Eric D. Kolaczyk, Lizhen Lin, Steven Rosenberg, Jackson Walters, Jie Xu.

Source: The Annals of Statistics, Volume 48, Number 1, 514--538.

Abstract:
It is becoming increasingly common to see large collections of network data objects, that is, data sets in which a network is viewed as a fundamental unit of observation. As a result, there is a pressing need to develop network-based analogues of even many of the most basic tools already standard for scalar and vector data. In this paper, our focus is on averages of unlabeled, undirected networks with edge weights. Specifically, we (i) characterize a certain notion of the space of all such networks, (ii) describe key topological and geometric properties of this space relevant to doing probability and statistics thereupon, and (iii) use these properties to establish the asymptotic behavior of a generalized notion of an empirical mean under sampling from a distribution supported on this space. Our results rely on a combination of tools from geometry, probability theory and statistical shape analysis. In particular, the lack of vertex labeling necessitates working with a quotient space modding out permutations of labels. This results in a nontrivial geometry for the space of unlabeled networks, which in turn is found to have important implications on the types of probabilistic and statistical results that may be obtained and the techniques needed to obtain them.

Changliang Zou, Guanghui Wang, Runze Li.

Source: The Annals of Statistics, Volume 48, Number 1, 413--439.

Abstract:
In multiple change-point analysis, one of the major challenges is to estimate the number of change-points. Most existing approaches attempt to minimize a Schwarz information criterion which balances a term quantifying model fit with a penalization term accounting for model complexity that increases with the number of change-points and limits overfitting. However, different penalization terms are required to adapt to different contexts of multiple change-point problems and the optimal penalization magnitude usually varies from the model and error distribution. We propose a data-driven selection criterion that is applicable to most kinds of popular change-point detection methods, including binary segmentation and optimal partitioning algorithms. The key idea is to select the number of change-points that minimizes the squared prediction error, which measures the fit of a specified model for a new sample. We develop a cross-validation estimation scheme based on an order-preserved sample-splitting strategy, and establish its asymptotic selection consistency under some mild conditions. Effectiveness of the proposed selection criterion is demonstrated on a variety of numerical experiments and real-data examples.

Han Hong, Jessie Li.

Source: The Annals of Statistics, Volume 48, Number 1, 397--412.

Abstract:
This paper proposes a numerical bootstrap method that is consistent in many cases where the standard bootstrap is known to fail and where the $m$-out-of-$n$ bootstrap and subsampling have been the most commonly used inference approaches. We provide asymptotic analysis under both fixed and drifting parameter sequences, and we compare the approximation error of the numerical bootstrap with that of the $m$-out-of-$n$ bootstrap and subsampling. Finally, we discuss applications of the numerical bootstrap, such as constrained and unconstrained M-estimators converging at both regular and nonstandard rates, Laplace-type estimators, and test statistics for partially identified models.

Hock Peng Chan.

Source: The Annals of Statistics, Volume 48, Number 1, 346--373.

Abstract:
Lai and Robbins ( Adv. in Appl. Math. 6 (1985) 4–22) and Lai ( Ann. Statist. 15 (1987) 1091–1114) provided efficient parametric solutions to the multi-armed bandit problem, showing that arm allocation via upper confidence bounds (UCB) achieves minimum regret. These bounds are constructed from the Kullback–Leibler information of the reward distributions, estimated from specified parametric families. In recent years, there has been renewed interest in the multi-armed bandit problem due to new applications in machine learning algorithms and data analytics. Nonparametric arm allocation procedures like $epsilon $-greedy, Boltzmann exploration and BESA were studied, and modified versions of the UCB procedure were also analyzed under nonparametric settings. However, unlike UCB these nonparametric procedures are not efficient under general parametric settings. In this paper, we propose efficient nonparametric procedures.

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.

Dimitris Bertsimas, Bart Van Parys.

Source: The Annals of Statistics, Volume 48, Number 1, 300--323.

Abstract:
We present a novel binary convex reformulation of the sparse regression problem that constitutes a new duality perspective. We devise a new cutting plane method and provide evidence that it can solve to provable optimality the sparse regression problem for sample sizes $n$ and number of regressors $p$ in the 100,000s, that is, two orders of magnitude better than the current state of the art, in seconds. The ability to solve the problem for very high dimensions allows us to observe new phase transition phenomena. Contrary to traditional complexity theory which suggests that the difficulty of a problem increases with problem size, the sparse regression problem has the property that as the number of samples $n$ increases the problem becomes easier in that the solution recovers 100% of the true signal, and our approach solves the problem extremely fast (in fact faster than Lasso), while for small number of samples $n$, our approach takes a larger amount of time to solve the problem, but importantly the optimal solution provides a statistically more relevant regressor. We argue that our exact sparse regression approach presents a superior alternative over heuristic methods available at present.

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.

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.

Guangyu Zhu, Zhihua Su.

Source: The Annals of Statistics, Volume 48, Number 1, 161--182.

Abstract:
Sparse partial least squares (SPLS) is widely used in applied sciences as a method that performs dimension reduction and variable selection simultaneously in linear regression. Several implementations of SPLS have been derived, among which the SPLS proposed in Chun and Keleş ( J. R. Stat. Soc. Ser. B. Stat. Methodol. 72 (2010) 3–25) is very popular and highly cited. However, for all of these implementations, the theoretical properties of SPLS are largely unknown. In this paper, we propose a new version of SPLS, called the envelope-based SPLS, using a connection between envelope models and partial least squares (PLS). We establish the consistency, oracle property and asymptotic normality of the envelope-based SPLS estimator. The large-sample scenario and high-dimensional scenario are both considered. We also develop the envelope-based SPLS estimators under the context of generalized linear models, and discuss its theoretical properties including consistency, oracle property and asymptotic distribution. Numerical experiments and examples show that the envelope-based SPLS estimator has better variable selection and prediction performance over the SPLS estimator ( J. R. Stat. Soc. Ser. B. Stat. Methodol. 72 (2010) 3–25).

Xiaoqin Wang, Li Yin.

Source: The Annals of Statistics, Volume 48, Number 1, 138--160.

Abstract:
In sequential causal inference, two types of causal effects are of practical interest, namely, the causal effect of the treatment regime (called the sequential causal effect) and the blip effect of treatment on the potential outcome after the last treatment. The well-known $G$-formula expresses these causal effects in terms of the standard parameters. In this article, we obtain a new $G$-formula that expresses these causal effects in terms of the point observable effects of treatments similar to treatment in the framework of single-point causal inference. Based on the new $G$-formula, we estimate these causal effects by maximum likelihood via point observable effects with methods extended from single-point causal inference. We are able to increase precision of the estimation without introducing biases by an unsaturated model imposing constraints on the point observable effects. We are also able to reduce the number of point observable effects in the estimation by treatment assignment conditions.

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.

John Goes, Gilad Lerman, Boaz Nadler.

Source: The Annals of Statistics, Volume 48, Number 1, 86--110.

Abstract:
Estimating a high-dimensional sparse covariance matrix from a limited number of samples is a fundamental task in contemporary data analysis. Most proposals to date, however, are not robust to outliers or heavy tails. Toward bridging this gap, in this work we consider estimating a sparse shape matrix from $n$ samples following a possibly heavy-tailed elliptical distribution. We propose estimators based on thresholding either Tyler’s M-estimator or its regularized variant. We prove that in the joint limit as the dimension $p$ and the sample size $n$ tend to infinity with $p/n ogamma>0$, our estimators are minimax rate optimal. Results on simulated data support our theoretical analysis.

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.

Victor Veitch, Daniel M. Roy.

Source: The Annals of Statistics, Volume 47, Number 6, 3274--3299.

Abstract:
Sparse exchangeable graphs on $mathbb{R}_{+}$, and the associated graphex framework for sparse graphs, generalize exchangeable graphs on $mathbb{N}$, and the associated graphon framework for dense graphs. We develop the graphex framework as a tool for statistical network analysis by identifying the sampling scheme that is naturally associated with the models of the framework, formalizing two natural notions of consistent estimation of the parameter (the graphex) underlying these models, and identifying general consistent estimators in each case. The sampling scheme is a modification of independent vertex sampling that throws away vertices that are isolated in the sampled subgraph. The estimators are variants of the empirical graphon estimator, which is known to be a consistent estimator for the distribution of dense exchangeable graphs; both can be understood as graph analogues to the empirical distribution in the i.i.d. sequence setting. Our results may be viewed as a generalization of consistent estimation via the empirical graphon from the dense graph regime to also include sparse graphs.

Jianqing Fan, Dong Wang, Kaizheng Wang, Ziwei Zhu.

Source: The Annals of Statistics, Volume 47, Number 6, 3009--3031.

Abstract:
Principal component analysis (PCA) is fundamental to statistical machine learning. It extracts latent principal factors that contribute to the most variation of the data. When data are stored across multiple machines, however, communication cost can prohibit the computation of PCA in a central location and distributed algorithms for PCA are thus needed. This paper proposes and studies a distributed PCA algorithm: each node machine computes the top $K$ eigenvectors and transmits them to the central server; the central server then aggregates the information from all the node machines and conducts a PCA based on the aggregated information. We investigate the bias and variance for the resulting distributed estimator of the top $K$ eigenvectors. In particular, we show that for distributions with symmetric innovation, the empirical top eigenspaces are unbiased, and hence the distributed PCA is “unbiased.” We derive the rate of convergence for distributed PCA estimators, which depends explicitly on the effective rank of covariance, eigengap, and the number of machines. We show that when the number of machines is not unreasonably large, the distributed PCA performs as well as the whole sample PCA, even without full access of whole data. The theoretical results are verified by an extensive simulation study. We also extend our analysis to the heterogeneous case where the population covariance matrices are different across local machines but share similar top eigenstructures.

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.

Bo Zhou, Ramon van den Akker, Bas J. M. Werker.

Source: The Annals of Statistics, Volume 47, Number 5, 2601--2638.

Abstract:
We propose a new class of unit root tests that exploits invariance properties in the Locally Asymptotically Brownian Functional limit experiment associated to the unit root model. The invariance structures naturally suggest tests that are based on the ranks of the increments of the observations, their average and an assumed reference density for the innovations. The tests are semiparametric in the sense that they are valid, that is, have the correct (asymptotic) size, irrespective of the true innovation density. For a correctly specified reference density, our test is point-optimal and nearly efficient. For arbitrary reference densities, we establish a Chernoff–Savage-type result, that is, our test performs as well as commonly used tests under Gaussian innovations but has improved power under other, for example, fat-tailed or skewed, innovation distributions. To avoid nonparametric estimation, we propose a simplified version of our test that exhibits the same asymptotic properties, except for the Chernoff–Savage result that we are only able to demonstrate by means of simulations.

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.

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.

Qian Qin, James P. Hobert.

Source: The Annals of Statistics, Volume 47, Number 4, 2320--2347.

Abstract:
The use of MCMC algorithms in high dimensional Bayesian problems has become routine. This has spurred so-called convergence complexity analysis, the goal of which is to ascertain how the convergence rate of a Monte Carlo Markov chain scales with sample size, $n$, and/or number of covariates, $p$. This article provides a thorough convergence complexity analysis of Albert and Chib’s [ J. Amer. Statist. Assoc. 88 (1993) 669–679] data augmentation algorithm for the Bayesian probit regression model. The main tools used in this analysis are drift and minorization conditions. The usual pitfalls associated with this type of analysis are avoided by utilizing centered drift functions, which are minimized in high posterior probability regions, and by using a new technique to suppress high-dimensionality in the construction of minorization conditions. The main result is that the geometric convergence rate of the underlying Markov chain is bounded below 1 both as $n ightarrowinfty$ (with $p$ fixed), and as $p ightarrowinfty$ (with $n$ fixed). Furthermore, the first computable bounds on the total variation distance to stationarity are byproducts of the asymptotic analysis.

Yuxin Chen, Jianqing Fan, Cong Ma, Kaizheng Wang.

Source: The Annals of Statistics, Volume 47, Number 4, 2204--2235.

Abstract:
This paper is concerned with the problem of top-$K$ ranking from pairwise comparisons. Given a collection of $n$ items and a few pairwise comparisons across them, one wishes to identify the set of $K$ items that receive the highest ranks. To tackle this problem, we adopt the logistic parametric model—the Bradley–Terry–Luce model, where each item is assigned a latent preference score, and where the outcome of each pairwise comparison depends solely on the relative scores of the two items involved. Recent works have made significant progress toward characterizing the performance (e.g., the mean square error for estimating the scores) of several classical methods, including the spectral method and the maximum likelihood estimator (MLE). However, where they stand regarding top-$K$ ranking remains unsettled. We demonstrate that under a natural random sampling model, the spectral method alone, or the regularized MLE alone, is minimax optimal in terms of the sample complexity—the number of paired comparisons needed to ensure exact top-$K$ identification, for the fixed dynamic range regime. This is accomplished via optimal control of the entrywise error of the score estimates. We complement our theoretical studies by numerical experiments, confirming that both methods yield low entrywise errors for estimating the underlying scores. Our theory is established via a novel leave-one-out trick, which proves effective for analyzing both iterative and noniterative procedures. Along the way, we derive an elementary eigenvector perturbation bound for probability transition matrices, which parallels the Davis–Kahan $mathop{mathrm{sin}} olimits Theta $ theorem for symmetric matrices. This also allows us to close the gap between the $ell_{2}$ error upper bound for the spectral method and the minimax lower limit.

James G. Scott, James O. Berger

Source: Ann. Statist., Volume 38, Number 5, 2587--2619.

Abstract:
This paper studies the multiplicity-correction effect of standard Bayesian variable-selection priors in linear regression. Our first goal is to clarify when, and how, multiplicity correction happens automatically in Bayesian analysis, and to distinguish this correction from the Bayesian Ockham’s-razor effect. Our second goal is to contrast empirical-Bayes and fully Bayesian approaches to variable selection through examples, theoretical results and simulations. Considerable differences between the two approaches are found. In particular, we prove a theorem that characterizes a surprising aymptotic discrepancy between fully Bayes and empirical Bayes. This discrepancy arises from a different source than the failure to account for hyperparameter uncertainty in the empirical-Bayes estimate. Indeed, even at the extreme, when the empirical-Bayes estimate converges asymptotically to the true variable-inclusion probability, the potential for a serious difference remains.

(Enterprise Service Bus) Universal integration backbone. An ESB acts as a shared messaging layer for connecting applications and other services throughout an enterprise computing infrastructure. It supplements its core asynchronous messaging backbone with intelligent transformation and routing to ensure messages are passed reliably. Services participate in the ESB using either web services messaging standards or the Java Message System (JMS). Originally defined by analysts at Gartner, ESB is increasingly seen as a core component in a service-oriented infrastructure.

Ability to work with each other. In the loosely coupled environment of a service-oriented architecture, separate resources don't need to know the details of how they each work, but they need to have enough common ground to reliably exchange messages without error or misunderstanding. Standardized specifications go a long way towards creating this common ground, but differences in implementation may still lead to breakdowns in communication. Interoperability is when services can interact with each other without encountering such problems.

Digital identity standards group. Set up at the instigation of Sun Microsystems in 2001, the Liberty Alliance Project is a consortium of technology vendors and consumer-facing enterprises formed "to establish an open standard for federated network identity." It aims to make it easier for consumers to access networked services from multiple suppliers while safeguarding security and privacy. Its specifications have been published in three phases: the Identity Federation Framework (ID-FF) came first; the Identity Web Services Framework (ID-WSF) followed in November 2003; and work is in progress on the Identity Services Interface Specifications (ID-SIS). Liberty Alliance specifications are closely linked to the SAML single sign-on standard, and overlap with elements of WS-Security.

(Common Object Request Broker Architecture) Pioneering integration architecture. Developed during the 1990s by the Object Management Group (OMG), CORBA was the first major attempt to define a platform-neutral architecture for combining heterogenous software resources across a network. A forerunner of today's service-oriented architectures, CORBA was designed for high-end, transaction-heavy, enterprise deployments, and thus it works best for tight coupling of software resources written in traditional programming languages such as C, C++, Java, Smalltalk and COBOL. Although the addition of IIOP (Internet Inter-ORB Protocol) extended CORBA to run over the Internet, it is less flexible than today's more loosely coupled SOAs, which are based on the exchange of XML documents using web services.

(COmmon Business Oriented Language) World's favorite mainframe programming language. Despite its venerable roots as one of the earliest high-level compiled languages, COBOL today still underpins some of the world's most important commercial and government operations, as it remains the most widely used programming language on mainframe computers. Created in 1959 by a cross-industry group of computer manufacturers under the auspices of the US Department of Defense, COBOL was designed as a machine-independent, industry-standard programming language for business data processing -- although in practice there were various incompatibilities between individual makers' versions. It has continued to evolve under the management of US and international standards bodies. The latest revision is COBOL 2002, with the next planned for 2008.

(OO) Structured around functional units. Object-oriented programming languages such as C++, SmallTalk and Java are designed to build software made up of objects: discrete bundles of functionality that can act on data only in certain pre-defined ways. This modular building-block approach makes complex software development tasks more flexible and easier to manage within a given programming environment. The emergence of object-oriented programming was a stepping stone to the development of componentization and subsequently of service-oriented architectures.

(Business Intelligence) Analysis of business data. BI is the name given to a class of software tools specifically designed to aid analysis of business data. BI tools have traditionally been associated with in-depth analysis of historical transaction data, supplied by either a data warehouse or an online analytical processing (OLAP) server linked to a database system. BI has a wide range of commercial and non-commercial applications, with the most common being the analysis of patterns such as sales and stock trends, pricing and customer behavior to inform business decision-making. For this reason it is sometimes referred to as decision support software.

(eXtensible Business Reporting Language) Standard format for reporting financial data. XBRL is an internationally agreed, open specification that uses XML to structure financial information for automated electronic processing. It is being adopted by major accounting standards bodies, regulators, tax authorities, banks and credit organizations around the world to streamline the reporting and analysis of statutory financial statements and other business financial information.

(Enterprise JavaBeans) Software components for networked Java applications. Defined by the Enterprise JavaBeans specification, EJBs are the basic building blocks of software applications on the J2EE platform, which has been the preferred choice for many enterprises when building large-scale, web-accessed applications. Recently, however, many developers have been turning away from the complexity of EJBs in favor of simpler alternatives. The new EJB 3.0 specification attempts to answer these criticisms by simplifying EJB development.

Trang Quynh Nguyen, Elizabeth A. Stuart.

Source: The Annals of Applied Statistics, Volume 14, Number 1, 518--520.

Boyu Ren, Sergio Bacallado, Stefano Favaro, Tommi Vatanen, Curtis Huttenhower, Lorenzo Trippa.

Source: The Annals of Applied Statistics, Volume 14, Number 1, 494--517.

Abstract:
Detecting associations between microbial compositions and sample characteristics is one of the most important tasks in microbiome studies. Most of the existing methods apply univariate models to single microbial species separately, with adjustments for multiple hypothesis testing. We propose a Bayesian analysis for a generalized mixed effects linear model tailored to this application. The marginal prior on each microbial composition is a Dirichlet process, and dependence across compositions is induced through a linear combination of individual covariates, such as disease biomarkers or the subject’s age, and latent factors. The latent factors capture residual variability and their dimensionality is learned from the data in a fully Bayesian procedure. The proposed model is tested in data analyses and simulation studies with zero-inflated compositions. In these settings and within each sample, a large proportion of counts per microbial species are equal to zero. In our Bayesian model a priori the probability of compositions with absent microbial species is strictly positive. We propose an efficient algorithm to sample from the posterior and visualizations of model parameters which reveal associations between covariates and microbial compositions. We evaluate the proposed method in simulation studies, and then analyze a microbiome dataset for infants with type 1 diabetes which contains a large proportion of zeros in the sample-specific microbial compositions.

Alex Diana, Eleni Matechou, Jim Griffin, Alison Johnston.

Source: The Annals of Applied Statistics, Volume 14, Number 1, 473--493.

Abstract:
Environmental changes in recent years have been linked to phenological shifts which in turn are linked to the survival of species. The work in this paper is motivated by capture-recapture data on blackcaps collected by the British Trust for Ornithology as part of the Constant Effort Sites monitoring scheme. Blackcaps overwinter abroad and migrate to the UK annually for breeding purposes. We propose a novel Bayesian nonparametric approach for expressing the bivariate density of individual arrival and departure times at different sites across a number of years as a mixture model. The new model combines the ideas of the hierarchical and the dependent Dirichlet process, allowing the estimation of site-specific weights and year-specific mixture locations, which are modelled as functions of environmental covariates using a multivariate extension of the Gaussian process. The proposed modelling framework is extremely general and can be used in any context where multivariate density estimation is performed jointly across different groups and in the presence of a continuous covariate.

Michael E. Sobel, Martin A. Lindquist.

Source: The Annals of Applied Statistics, Volume 14, Number 1, 452--472.

Abstract:
Neuroscientists often use functional magnetic resonance imaging (fMRI) to infer effects of treatments on neural activity in brain regions. In a typical fMRI experiment, each subject is observed at several hundred time points. At each point, the blood oxygenation level dependent (BOLD) response is measured at 100,000 or more locations (voxels). Typically, these responses are modeled treating each voxel separately, and no rationale for interpreting associations as effects is given. Building on Sobel and Lindquist ( J. Amer. Statist. Assoc. 109 (2014) 967–976), who used potential outcomes to define unit and average effects at each voxel and time point, we define and estimate both “point” and “cumulated” effects for brain regions. Second, we construct a multisubject, multivoxel, multirun whole brain causal model with explicit parameters for regions. We justify estimation using BOLD responses averaged over voxels within regions, making feasible estimation for all regions simultaneously, thereby also facilitating inferences about association between effects in different regions. We apply the model to a study of pain, finding effects in standard pain regions. We also observe more cerebellar activity than observed in previous studies using prevailing methods.

Zhonghua Liu, Ian Barnett, Xihong Lin.

Source: The Annals of Applied Statistics, Volume 14, Number 1, 433--451.

Abstract:
Principal component analysis (PCA) is a popular method for dimension reduction in unsupervised multivariate analysis. However, existing ad hoc uses of PCA in both multivariate regression (multiple outcomes) and multiple regression (multiple predictors) lack theoretical justification. The differences in the statistical properties of PCAs in these two regression settings are not well understood. In this paper we provide theoretical results on the power of PCA in genetic association testings in both multiple phenotype and SNP-set settings. The multiple phenotype setting refers to the case when one is interested in studying the association between a single SNP and multiple phenotypes as outcomes. The SNP-set setting refers to the case when one is interested in studying the association between multiple SNPs in a SNP set and a single phenotype as the outcome. We demonstrate analytically that the properties of the PC-based analysis in these two regression settings are substantially different. We show that the lower order PCs, that is, PCs with large eigenvalues, are generally preferred and lead to a higher power in the SNP-set setting, while the higher-order PCs, that is, PCs with small eigenvalues, are generally preferred in the multiple phenotype setting. We also investigate the power of three other popular statistical methods, the Wald test, the variance component test and the minimum $p$-value test, in both multiple phenotype and SNP-set settings. We use theoretical power, simulation studies, and two real data analyses to validate our findings.

Yicheng Li, Adrian E. Raftery.

Source: The Annals of Applied Statistics, Volume 14, Number 1, 381--408.

Abstract:
Smoking is one of the leading preventable threats to human health and a major risk factor for lung cancer, upper aerodigestive cancer and chronic obstructive pulmonary disease. Estimating and forecasting the smoking attributable fraction (SAF) of mortality can yield insights into smoking epidemics and also provide a basis for more accurate mortality and life expectancy projection. Peto et al. ( Lancet 339 (1992) 1268–1278) proposed a method to estimate the SAF using the lung cancer mortality rate as an indicator of exposure to smoking in the population of interest. Here, we use the same method to estimate the all-age SAF (ASAF) for both genders for over 60 countries. We document a strong and cross-nationally consistent pattern of the evolution of the SAF over time. We use this as the basis for a new Bayesian hierarchical model to project future male and female ASAF from over 60 countries simultaneously. This gives forecasts as well as predictive distributions that can be used to find uncertainty intervals for any quantity of interest. We assess the model using out-of-sample predictive validation and find that it provides good forecasts and well-calibrated forecast intervals, comparing favorably with other methods.

Peng Shi, Zifeng Zhao.

Source: The Annals of Applied Statistics, Volume 14, Number 1, 357--380.

Abstract:
In actuarial research a task of particular interest and importance is to predict the loss cost for individual risks so that informative decisions are made in various insurance operations such as underwriting, ratemaking and capital management. The loss cost is typically viewed to follow a compound distribution where the summation of the severity variables is stopped by the frequency variable. A challenging issue in modeling such outcomes is to accommodate the potential dependence between the number of claims and the size of each individual claim. In this article we introduce a novel regression framework for compound distributions that uses a copula to accommodate the association between the frequency and the severity variables and, thus, allows for arbitrary dependence between the two components. We further show that the new model is very flexible and is easily modified to account for incomplete data due to censoring or truncation. The flexibility of the proposed model is illustrated using both simulated and real data sets. In the analysis of granular claims data from property insurance, we find substantive negative relationship between the number and the size of insurance claims. In addition, we demonstrate that ignoring the frequency-severity association could lead to biased decision-making in insurance operations.

Jared D. Fisher, Davide Pettenuzzo, Carlos M. Carvalho.

Source: The Annals of Applied Statistics, Volume 14, Number 1, 299--338.

Abstract:
We introduce a fast, closed-form, simulation-free method to model and forecast multiple asset returns and employ it to investigate the optimal ensemble of features to include when jointly predicting monthly stock and bond excess returns. Our approach builds on the Bayesian dynamic linear models of West and Harrison ( Bayesian Forecasting and Dynamic Models (1997) Springer), and it can objectively determine, through a fully automated procedure, both the optimal set of regressors to include in the predictive system and the degree to which the model coefficients, volatilities and covariances should vary over time. When applied to a portfolio of five stock and bond returns, we find that our method leads to large forecast gains, both in statistical and economic terms. In particular, we find that relative to a standard no-predictability benchmark, the optimal combination of predictors, stochastic volatility and time-varying covariances increases the annualized certainty equivalent returns of a leverage-constrained power utility investor by more than 500 basis points.

Wanghuan Chu, Runze Li, Jingyuan Liu, Matthew Reimherr.

Source: The Annals of Applied Statistics, Volume 14, Number 1, 276--298.

Abstract:
Motivated by an empirical analysis of data from a genome-wide association study on obesity, measured by the body mass index (BMI), we propose a two-step gene-detection procedure for generalized varying coefficient mixed-effects models with ultrahigh dimensional covariates. The proposed procedure selects significant single nucleotide polymorphisms (SNPs) impacting the mean BMI trend, some of which have already been biologically proven to be “fat genes.” The method also discovers SNPs that significantly influence the age-dependent variability of BMI. The proposed procedure takes into account individual variations of genetic effects and can also be directly applied to longitudinal data with continuous, binary or count responses. We employ Monte Carlo simulation studies to assess the performance of the proposed method and further carry out causal inference for the selected SNPs.

Joseph Antonelli, Maitreyi Mazumdar, David Bellinger, David Christiani, Robert Wright, Brent Coull.

Source: The Annals of Applied Statistics, Volume 14, Number 1, 257--275.

Abstract:
Humans are routinely exposed to mixtures of chemical and other environmental factors, making the quantification of health effects associated with environmental mixtures a critical goal for establishing environmental policy sufficiently protective of human health. The quantification of the effects of exposure to an environmental mixture poses several statistical challenges. It is often the case that exposure to multiple pollutants interact with each other to affect an outcome. Further, the exposure-response relationship between an outcome and some exposures, such as some metals, can exhibit complex, nonlinear forms, since some exposures can be beneficial and detrimental at different ranges of exposure. To estimate the health effects of complex mixtures, we propose a flexible Bayesian approach that allows exposures to interact with each other and have nonlinear relationships with the outcome. We induce sparsity using multivariate spike and slab priors to determine which exposures are associated with the outcome and which exposures interact with each other. The proposed approach is interpretable, as we can use the posterior probabilities of inclusion into the model to identify pollutants that interact with each other. We utilize our approach to study the impact of exposure to metals on child neurodevelopment in Bangladesh and find a nonlinear, interactive relationship between arsenic and manganese.

Tsuyoshi Kunihama, Zehang Richard Li, Samuel J. Clark, Tyler H. McCormick.

Source: The Annals of Applied Statistics, Volume 14, Number 1, 241--256.

Abstract:
The distribution of deaths by cause provides crucial information for public health planning, response and evaluation. About 60% of deaths globally are not registered or given a cause, limiting our ability to understand disease epidemiology. Verbal autopsy (VA) surveys are increasingly used in such settings to collect information on the signs, symptoms and medical history of people who have recently died. This article develops a novel Bayesian method for estimation of population distributions of deaths by cause using verbal autopsy data. The proposed approach is based on a multivariate probit model where associations among items in questionnaires are flexibly induced by latent factors. Using the Population Health Metrics Research Consortium labeled data that include both VA and medically certified causes of death, we assess performance of the proposed method. Further, we estimate important questionnaire items that are highly associated with causes of death. This framework provides insights that will simplify future data

Mohamad Elmasri, Maxwell J. Farrell, T. Jonathan Davies, David A. Stephens.

Source: The Annals of Applied Statistics, Volume 14, Number 1, 221--240.

Abstract:
Identifying undocumented or potential future interactions among species is a challenge facing modern ecologists. Recent link prediction methods rely on trait data; however, large species interaction databases are typically sparse and covariates are limited to only a fraction of species. On the other hand, evolutionary relationships, encoded as phylogenetic trees, can act as proxies for underlying traits and historical patterns of parasite sharing among hosts. We show that, using a network-based conditional model, phylogenetic information provides strong predictive power in a recently published global database of host-parasite interactions. By scaling the phylogeny using an evolutionary model, our method allows for biological interpretation often missing from latent variable models. To further improve on the phylogeny-only model, we combine a hierarchical Bayesian latent score framework for bipartite graphs that accounts for the number of interactions per species with host dependence informed by phylogeny. Combining the two information sources yields significant improvement in predictive accuracy over each of the submodels alone. As many interaction networks are constructed from presence-only data, we extend the model by integrating a correction mechanism for missing interactions which proves valuable in reducing uncertainty in unobserved interactions.

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.

Bryan D. Martin, Daniela Witten, Amy D. Willis.

Source: The Annals of Applied Statistics, Volume 14, Number 1, 94--115.

Abstract:
Using a sample from a population to estimate the proportion of the population with a certain category label is a broadly important problem. In the context of microbiome studies, this problem arises when researchers wish to use a sample from a population of microbes to estimate the population proportion of a particular taxon, known as the taxon’s relative abundance . In this paper, we propose a beta-binomial model for this task. Like existing models, our model allows for a taxon’s relative abundance to be associated with covariates of interest. However, unlike existing models, our proposal also allows for the overdispersion in the taxon’s counts to be associated with covariates of interest. We exploit this model in order to propose tests not only for differential relative abundance, but also for differential variability. The latter is particularly valuable in light of speculation that dysbiosis , the perturbation from a normal microbiome that can occur in certain disease conditions, may manifest as a loss of stability, or increase in variability, of the counts associated with each taxon. We demonstrate the performance of our proposed model using a simulation study and an application to soil microbial data.

Paul J. Birrell, Lorenz Wernisch, Brian D. M. Tom, Leonhard Held, Gareth O. Roberts, Richard G. Pebody, Daniela De Angelis.

Source: The Annals of Applied Statistics, Volume 14, Number 1, 74--93.

Abstract:
A prompt public health response to a new epidemic relies on the ability to monitor and predict its evolution in real time as data accumulate. The 2009 A/H1N1 outbreak in the UK revealed pandemic data as noisy, contaminated, potentially biased and originating from multiple sources. This seriously challenges the capacity for real-time monitoring. Here, we assess the feasibility of real-time inference based on such data by constructing an analytic tool combining an age-stratified SEIR transmission model with various observation models describing the data generation mechanisms. As batches of data become available, a sequential Monte Carlo (SMC) algorithm is developed to synthesise multiple imperfect data streams, iterate epidemic inferences and assess model adequacy amidst a rapidly evolving epidemic environment, substantially reducing computation time in comparison to standard MCMC, to ensure timely delivery of real-time epidemic assessments. In application to simulated data designed to mimic the 2009 A/H1N1 epidemic, SMC is shown to have additional benefits in terms of assessing predictive performance and coping with parameter nonidentifiability.

Jennifer E. Starling, Jared S. Murray, Carlos M. Carvalho, Radek K. Bukowski, James G. Scott.

Source: The Annals of Applied Statistics, Volume 14, Number 1, 28--50.

Abstract:
This article introduces BART with Targeted Smoothing, or tsBART, a new Bayesian tree-based model for nonparametric regression. The goal of tsBART is to introduce smoothness over a single target covariate $t$ while not necessarily requiring smoothness over other covariates $x$. tsBART is based on the Bayesian Additive Regression Trees (BART) model, an ensemble of regression trees. tsBART extends BART by parameterizing each tree’s terminal nodes with smooth functions of $t$ rather than independent scalars. Like BART, tsBART captures complex nonlinear relationships and interactions among the predictors. But unlike BART, tsBART guarantees that the response surface will be smooth in the target covariate. This improves interpretability and helps to regularize the estimate. After introducing and benchmarking the tsBART model, we apply it to our motivating example—pregnancy outcomes data from the National Center for Health Statistics. Our aim is to provide patient-specific estimates of stillbirth risk across gestational age $(t)$ and based on maternal and fetal risk factors $(x)$. Obstetricians expect stillbirth risk to vary smoothly over gestational age but not necessarily over other covariates, and tsBART has been designed precisely to reflect this structural knowledge. The results of our analysis show the clear superiority of the tsBART model for quantifying stillbirth risk, thereby providing patients and doctors with better information for managing the risk of fetal mortality. All methods described here are implemented in the R package tsbart .