Verdebout.

Source: The Annals of Statistics, Volume 48, Number 1, 324--345.

Abstract:
We consider the problem of testing, on the basis of a $p$-variate Gaussian random sample, the null hypothesis $mathcal{H}_{0}:oldsymbol{ heta}_{1}=oldsymbol{ heta}_{1}^{0}$ against the alternative $mathcal{H}_{1}:oldsymbol{ heta}_{1} eq oldsymbol{ heta}_{1}^{0}$, where $oldsymbol{ heta}_{1}$ is the “first” eigenvector of the underlying covariance matrix and $oldsymbol{ heta}_{1}^{0}$ is a fixed unit $p$-vector. In the classical setup where eigenvalues $lambda_{1}>lambda_{2}geq cdots geq lambda_{p}$ are fixed, the Anderson ( Ann. Math. Stat. 34 (1963) 122–148) likelihood ratio test (LRT) and the Hallin, Paindaveine and Verdebout ( Ann. Statist. 38 (2010) 3245–3299) Le Cam optimal test for this problem are asymptotically equivalent under the null hypothesis, hence also under sequences of contiguous alternatives. We show that this equivalence does not survive asymptotic scenarios where $lambda_{n1}/lambda_{n2}=1+O(r_{n})$ with $r_{n}=O(1/sqrt{n})$. For such scenarios, the Le Cam optimal test still asymptotically meets the nominal level constraint, whereas the LRT severely overrejects the null hypothesis. Consequently, the former test should be favored over the latter one whenever the two largest sample eigenvalues are close to each other. By relying on the Le Cam’s asymptotic theory of statistical experiments, we study the non-null and optimality properties of the Le Cam optimal test in the aforementioned asymptotic scenarios and show that the null robustness of this test is not obtained at the expense of power. Our asymptotic investigation is extensive in the sense that it allows $r_{n}$ to converge to zero at an arbitrary rate. While we restrict to single-spiked spectra of the form $lambda_{n1}>lambda_{n2}=cdots =lambda_{np}$ to make our results as striking as possible, we extend our results to the more general elliptical case. Finally, we present an illustrative real data example.

Stephen M. S. Lee, Puyudi Yang.

Source: The Annals of Statistics, Volume 48, Number 1, 274--299.

Abstract:
Suppose that a confidence region is desired for a subvector $ heta $ of a multidimensional parameter $xi =( heta ,psi )$, based on an M-estimator $hat{xi }_{n}=(hat{ heta }_{n},hat{psi }_{n})$ calculated from a random sample of size $n$. Under nonstandard conditions $hat{xi }_{n}$ often converges at a nonregular rate $r_{n}$, in which case consistent estimation of the distribution of $r_{n}(hat{ heta }_{n}- heta )$, a pivot commonly chosen for confidence region construction, is most conveniently effected by the $m$ out of $n$ bootstrap. The above choice of pivot has three drawbacks: (i) the shape of the region is either subjectively prescribed or controlled by a computationally intensive depth function; (ii) the region is not transformation equivariant; (iii) $hat{xi }_{n}$ may not be uniquely defined. To resolve the above difficulties, we propose a one-dimensional pivot derived from the criterion function, and prove that its distribution can be consistently estimated by the $m$ out of $n$ bootstrap, or by a modified version of the perturbation bootstrap. This leads to a new method for constructing confidence regions which are transformation equivariant and have shapes driven solely by the criterion function. A subsampling procedure is proposed for selecting $m$ in practice. Empirical performance of the new method is illustrated with examples drawn from different nonstandard M-estimation settings. Extension of our theory to row-wise independent triangular arrays is also explored.

Xi Chen, Jason D. Lee, Xin T. Tong, Yichen Zhang.

Source: The Annals of Statistics, Volume 48, Number 1, 251--273.

Abstract:
The stochastic gradient descent (SGD) algorithm has been widely used in statistical estimation for large-scale data due to its computational and memory efficiency. While most existing works focus on the convergence of the objective function or the error of the obtained solution, we investigate the problem of statistical inference of true model parameters based on SGD when the population loss function is strongly convex and satisfies certain smoothness conditions. Our main contributions are twofold. First, in the fixed dimension setup, we propose two consistent estimators of the asymptotic covariance of the average iterate from SGD: (1) a plug-in estimator, and (2) a batch-means estimator, which is computationally more efficient and only uses the iterates from SGD. Both proposed estimators allow us to construct asymptotically exact confidence intervals and hypothesis tests. Second, for high-dimensional linear regression, using a variant of the SGD algorithm, we construct a debiased estimator of each regression coefficient that is asymptotically normal. This gives a one-pass algorithm for computing both the sparse regression coefficients and confidence intervals, which is computationally attractive and applicable to online data.

Subhadeep Paul, Yuguo Chen.

Source: The Annals of Statistics, Volume 48, Number 1, 230--250.

Abstract:
We consider the problem of estimating a consensus community structure by combining information from multiple layers of a multi-layer network using methods based on the spectral clustering or a low-rank matrix factorization. As a general theme, these “intermediate fusion” methods involve obtaining a low column rank matrix by optimizing an objective function and then using the columns of the matrix for clustering. However, the theoretical properties of these methods remain largely unexplored. In the absence of statistical guarantees on the objective functions, it is difficult to determine if the algorithms optimizing the objectives will return good community structures. We investigate the consistency properties of the global optimizer of some of these objective functions under the multi-layer stochastic blockmodel. For this purpose, we derive several new asymptotic results showing consistency of the intermediate fusion techniques along with the spectral clustering of mean adjacency matrix under a high dimensional setup, where the number of nodes, the number of layers and the number of communities of the multi-layer graph grow. Our numerical study shows that the intermediate fusion techniques outperform late fusion methods, namely spectral clustering on aggregate spectral kernel and module allegiance matrix in sparse networks, while they outperform the spectral clustering of mean adjacency matrix in multi-layer networks that contain layers with both homophilic and heterophilic communities.

Adityanand Guntuboyina, Donovan Lieu, Sabyasachi Chatterjee, Bodhisattva Sen.

Source: The Annals of Statistics, Volume 48, Number 1, 205--229.

Abstract:
We study trend filtering, a relatively recent method for univariate nonparametric regression. For a given integer $rgeq1$, the $r$th order trend filtering estimator is defined as the minimizer of the sum of squared errors when we constrain (or penalize) the sum of the absolute $r$th order discrete derivatives of the fitted function at the design points. For $r=1$, the estimator reduces to total variation regularization which has received much attention in the statistics and image processing literature. In this paper, we study the performance of the trend filtering estimator for every $rgeq1$, both in the constrained and penalized forms. Our main results show that in the strong sparsity setting when the underlying function is a (discrete) spline with few “knots,” the risk (under the global squared error loss) of the trend filtering estimator (with an appropriate choice of the tuning parameter) achieves the parametric $n^{-1}$-rate, up to a logarithmic (multiplicative) factor. Our results therefore provide support for the use of trend filtering, for every $rgeq1$, in the strong sparsity setting.

Min Xu, Varun Jog, Po-Ling Loh.

Source: The Annals of Statistics, Volume 48, Number 1, 183--204.

Abstract:
Community identification in a network is an important problem in fields such as social science, neuroscience and genetics. Over the past decade, stochastic block models (SBMs) have emerged as a popular statistical framework for this problem. However, SBMs have an important limitation in that they are suited only for networks with unweighted edges; in various scientific applications, disregarding the edge weights may result in a loss of valuable information. We study a weighted generalization of the SBM, in which observations are collected in the form of a weighted adjacency matrix and the weight of each edge is generated independently from an unknown probability density determined by the community membership of its endpoints. We characterize the optimal rate of misclustering error of the weighted SBM in terms of the Renyi divergence of order 1/2 between the weight distributions of within-community and between-community edges, substantially generalizing existing results for unweighted SBMs. Furthermore, we present a computationally tractable algorithm based on discretization that achieves the optimal error rate. Our method is adaptive in the sense that the algorithm, without assuming knowledge of the weight densities, performs as well as the best algorithm that knows the weight densities.

Florentina Bunea, Christophe Giraud, Xi Luo, Martin Royer, Nicolas Verzelen.

Source: The Annals of Statistics, Volume 48, Number 1, 111--137.

Abstract:
The problem of variable clustering is that of estimating groups of similar components of a $p$-dimensional vector $X=(X_{1},ldots ,X_{p})$ from $n$ independent copies of $X$. There exists a large number of algorithms that return data-dependent groups of variables, but their interpretation is limited to the algorithm that produced them. An alternative is model-based clustering, in which one begins by defining population level clusters relative to a model that embeds notions of similarity. Algorithms tailored to such models yield estimated clusters with a clear statistical interpretation. We take this view here and introduce the class of $G$-block covariance models as a background model for variable clustering. In such models, two variables in a cluster are deemed similar if they have similar associations will all other variables. This can arise, for instance, when groups of variables are noise corrupted versions of the same latent factor. We quantify the difficulty of clustering data generated from a $G$-block covariance model in terms of cluster proximity, measured with respect to two related, but different, cluster separation metrics. We derive minimax cluster separation thresholds, which are the metric values below which no algorithm can recover the model-defined clusters exactly, and show that they are different for the two metrics. We therefore develop two algorithms, COD and PECOK, tailored to $G$-block covariance models, and study their minimax-optimality with respect to each metric. Of independent interest is the fact that the analysis of the PECOK algorithm, which is based on a corrected convex relaxation of the popular $K$-means algorithm, provides the first statistical analysis of such algorithms for variable clustering. Additionally, we compare our methods with another popular clustering method, spectral clustering. Extensive simulation studies, as well as our data analyses, confirm the applicability of our approach.

Holger Dette, Weichi Wu.

Source: The Annals of Statistics, Volume 47, Number 6, 3578--3608.

Abstract:
This paper considers the problem of testing if a sequence of means $(mu_{t})_{t=1,ldots ,n}$ of a nonstationary time series $(X_{t})_{t=1,ldots ,n}$ is stable in the sense that the difference of the means $mu_{1}$ and $mu_{t}$ between the initial time $t=1$ and any other time is smaller than a given threshold, that is $|mu_{1}-mu_{t}|leq c$ for all $t=1,ldots ,n$. A test for hypotheses of this type is developed using a bias corrected monotone rearranged local linear estimator and asymptotic normality of the corresponding test statistic is established. As the asymptotic variance depends on the location of the roots of the equation $|mu_{1}-mu_{t}|=c$ a new bootstrap procedure is proposed to obtain critical values and its consistency is established. As a consequence we are able to quantitatively describe relevant deviations of a nonstationary sequence from its initial value. The results are illustrated by means of a simulation study and by analyzing data examples.

Kyoungjae Lee, Jaeyong Lee, Lizhen Lin.

Source: The Annals of Statistics, Volume 47, Number 6, 3413--3437.

Abstract:
In this paper we study the high-dimensional sparse directed acyclic graph (DAG) models under the empirical sparse Cholesky prior. Among our results, strong model selection consistency or graph selection consistency is obtained under more general conditions than those in the existing literature. Compared to Cao, Khare and Ghosh [ Ann. Statist. (2019) 47 319–348], the required conditions are weakened in terms of the dimensionality, sparsity and lower bound of the nonzero elements in the Cholesky factor. Furthermore, our result does not require the irrepresentable condition, which is necessary for Lasso-type methods. We also derive the posterior convergence rates for precision matrices and Cholesky factors with respect to various matrix norms. The obtained posterior convergence rates are the fastest among those of the existing Bayesian approaches. In particular, we prove that our posterior convergence rates for Cholesky factors are the minimax or at least nearly minimax depending on the relative size of true sparseness for the entire dimension. The simulation study confirms that the proposed method outperforms the competing methods.

Yi Lin, Ryan Martin, Min Yang.

Source: The Annals of Statistics, Volume 47, Number 6, 3335--3359.

Abstract:
Classically, Fisher information is the relevant object in defining optimal experimental designs. However, for models that lack certain regularity, the Fisher information does not exist, and hence, there is no notion of design optimality available in the literature. This article seeks to fill the gap by proposing a so-called Hellinger information , which generalizes Fisher information in the sense that the two measures agree in regular problems, but the former also exists for certain types of nonregular problems. We derive a Hellinger information inequality, showing that Hellinger information defines a lower bound on the local minimax risk of estimators. This provides a connection between features of the underlying model—in particular, the design—and the performance of estimators, motivating the use of this new Hellinger information for nonregular optimal design problems. Hellinger optimal designs are derived for several nonregular regression problems, with numerical results empirically demonstrating the efficiency of these designs compared to alternatives.

Xi Chen, Weidong Liu, Yichen Zhang.

Source: The Annals of Statistics, Volume 47, Number 6, 3244--3273.

Abstract:
This paper studies the inference problem in quantile regression (QR) for a large sample size $n$ but under a limited memory constraint, where the memory can only store a small batch of data of size $m$. A natural method is the naive divide-and-conquer approach, which splits data into batches of size $m$, computes the local QR estimator for each batch and then aggregates the estimators via averaging. However, this method only works when $n=o(m^{2})$ and is computationally expensive. This paper proposes a computationally efficient method, which only requires an initial QR estimator on a small batch of data and then successively refines the estimator via multiple rounds of aggregations. Theoretically, as long as $n$ grows polynomially in $m$, we establish the asymptotic normality for the obtained estimator and show that our estimator with only a few rounds of aggregations achieves the same efficiency as the QR estimator computed on all the data. Moreover, our result allows the case that the dimensionality $p$ goes to infinity. The proposed method can also be applied to address the QR problem under distributed computing environment (e.g., in a large-scale sensor network) or for real-time streaming data.

Ke Zhu.

Source: The Annals of Statistics, Volume 47, Number 6, 3185--3215.

Abstract:
This paper provides an entire inference procedure for the autoregressive model under (conditional) heteroscedasticity of unknown form with a finite variance. We first establish the asymptotic normality of the weighted least absolute deviations estimator (LADE) for the model. Second, we develop the random weighting (RW) method to estimate its asymptotic covariance matrix, leading to the implementation of the Wald test. Third, we construct a portmanteau test for model checking, and use the RW method to obtain its critical values. As a special weighted LADE, the feasible adaptive LADE (ALADE) is proposed and proved to have the same efficiency as its infeasible counterpart. The importance of our entire methodology based on the feasible ALADE is illustrated by simulation results and the real data analysis on three U.S. economic data sets.

Xin Bing, Marten H. Wegkamp.

Source: The Annals of Statistics, Volume 47, Number 6, 3157--3184.

Abstract:
We consider the multivariate response regression problem with a regression coefficient matrix of low, unknown rank. In this setting, we analyze a new criterion for selecting the optimal reduced rank. This criterion differs notably from the one proposed in Bunea, She and Wegkamp ( Ann. Statist. 39 (2011) 1282–1309) in that it does not require estimation of the unknown variance of the noise, nor does it depend on a delicate choice of a tuning parameter. We develop an iterative, fully data-driven procedure, that adapts to the optimal signal-to-noise ratio. This procedure finds the true rank in a few steps with overwhelming probability. At each step, our estimate increases, while at the same time it does not exceed the true rank. Our finite sample results hold for any sample size and any dimension, even when the number of responses and of covariates grow much faster than the number of observations. We perform an extensive simulation study that confirms our theoretical findings. The new method performs better and is more stable than the procedure of Bunea, She and Wegkamp ( Ann. Statist. 39 (2011) 1282–1309) in both low- and high-dimensional settings.

Veeranjaneyulu Sadhanala, Ryan J. Tibshirani.

Source: The Annals of Statistics, Volume 47, Number 6, 3032--3068.

Abstract:
We study additive models built with trend filtering, that is, additive models whose components are each regularized by the (discrete) total variation of their $k$th (discrete) derivative, for a chosen integer $kgeq0$. This results in $k$th degree piecewise polynomial components, (e.g., $k=0$ gives piecewise constant components, $k=1$ gives piecewise linear, $k=2$ gives piecewise quadratic, etc.). Analogous to its advantages in the univariate case, additive trend filtering has favorable theoretical and computational properties, thanks in large part to the localized nature of the (discrete) total variation regularizer that it uses. On the theory side, we derive fast error rates for additive trend filtering estimates, and show these rates are minimax optimal when the underlying function is additive and has component functions whose derivatives are of bounded variation. We also show that these rates are unattainable by additive smoothing splines (and by additive models built from linear smoothers, in general). On the computational side, we use backfitting, to leverage fast univariate trend filtering solvers; we also describe a new backfitting algorithm whose iterations can be run in parallel, which (as far as we can tell) is the first of its kind. Lastly, we present a number of experiments to examine the empirical performance of trend filtering.

Taras Bodnar, Holger Dette, Nestor Parolya.

Source: The Annals of Statistics, Volume 47, Number 5, 2977--3008.

Abstract:
In this paper, new tests for the independence of two high-dimensional vectors are investigated. We consider the case where the dimension of the vectors increases with the sample size and propose multivariate analysis of variance-type statistics for the hypothesis of a block diagonal covariance matrix. The asymptotic properties of the new test statistics are investigated under the null hypothesis and the alternative hypothesis using random matrix theory. For this purpose, we study the weak convergence of linear spectral statistics of central and (conditionally) noncentral Fisher matrices. In particular, a central limit theorem for linear spectral statistics of large dimensional (conditionally) noncentral Fisher matrices is derived which is then used to analyse the power of the tests under the alternative. The theoretical results are illustrated by means of a simulation study where we also compare the new tests with several alternative, in particular with the commonly used corrected likelihood ratio test. It is demonstrated that the latter test does not keep its nominal level, if the dimension of one sub-vector is relatively small compared to the dimension of the other sub-vector. On the other hand, the tests proposed in this paper provide a reasonable approximation of the nominal level in such situations. Moreover, we observe that one of the proposed tests is most powerful under a variety of correlation scenarios.

Charles R. Doss, Jon A. Wellner.

Source: The Annals of Statistics, Volume 47, Number 5, 2950--2976.

Abstract:
We study a likelihood ratio test for the location of the mode of a log-concave density. Our test is based on comparison of the log-likelihoods corresponding to the unconstrained maximum likelihood estimator of a log-concave density and the constrained maximum likelihood estimator where the constraint is that the mode of the density is fixed, say at $m$. The constrained estimation problem is studied in detail in Doss and Wellner (2018). Here, the results of that paper are used to show that, under the null hypothesis (and strict curvature of $-log f$ at the mode), the likelihood ratio statistic is asymptotically pivotal: that is, it converges in distribution to a limiting distribution which is free of nuisance parameters, thus playing the role of the $chi_{1}^{2}$ distribution in classical parametric statistical problems. By inverting this family of tests, we obtain new (likelihood ratio based) confidence intervals for the mode of a log-concave density $f$. These new intervals do not depend on any smoothing parameters. We study the new confidence intervals via Monte Carlo methods and illustrate them with two real data sets. The new intervals seem to have several advantages over existing procedures. Software implementing the test and confidence intervals is available in the R package verb+logcondens.mode+.

Heng Lian, Kaifeng Zhao, Shaogao Lv.

Source: The Annals of Statistics, Volume 47, Number 5, 2922--2949.

Abstract:
In this paper, we consider the local asymptotics of the nonparametric function in a partially linear model, within the framework of the divide-and-conquer estimation. Unlike the fixed-dimensional setting in which the parametric part does not affect the nonparametric part, the high-dimensional setting makes the issue more complicated. In particular, when a sparsity-inducing penalty such as lasso is used to make the estimation of the linear part feasible, the bias introduced will propagate to the nonparametric part. We propose a novel approach for estimation of the nonparametric function and establish the local asymptotics of the estimator. The result is useful for massive data with possibly different linear coefficients in each subpopulation but common nonparametric function. Some numerical illustrations are also presented.

Zhou Fan, Iain M. Johnstone.

Source: The Annals of Statistics, Volume 47, Number 5, 2855--2886.

Abstract:
We study the spectra of MANOVA estimators for variance component covariance matrices in multivariate random effects models. When the dimensionality of the observations is large and comparable to the number of realizations of each random effect, we show that the empirical spectra of such estimators are well approximated by deterministic laws. The Stieltjes transforms of these laws are characterized by systems of fixed-point equations, which are numerically solvable by a simple iterative procedure. Our proof uses operator-valued free probability theory, and we establish a general asymptotic freeness result for families of rectangular orthogonally invariant random matrices, which is of independent interest. Our work is motivated in part by the estimation of components of covariance between multiple phenotypic traits in quantitative genetics, and we specialize our results to common experimental designs that arise in this application.

Aryeh Kontorovich, Iosif Pinelis.

Source: The Annals of Statistics, Volume 47, Number 5, 2822--2854.

Abstract:
We provide an exact nonasymptotic lower bound on the minimax expected excess risk (EER) in the agnostic probably-approximately-correct (PAC) machine learning classification model and identify minimax learning algorithms as certain maximally symmetric and minimally randomized “voting” procedures. Based on this result, an exact asymptotic lower bound on the minimax EER is provided. This bound is of the simple form $c_{infty}/sqrt{ u}$ as $ u oinfty$, where $c_{infty}=0.16997dots$ is a universal constant, $ u=m/d$, $m$ is the size of the training sample and $d$ is the Vapnik–Chervonenkis dimension of the hypothesis class. It is shown that the differences between these asymptotic and nonasymptotic bounds, as well as the differences between these two bounds and the maximum EER of any learning algorithms that minimize the empirical risk, are asymptotically negligible, and all these differences are due to ties in the mentioned “voting” procedures. A few easy to compute nonasymptotic lower bounds on the minimax EER are also obtained, which are shown to be close to the exact asymptotic lower bound $c_{infty}/sqrt{ u}$ even for rather small values of the ratio $ u=m/d$. As an application of these results, we substantially improve existing lower bounds on the tail probability of the excess risk. Among the tools used are Bayes estimation and apparently new identities and inequalities for binomial distributions.

Björn Böttcher, Martin Keller-Ressel, René L. Schilling.

Source: The Annals of Statistics, Volume 47, Number 5, 2757--2789.

Abstract:
We introduce two new measures for the dependence of $nge2$ random variables: distance multivariance and total distance multivariance . Both measures are based on the weighted $L^{2}$-distance of quantities related to the characteristic functions of the underlying random variables. These extend distance covariance (introduced by Székely, Rizzo and Bakirov) from pairs of random variables to $n$-tuplets of random variables. We show that total distance multivariance can be used to detect the independence of $n$ random variables and has a simple finite-sample representation in terms of distance matrices of the sample points, where distance is measured by a continuous negative definite function. Under some mild moment conditions, this leads to a test for independence of multiple random vectors which is consistent against all alternatives.

Wei-Kuo Chen.

Source: The Annals of Statistics, Volume 47, Number 5, 2734--2756.

Abstract:
We consider the problem of detecting a deformation from a symmetric Gaussian random $p$-tensor $(pgeq3)$ with a rank-one spike sampled from the Rademacher prior. Recently, in Lesieur et al. (Barbier, Krzakala, Macris, Miolane and Zdeborová (2017)), it was proved that there exists a critical threshold $eta_{p}$ so that when the signal-to-noise ratio exceeds $eta_{p}$, one can distinguish the spiked and unspiked tensors and weakly recover the prior via the minimal mean-square-error method. On the other side, Perry, Wein and Bandeira (Perry, Wein and Bandeira (2017)) proved that there exists a $eta_{p}'<eta_{p}$ such that any statistical hypothesis test cannot distinguish these two tensors, in the sense that their total variation distance asymptotically vanishes, when the signa-to-noise ratio is less than $eta_{p}'$. In this work, we show that $eta_{p}$ is indeed the critical threshold that strictly separates the distinguishability and indistinguishability between the two tensors under the total variation distance. Our approach is based on a subtle analysis of the high temperature behavior of the pure $p$-spin model with Ising spin, arising initially from the field of spin glasses. In particular, we identify the signal-to-noise criticality $eta_{p}$ as the critical temperature, distinguishing the high and low temperature behavior, of the Ising pure $p$-spin mean-field spin glass model.

Robert A. Vandermeulen, Clayton D. Scott.

Source: The Annals of Statistics, Volume 47, Number 5, 2704--2733.

Abstract:
When estimating finite mixture models, it is common to make assumptions on the mixture components, such as parametric assumptions. In this work, we make no distributional assumptions on the mixture components and instead assume that observations from the mixture model are grouped, such that observations in the same group are known to be drawn from the same mixture component. We precisely characterize the number of observations $n$ per group needed for the mixture model to be identifiable, as a function of the number $m$ of mixture components. In addition to our assumption-free analysis, we also study the settings where the mixture components are either linearly independent or jointly irreducible. Furthermore, our analysis considers two kinds of identifiability, where the mixture model is the simplest one explaining the data, and where it is the only one. As an application of these results, we precisely characterize identifiability of multinomial mixture models. Our analysis relies on an operator-theoretic framework that associates mixture models in the grouped-sample setting with certain infinite-dimensional tensors. Based on this framework, we introduce a general spectral algorithm for recovering the mixture components.

Chengchun Shi, Rui Song, Zhao Chen, Runze Li.

Source: The Annals of Statistics, Volume 47, Number 5, 2671--2703.

Abstract:
This paper is concerned with testing linear hypotheses in high dimensional generalized linear models. To deal with linear hypotheses, we first propose the constrained partial regularization method and study its statistical properties. We further introduce an algorithm for solving regularization problems with folded-concave penalty functions and linear constraints. To test linear hypotheses, we propose a partial penalized likelihood ratio test, a partial penalized score test and a partial penalized Wald test. We show that the limiting null distributions of these three test statistics are $chi^{2}$ distribution with the same degrees of freedom, and under local alternatives, they asymptotically follow noncentral $chi^{2}$ distributions with the same degrees of freedom and noncentral parameter, provided the number of parameters involved in the test hypothesis grows to $infty$ at a certain rate. Simulation studies are conducted to examine the finite sample performance of the proposed tests. Empirical analysis of a real data example is used to illustrate the proposed testing procedures.

Matey Neykov, Han Liu.

Source: The Annals of Statistics, Volume 47, Number 5, 2472--2503.

Abstract:
This paper explores the information-theoretic limitations of graph property testing in zero-field Ising models. Instead of learning the entire graph structure, sometimes testing a basic graph property such as connectivity, cycle presence or maximum clique size is a more relevant and attainable objective. Since property testing is more fundamental than graph recovery, any necessary conditions for property testing imply corresponding conditions for graph recovery, while custom property tests can be statistically and/or computationally more efficient than graph recovery based algorithms. Understanding the statistical complexity of property testing requires the distinction of ferromagnetic (i.e., positive interactions only) and general Ising models. Using combinatorial constructs such as graph packing and strong monotonicity, we characterize how target properties affect the corresponding minimax upper and lower bounds within the realm of ferromagnets. On the other hand, by studying the detection of an antiferromagnetic (i.e., negative interactions only) Curie–Weiss model buried in Rademacher noise, we show that property testing is strictly more challenging over general Ising models. In terms of methodological development, we propose two types of correlation based tests: computationally efficient screening for ferromagnets, and score type tests for general models, including a fast cycle presence test. Our correlation screening tests match the information-theoretic bounds for property testing in ferromagnets in certain regimes.