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.

Emmanuel J. Candès, Pragya Sur.

Source: The Annals of Statistics, Volume 48, Number 1, 27--42.

Abstract:
This paper rigorously establishes that the existence of the maximum likelihood estimate (MLE) in high-dimensional logistic regression models with Gaussian covariates undergoes a sharp “phase transition.” We introduce an explicit boundary curve $h_{mathrm{MLE}}$, parameterized by two scalars measuring the overall magnitude of the unknown sequence of regression coefficients, with the following property: in the limit of large sample sizes $n$ and number of features $p$ proportioned in such a way that $p/n ightarrow kappa $, we show that if the problem is sufficiently high dimensional in the sense that $kappa >h_{mathrm{MLE}}$, then the MLE does not exist with probability one. Conversely, if $kappa <h_{mathrm{MLE}}$, the MLE asymptotically exists with probability one.

Alessandro Rinaldo, Larry Wasserman, Max G’Sell.

Source: The Annals of Statistics, Volume 47, Number 6, 3438--3469.

Abstract:
Several new methods have been recently proposed for performing valid inference after model selection. An older method is sample splitting: use part of the data for model selection and the rest for inference. In this paper, we revisit sample splitting combined with the bootstrap (or the Normal approximation). We show that this leads to a simple, assumption-lean approach to inference and we establish results on the accuracy of the method. In fact, we find new bounds on the accuracy of the bootstrap and the Normal approximation for general nonlinear parameters with increasing dimension which we then use to assess the accuracy of regression inference. We define new parameters that measure variable importance and that can be inferred with greater accuracy than the usual regression coefficients. Finally, we elucidate an inference-prediction trade-off: splitting increases the accuracy and robustness of inference but can decrease the accuracy of the predictions.

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.

Zeng Li, Clifford Lam, Jianfeng Yao, Qiwei Yao.

Source: The Annals of Statistics, Volume 47, Number 6, 3382--3412.

Abstract:
Testing for white noise is a classical yet important problem in statistics, especially for diagnostic checks in time series modeling and linear regression. For high-dimensional time series in the sense that the dimension $p$ is large in relation to the sample size $T$, the popular omnibus tests including the multivariate Hosking and Li–McLeod tests are extremely conservative, leading to substantial power loss. To develop more relevant tests for high-dimensional cases, we propose a portmanteau-type test statistic which is the sum of squared singular values of the first $q$ lagged sample autocovariance matrices. It, therefore, encapsulates all the serial correlations (up to the time lag $q$) within and across all component series. Using the tools from random matrix theory and assuming both $p$ and $T$ diverge to infinity, we derive the asymptotic normality of the test statistic under both the null and a specific VMA(1) alternative hypothesis. As the actual implementation of the test requires the knowledge of three characteristic constants of the population cross-sectional covariance matrix and the value of the fourth moment of the standardized innovations, nontrivial estimations are proposed for these parameters and their integration leads to a practically usable test. Extensive simulation confirms the excellent finite-sample performance of the new test with accurate size and satisfactory power for a large range of finite $(p,T)$ combinations, therefore, ensuring wide applicability in practice. In particular, the new tests are consistently superior to the traditional Hosking and Li–McLeod tests.

Benjamin Eltzner, Stephan F. Huckemann.

Source: The Annals of Statistics, Volume 47, Number 6, 3360--3381.

Abstract:
The (CLT) central limit theorems for generalized Fréchet means (data descriptors assuming values in manifolds, such as intrinsic means, geodesics, etc.) on manifolds from the literature are only valid if a certain empirical process of Hessians of the Fréchet function converges suitably, as in the proof of the prototypical BP-CLT [ Ann. Statist. 33 (2005) 1225–1259]. This is not valid in many realistic scenarios and we provide for a new very general CLT. In particular, this includes scenarios where, in a suitable chart, the sample mean fluctuates asymptotically at a scale $n^{alpha }$ with exponents $alpha <1/2$ with a nonnormal distribution. As the BP-CLT yields only fluctuations that are, rescaled with $n^{1/2}$, asymptotically normal, just as the classical CLT for random vectors, these lower rates, somewhat loosely called smeariness, had to date been observed only on the circle. We make the concept of smeariness on manifolds precise, give an example for two-smeariness on spheres of arbitrary dimension, and show that smeariness, although “almost never” occurring, may have serious statistical implications on a continuum of sample scenarios nearby. In fact, this effect increases with dimension, striking in particular in high dimension low sample size scenarios.

Shurong Zheng, Zhao Chen, Hengjian Cui, Runze Li.

Source: The Annals of Statistics, Volume 47, Number 6, 3300--3334.

Abstract:
This paper is concerned with test of significance on high-dimensional covariance structures, and aims to develop a unified framework for testing commonly used linear covariance structures. We first construct a consistent estimator for parameters involved in the linear covariance structure, and then develop two tests for the linear covariance structures based on entropy loss and quadratic loss used for covariance matrix estimation. To study the asymptotic properties of the proposed tests, we study related high-dimensional random matrix theory, and establish several highly useful asymptotic results. With the aid of these asymptotic results, we derive the limiting distributions of these two tests under the null and alternative hypotheses. We further show that the quadratic loss based test is asymptotically unbiased. We conduct Monte Carlo simulation study to examine the finite sample performance of the two tests. Our simulation results show that the limiting null distributions approximate their null distributions quite well, and the corresponding asymptotic critical values keep Type I error rate very well. Our numerical comparison implies that the proposed tests outperform existing ones in terms of controlling Type I error rate and power. Our simulation indicates that the test based on quadratic loss seems to have better power than the test based on entropy loss.

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.

Xiaohui Chen, Kengo Kato.

Source: The Annals of Statistics, Volume 47, Number 6, 3127--3156.

Abstract:
This paper studies inference for the mean vector of a high-dimensional $U$-statistic. In the era of big data, the dimension $d$ of the $U$-statistic and the sample size $n$ of the observations tend to be both large, and the computation of the $U$-statistic is prohibitively demanding. Data-dependent inferential procedures such as the empirical bootstrap for $U$-statistics is even more computationally expensive. To overcome such a computational bottleneck, incomplete $U$-statistics obtained by sampling fewer terms of the $U$-statistic are attractive alternatives. In this paper, we introduce randomized incomplete $U$-statistics with sparse weights whose computational cost can be made independent of the order of the $U$-statistic. We derive nonasymptotic Gaussian approximation error bounds for the randomized incomplete $U$-statistics in high dimensions, namely in cases where the dimension $d$ is possibly much larger than the sample size $n$, for both nondegenerate and degenerate kernels. In addition, we propose generic bootstrap methods for the incomplete $U$-statistics that are computationally much less demanding than existing bootstrap methods, and establish finite sample validity of the proposed bootstrap methods. Our methods are illustrated on the application to nonparametric testing for the pairwise independence of a high-dimensional random vector under weaker assumptions than those appearing in the literature.

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.

Shurong Zheng, Guanghui Cheng, Jianhua Guo, Hongtu Zhu.

Source: The Annals of Statistics, Volume 47, Number 5, 2887--2921.

Abstract:
Testing correlation structures has attracted extensive attention in the literature due to both its importance in real applications and several major theoretical challenges. The aim of this paper is to develop a general framework of testing correlation structures for the one , two and multiple sample testing problems under a high-dimensional setting when both the sample size and data dimension go to infinity. Our test statistics are designed to deal with both the dense and sparse alternatives. We systematically investigate the asymptotic null distribution, power function and unbiasedness of each test statistic. Theoretically, we make great efforts to deal with the nonindependency of all random matrices of the sample correlation matrices. We use simulation studies and real data analysis to illustrate the versatility and practicability of our test statistics.

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.

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.

Zhiqiang Tan, Cun-Hui Zhang.

Source: The Annals of Statistics, Volume 47, Number 5, 2567--2600.

Abstract:
Additive regression provides an extension of linear regression by modeling the signal of a response as a sum of functions of covariates of relatively low complexity. We study penalized estimation in high-dimensional nonparametric additive regression where functional semi-norms are used to induce smoothness of component functions and the empirical $L_{2}$ norm is used to induce sparsity. The functional semi-norms can be of Sobolev or bounded variation types and are allowed to be different amongst individual component functions. We establish oracle inequalities for the predictive performance of such methods under three simple technical conditions: a sub-Gaussian condition on the noise, a compatibility condition on the design and the functional classes under consideration and an entropy condition on the functional classes. For random designs, the sample compatibility condition can be replaced by its population version under an additional condition to ensure suitable convergence of empirical norms. In homogeneous settings where the complexities of the component functions are of the same order, our results provide a spectrum of minimax convergence rates, from the so-called slow rate without requiring the compatibility condition to the fast rate under the hard sparsity or certain $L_{q}$ sparsity to allow many small components in the true regression function. These results significantly broaden and sharpen existing ones in the literature.

Rina Foygel Barber, Emmanuel J. Candès.

Source: The Annals of Statistics, Volume 47, Number 5, 2504--2537.

Abstract:
This paper develops a framework for testing for associations in a possibly high-dimensional linear model where the number of features/variables may far exceed the number of observational units. In this framework, the observations are split into two groups, where the first group is used to screen for a set of potentially relevant variables, whereas the second is used for inference over this reduced set of variables; we also develop strategies for leveraging information from the first part of the data at the inference step for greater power. In our work, the inferential step is carried out by applying the recently introduced knockoff filter, which creates a knockoff copy—a fake variable serving as a control—for each screened variable. We prove that this procedure controls the directional false discovery rate (FDR) in the reduced model controlling for all screened variables; this says that our high-dimensional knockoff procedure “discovers” important variables as well as the directions (signs) of their effects, in such a way that the expected proportion of wrongly chosen signs is below the user-specified level (thereby controlling a notion of Type S error averaged over the selected set). This result is nonasymptotic, and holds for any distribution of the original features and any values of the unknown regression coefficients, so that inference is not calibrated under hypothesized values of the effect sizes. We demonstrate the performance of our general and flexible approach through numerical studies, showing more power than existing alternatives. Finally, we apply our method to a genome-wide association study to find locations on the genome that are possibly associated with a continuous phenotype.

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.

Joshua Cape, Minh Tang, Carey E. Priebe.

Source: The Annals of Statistics, Volume 47, Number 5, 2405--2439.

Abstract:
The singular value matrix decomposition plays a ubiquitous role throughout statistics and related fields. Myriad applications including clustering, classification, and dimensionality reduction involve studying and exploiting the geometric structure of singular values and singular vectors. This paper provides a novel collection of technical and theoretical tools for studying the geometry of singular subspaces using the two-to-infinity norm. Motivated by preliminary deterministic Procrustes analysis, we consider a general matrix perturbation setting in which we derive a new Procrustean matrix decomposition. Together with flexible machinery developed for the two-to-infinity norm, this allows us to conduct a refined analysis of the induced perturbation geometry with respect to the underlying singular vectors even in the presence of singular value multiplicity. Our analysis yields singular vector entrywise perturbation bounds for a range of popular matrix noise models, each of which has a meaningful associated statistical inference task. In addition, we demonstrate how the two-to-infinity norm is the preferred norm in certain statistical settings. Specific applications discussed in this paper include covariance estimation, singular subspace recovery, and multiple graph inference. Both our Procrustean matrix decomposition and the technical machinery developed for the two-to-infinity norm may be of independent interest.

Hong Zhang, Tiejun Tong, John Landers, Zheyang Wu.

Source: The Annals of Applied Statistics, Volume 14, Number 1, 178--201.

Abstract:
The $p$-value combination approach is an important statistical strategy for testing global hypotheses with broad applications in signal detection, meta-analysis, data integration, etc. In this paper we extend the classic Fisher’s combination method to a unified family of statistics, called TFisher, which allows a general truncation-and-weighting scheme of input $p$-values. TFisher can significantly improve statistical power over the Fisher and related truncation-only methods for detecting both rare and dense “signals.” To address wide applications, analytical calculations for TFisher’s size and power are deduced under any two continuous distributions in the null and the alternative hypotheses. The corresponding omnibus test (oTFisher) and its size calculation are also provided for data-adaptive analysis. We study the asymptotic optimal parameters of truncation and weighting based on Bahadur efficiency (BE). A new asymptotic measure, called the asymptotic power efficiency (APE), is also proposed for better reflecting the statistics’ performance in real data analysis. Interestingly, under the Gaussian mixture model in the signal detection problem, both BE and APE indicate that the soft-thresholding scheme is the best, the truncation and weighting parameters should be equal. By simulations of various signal patterns, we systematically compare the power of statistics within TFisher family as well as some rare-signal-optimal tests. We illustrate the use of TFisher in an exome-sequencing analysis for detecting novel genes of amyotrophic lateral sclerosis. Relevant computation has been implemented into an R package TFisher published on the Comprehensive R Archive Network to cater for applications.

Seth Flaxman, Michael Chirico, Pau Pereira, Charles Loeffler.

Source: The Annals of Applied Statistics, Volume 13, Number 4, 2564--2585.

Abstract:
We propose a generic spatiotemporal event forecasting method which we developed for the National Institute of Justice’s (NIJ) Real-Time Crime Forecasting Challenge (National Institute of Justice (2017)). Our method is a spatiotemporal forecasting model combining scalable randomized Reproducing Kernel Hilbert Space (RKHS) methods for approximating Gaussian processes with autoregressive smoothing kernels in a regularized supervised learning framework. While the smoothing kernels capture the two main approaches in current use in the field of crime forecasting, kernel density estimation (KDE) and self-exciting point process (SEPP) models, the RKHS component of the model can be understood as an approximation to the popular log-Gaussian Cox Process model. For inference, we discretize the spatiotemporal point pattern and learn a log-intensity function using the Poisson likelihood and highly efficient gradient-based optimization methods. Model hyperparameters including quality of RKHS approximation, spatial and temporal kernel lengthscales, number of autoregressive lags and bandwidths for smoothing kernels as well as cell shape, size and rotation, were learned using cross validation. Resulting predictions significantly exceeded baseline KDE estimates and SEPP models for sparse events.

Fan Li, Fan Li.

Source: The Annals of Applied Statistics, Volume 13, Number 4, 2389--2415.

Abstract:
Causal or unconfounded descriptive comparisons between multiple groups are common in observational studies. Motivated from a racial disparity study in health services research, we propose a unified propensity score weighting framework, the balancing weights, for estimating causal effects with multiple treatments. These weights incorporate the generalized propensity scores to balance the weighted covariate distribution of each treatment group, all weighted toward a common prespecified target population. The class of balancing weights include several existing approaches such as the inverse probability weights and trimming weights as special cases. Within this framework, we propose a set of target estimands based on linear contrasts. We further develop the generalized overlap weights, constructed as the product of the inverse probability weights and the harmonic mean of the generalized propensity scores. The generalized overlap weighting scheme corresponds to the target population with the most overlap in covariates across the multiple treatments. These weights are bounded and thus bypass the problem of extreme propensities. We show that the generalized overlap weights minimize the total asymptotic variance of the moment weighting estimators for the pairwise contrasts within the class of balancing weights. We consider two balance check criteria and propose a new sandwich variance estimator for estimating the causal effects with generalized overlap weights. We apply these methods to study the racial disparities in medical expenditure between several racial groups using the 2009 Medical Expenditure Panel Survey (MEPS) data. Simulations were carried out to compare with existing methods.

Pavel Zorin-Kranich.

Source: Bernoulli, Volume 26, Number 3, 2311--2318.

Abstract:
We prove an estimate for weighted $p$th moments of the pathwise $r$-variation of a martingale in terms of the $A_{p}$ characteristic of the weight. The novelty of the proof is that we avoid real interpolation techniques.

Arun K. Kuchibhotla, Rohit K. Patra.

Source: Bernoulli, Volume 26, Number 2, 1587--1618.

Abstract:
We consider estimation and inference in a single index regression model with an unknown but smooth link function. In contrast to the standard approach of using kernels or regression splines, we use smoothing splines to estimate the smooth link function. We develop a method to compute the penalized least squares estimators (PLSEs) of the parametric and the nonparametric components given independent and identically distributed (i.i.d.) data. We prove the consistency and find the rates of convergence of the estimators. We establish asymptotic normality under mild assumption and prove asymptotic efficiency of the parametric component under homoscedastic errors. A finite sample simulation corroborates our asymptotic theory. We also analyze a car mileage data set and a Ozone concentration data set. The identifiability and existence of the PLSEs are also investigated.

Binyan Jiang, Ziqi Chen, Chenlei Leng.

Source: Bernoulli, Volume 26, Number 2, 1234--1268.

Abstract:
High-dimensional data that evolve dynamically feature predominantly in the modern data era. As a partial response to this, recent years have seen increasing emphasis to address the dimensionality challenge. However, the non-static nature of these datasets is largely ignored. This paper addresses both challenges by proposing a novel yet simple dynamic linear programming discriminant (DLPD) rule for binary classification. Different from the usual static linear discriminant analysis, the new method is able to capture the changing distributions of the underlying populations by modeling their means and covariances as smooth functions of covariates of interest. Under an approximate sparse condition, we show that the conditional misclassification rate of the DLPD rule converges to the Bayes risk in probability uniformly over the range of the variables used for modeling the dynamics, when the dimensionality is allowed to grow exponentially with the sample size. The minimax lower bound of the estimation of the Bayes risk is also established, implying that the misclassification rate of our proposed rule is minimax-rate optimal. The promising performance of the DLPD rule is illustrated via extensive simulation studies and the analysis of a breast cancer dataset.

Giacomo Aletti, Irene Crimaldi, Andrea Ghiglietti.

Source: Bernoulli, Volume 26, Number 2, 1098--1138.

Abstract:
This work deals with a system of interacting reinforced stochastic processes , where each process $X^{j}=(X_{n,j})_{n}$ is located at a vertex $j$ of a finite weighted directed graph, and it can be interpreted as the sequence of “actions” adopted by an agent $j$ of the network. The interaction among the dynamics of these processes depends on the weighted adjacency matrix $W$ associated to the underlying graph: indeed, the probability that an agent $j$ chooses a certain action depends on its personal “inclination” $Z_{n,j}$ and on the inclinations $Z_{n,h}$, with $h eq j$, of the other agents according to the entries of $W$. The best known example of reinforced stochastic process is the Pólya urn. The present paper focuses on the weighted empirical means $N_{n,j}=sum_{k=1}^{n}q_{n,k}X_{k,j}$, since, for example, the current experience is more important than the past one in reinforced learning. Their almost sure synchronization and some central limit theorems in the sense of stable convergence are proven. The new approach with weighted means highlights the key points in proving some recent results for the personal inclinations $Z^{j}=(Z_{n,j})_{n}$ and for the empirical means $overline{X}^{j}=(sum_{k=1}^{n}X_{k,j}/n)_{n}$ given in recent papers (e.g. Aletti, Crimaldi and Ghiglietti (2019), Ann. Appl. Probab. 27 (2017) 3787–3844, Crimaldi et al. Stochastic Process. Appl. 129 (2019) 70–101). In fact, with a more sophisticated decomposition of the considered processes, we can understand how the different convergence rates of the involved stochastic processes combine. From an application point of view, we provide confidence intervals for the common limit inclination of the agents and a test statistics to make inference on the matrix $W$, based on the weighted empirical means. In particular, we answer a research question posed in Aletti, Crimaldi and Ghiglietti (2019).

Keisuke Yano, Kengo Kato.

Source: Bernoulli, Volume 26, Number 1, 616--641.

Abstract:
In this paper, we study frequentist coverage errors of Bayesian credible sets for an approximately linear regression model with (moderately) high dimensional regressors, where the dimension of the regressors may increase with but is smaller than the sample size. Specifically, we consider quasi-Bayesian inference on the slope vector under the quasi-likelihood with Gaussian error distribution. Under this setup, we derive finite sample bounds on frequentist coverage errors of Bayesian credible rectangles. Derivation of those bounds builds on a novel Berry–Esseen type bound on quasi-posterior distributions and recent results on high-dimensional CLT on hyperrectangles. We use this general result to quantify coverage errors of Castillo–Nickl and $L^{infty}$-credible bands for Gaussian white noise models, linear inverse problems, and (possibly non-Gaussian) nonparametric regression models. In particular, we show that Bayesian credible bands for those nonparametric models have coverage errors decaying polynomially fast in the sample size, implying advantages of Bayesian credible bands over confidence bands based on extreme value theory.

Nicolas Privault, Grzegorz Serafin.

Source: Bernoulli, Volume 26, Number 1, 587--615.

Abstract:
We derive normal approximation bounds in the Kolmogorov distance for sums of discrete multiple integrals and weighted $U$-statistics made of independent Bernoulli random variables. Such bounds are applied to normal approximation for the renormalized subgraph counts in the Erdős–Rényi random graph. This approach completely solves a long-standing conjecture in the general setting of arbitrary graph counting, while recovering recent results obtained for triangles and improving other bounds in the Wasserstein distance.

Xiucai Ding.

Source: Bernoulli, Volume 26, Number 1, 387--417.

Abstract:
We consider the recovery of a low rank $M imes N$ matrix $S$ from its noisy observation $ ilde{S}$ in the high dimensional framework when $M$ is comparable to $N$. We propose two efficient estimators for $S$ under two different regimes. Our analysis relies on the local asymptotics of the eigenstructure of large dimensional rectangular matrices with finite rank perturbation. We derive the convergent limits and rates for the singular values and vectors for such matrices.

St. Philip and St. James Church (Noarlunga, S.A.)

Hewish Henry -- Family.

St. Philip and St. James Church (Noarlunga, S.A.)

Urlwin (Family).

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The CEO of Move This World isn't big on screen time, but in the midst of the coronavirus pandemic, technology--when used with care--can help strengthen relationships.

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Boeing CEO Dave Calhoun said the company was aiming to resume production this month, despite the ongoing grounding and coronavirus pandemic.

The ongoing investigation of the fatal shooting in Brunswick, Georgia, will also look at a neighbor of suspects Gregory and Travis McMichael who recorded video of the incident, authorities said.

Xuan Cao, Kshitij Khare, Malay Ghosh.

Source: Bayesian Analysis, Volume 15, Number 1, 241--262.

Abstract:
The choice of tuning parameters in Bayesian variable selection is a critical problem in modern statistics. In particular, for Bayesian linear regression with non-local priors, the scale parameter in the non-local prior density is an important tuning parameter which reflects the dispersion of the non-local prior density around zero, and implicitly determines the size of the regression coefficients that will be shrunk to zero. Current approaches treat the scale parameter as given, and suggest choices based on prior coverage/asymptotic considerations. In this paper, we consider the fully Bayesian approach introduced in (Wu, 2016) with the pMOM non-local prior and an appropriate Inverse-Gamma prior on the tuning parameter to analyze the underlying theoretical property. Under standard regularity assumptions, we establish strong model selection consistency in a high-dimensional setting, where $p$ is allowed to increase at a polynomial rate with $n$ or even at a sub-exponential rate with $n$ . Through simulation studies, we demonstrate that our model selection procedure can outperform other Bayesian methods which treat the scale parameter as given, and commonly used penalized likelihood methods, in a range of simulation settings.

Gemma E. Moran, Veronika Ročková, Edward I. George.

Source: Bayesian Analysis, Volume 14, Number 4, 1091--1119.

Abstract:
Consider the problem of high dimensional variable selection for the Gaussian linear model when the unknown error variance is also of interest. In this paper, we show that the use of conjugate shrinkage priors for Bayesian variable selection can have detrimental consequences for such variance estimation. Such priors are often motivated by the invariance argument of Jeffreys (1961). Revisiting this work, however, we highlight a caveat that Jeffreys himself noticed; namely that biased estimators can result from inducing dependence between parameters a priori . In a similar way, we show that conjugate priors for linear regression, which induce prior dependence, can lead to such underestimation in the Bayesian high-dimensional regression setting. Following Jeffreys, we recommend as a remedy to treat regression coefficients and the error variance as independent a priori . Using such an independence prior framework, we extend the Spike-and-Slab Lasso of Ročková and George (2018) to the unknown variance case. This extended procedure outperforms both the fixed variance approach and alternative penalized likelihood methods on simulated data. On the protein activity dataset of Clyde and Parmigiani (1998), the Spike-and-Slab Lasso with unknown variance achieves lower cross-validation error than alternative penalized likelihood methods, demonstrating the gains in predictive accuracy afforded by simultaneous error variance estimation. The unknown variance implementation of the Spike-and-Slab Lasso is provided in the publicly available R package SSLASSO (Ročková and Moran, 2017).

Joseph Antonelli, Giovanni Parmigiani, Francesca Dominici.

Source: Bayesian Analysis, Volume 14, Number 3, 825--848.

Abstract:
In observational studies, estimation of a causal effect of a treatment on an outcome relies on proper adjustment for confounding. If the number of the potential confounders ( $p$ ) is larger than the number of observations ( $n$ ), then direct control for all potential confounders is infeasible. Existing approaches for dimension reduction and penalization are generally aimed at predicting the outcome, and are less suited for estimation of causal effects. Under standard penalization approaches (e.g. Lasso), if a variable $X_{j}$ is strongly associated with the treatment $T$ but weakly with the outcome $Y$ , the coefficient $eta_{j}$ will be shrunk towards zero thus leading to confounding bias. Under the assumption of a linear model for the outcome and sparsity, we propose continuous spike and slab priors on the regression coefficients $eta_{j}$ corresponding to the potential confounders $X_{j}$ . Specifically, we introduce a prior distribution that does not heavily shrink to zero the coefficients ( $eta_{j}$ s) of the $X_{j}$ s that are strongly associated with $T$ but weakly associated with $Y$ . We compare our proposed approach to several state of the art methods proposed in the literature. Our proposed approach has the following features: 1) it reduces confounding bias in high dimensional settings; 2) it shrinks towards zero coefficients of instrumental variables; and 3) it achieves good coverages even in small sample sizes. We apply our approach to the National Health and Nutrition Examination Survey (NHANES) data to estimate the causal effects of persistent pesticide exposure on triglyceride levels.

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