Didier Chauveau, David R. Hunter, Michael Levine.

Source: Statistics Surveys, Volume 9, 1--31.

Abstract:
The conditional independence assumption for nonparametric multivariate finite mixture models, a weaker form of the well-known conditional independence assumption for random effects models for longitudinal data, is the subject of an increasing number of theoretical and algorithmic developments in the statistical literature. After presenting a survey of this literature, including an in-depth discussion of the all-important identifiability results, this article describes and extends an algorithm for estimation of the parameters in these models. The algorithm works for any number of components in three or more dimensions. It possesses a descent property and can be easily adapted to situations where the data are grouped in blocks of conditionally independent variables. We discuss how to adapt this algorithm to various location-scale models that link component densities, and we even adapt it to a particular class of univariate mixture problems in which the components are assumed symmetric. We give a bandwidth selection procedure for our algorithm. Finally, we demonstrate the effectiveness of our algorithm using a simulation study and two psychometric datasets.

Aki Vehtari, Janne Ojanen.

Source: Statistics Surveys, Volume 8, , 1--1.

Abstract:
Errata for “A survey of Bayesian predictive methods for model assessment, selection and comparison” by A. Vehtari and J. Ojanen, Statistics Surveys , 6 (2012), 142–228. doi:10.1214/12-SS102.

Aki Vehtari, Janne Ojanen

Source: Statist. Surv., Volume 6, 142--228.

Abstract:
To date, several methods exist in the statistical literature for model assessment, which purport themselves specifically as Bayesian predictive methods. The decision theoretic assumptions on which these methods are based are not always clearly stated in the original articles, however. The aim of this survey is to provide a unified review of Bayesian predictive model assessment and selection methods, and of methods closely related to them. We review the various assumptions that are made in this context and discuss the connections between different approaches, with an emphasis on how each method approximates the expected utility of using a Bayesian model for the purpose of predicting future data.

Scott H. Holan, Robert Lund, Ginger Davis

Source: Statist. Surv., Volume 4, 232--274.

Abstract:
Autoregressive moving-average (ARMA) difference equations are ubiquitous models for short memory time series and have parsimoniously described many stationary series. Variants of ARMA models have been proposed to describe more exotic series features such as long memory autocovariances, periodic autocovariances, and count support set structures. This review paper enumerates, compares, and contrasts the common variants of ARMA models in today’s literature. After the basic properties of ARMA models are reviewed, we tour ARMA variants that describe seasonal features, long memory behavior, multivariate series, changing variances (stochastic volatility) and integer counts. A list of ARMA variant acronyms is provided.

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Sophie Achard, Jean-François Coeurjolly

Source: Statist. Surv., Volume 4, 117--147.

Abstract:
This paper gives an overview of the problem of estimating the Hurst parameter of a fractional Brownian motion when the data are observed with outliers and/or with an additive noise by using methods based on discrete variations. We show that the classical estimation procedure based on the log-linearity of the variogram of dilated series is made more robust to outliers and/or an additive noise by considering sample quantiles and trimmed means of the squared series or differences of empirical variances. These different procedures are compared and discussed through a large simulation study and are implemented in the R package dvfBm.

Building a scalable machine learning system for unsupervised anomaly detection via representation learning is highly desirable. One of the prevalent methods is using a reconstruction error from variational autoencoder (VAE) via maximizing the evidence lower bound. We revisit VAE from the perspective of information theory to provide some theoretical foundations on using the reconstruction error, and finally arrive at a simpler and more effective model for anomaly detection. In addition, to enhance the effectiveness of detecting anomalies, we incorporate a practical model uncertainty measure into the metric. We show empirically the competitive performance of our approach on benchmark datasets.

Spatial trajectories are ubiquitous and complex signals. Their analysis is crucial in many research fields, from urban planning to neuroscience. Several approaches have been proposed to cluster trajectories. They rely on hand-crafted features, which struggle to capture the spatio-temporal complexity of the signal, or on Artificial Neural Networks (ANNs) which can be more efficient but less interpretable. In this paper we present a novel ANN architecture designed to capture the spatio-temporal patterns characteristic of a set of trajectories, while taking into account the demographics of the navigators. Hence, our model extracts markers linked to both behaviour and demographics. We propose a composite signal analyser (CompSNN) combining three simple ANN modules. Each of these modules uses different signal representations of the trajectory while remaining interpretable. Our CompSNN performs significantly better than its modules taken in isolation and allows to visualise which parts of the signal were most useful to discriminate the trajectories.

We focus on the challenge of finding a diverse collection of quality solutions on complex continuous domains. While quality diver-sity (QD) algorithms like Novelty Search with Local Competition (NSLC) and MAP-Elites are designed to generate a diverse range of solutions, these algorithms require a large number of evaluations for exploration of continuous spaces. Meanwhile, variants of the Covariance Matrix Adaptation Evolution Strategy (CMA-ES) are among the best-performing derivative-free optimizers in single-objective continuous domains. This paper proposes a new QD algorithm called Covariance Matrix Adaptation MAP-Elites (CMA-ME). Our new algorithm combines the self-adaptation techniques of CMA-ES with archiving and mapping techniques for maintaining diversity in QD. Results from experiments based on standard continuous optimization benchmarks show that CMA-ME finds better-quality solutions than MAP-Elites; similarly, results on the strategic game Hearthstone show that CMA-ME finds both a higher overall quality and broader diversity of strategies than both CMA-ES and MAP-Elites. Overall, CMA-ME more than doubles the performance of MAP-Elites using standard QD performance metrics. These results suggest that QD algorithms augmented by operators from state-of-the-art optimization algorithms can yield high-performing methods for simultaneously exploring and optimizing continuous search spaces, with significant applications to design, testing, and reinforcement learning among other domains.

This paper develops a general framework for analyzing asymptotics of $V$-statistics. Previous literature on limiting distribution mainly focuses on the cases when $n o infty$ with fixed kernel size $k$. Under some regularity conditions, we demonstrate asymptotic normality when $k$ grows with $n$ by utilizing existing results for $U$-statistics. The key in our approach lies in a mathematical reduction to $U$-statistics by designing an equivalent kernel for $V$-statistics. We also provide a unified treatment on variance estimation for both $U$- and $V$-statistics by observing connections to existing methods and proposing an empirically more accurate estimator. Ensemble methods such as random forests, where multiple base learners are trained and aggregated for prediction purposes, serve as a running example throughout the paper because they are a natural and flexible application of $V$-statistics.

Factor analysis is a flexible technique for assessment of multivariate dependence and codependence. Besides being an exploratory tool used to reduce the dimensionality of multivariate data, it allows estimation of common factors that often have an interesting theoretical interpretation in real problems. However, standard factor analysis is only applicable when the variables are scaled, which is often inappropriate, for example, in data obtained from questionnaires in the field of psychology,where the variables are often categorical. In this framework, we propose a factor model for the analysis of multivariate ordered and non-ordered polychotomous data. The inference procedure is done under the Bayesian approach via Markov chain Monte Carlo methods. Two Monte-Carlo simulation studies are presented to investigate the performance of this approach in terms of estimation bias, precision and assessment of the number of factors. We also illustrate the proposed method to analyze participants' responses to the Motivational State Questionnaire dataset, developed to study emotions in laboratory and field settings.

We study the interplay of two important issues on Bayesian model selection (BMS): censoring and model misspecification. We consider additive accelerated failure time (AAFT), Cox proportional hazards and probit models, and a more general concave log-likelihood structure. A fundamental question is what solution can one hope BMS to provide, when (inevitably) models are misspecified. We show that asymptotically BMS keeps any covariate with predictive power for either the outcome or censoring times, and discards other covariates. Misspecification refers to assuming the wrong model or functional effect on the response, including using a finite basis for a truly non-parametric effect, or omitting truly relevant covariates. We argue for using simple models that are computationally practical yet attain good power to detect potentially complex effects, despite misspecification. Misspecification and censoring both have an asymptotically negligible effect on (suitably-defined) false positives, but their impact on power is exponential. We portray these issues via simple descriptions of early/late censoring and the drop in predictive accuracy due to misspecification. From a methods point of view, we consider local priors and a novel structure that combines local and non-local priors to enforce sparsity. We develop algorithms to capitalize on the AAFT tractability, approximations to AAFT and probit likelihoods giving significant computational gains, a simple augmented Gibbs sampler to hierarchically explore linear and non-linear effects, and an implementation in the R package mombf. We illustrate the proposed methods and others based on likelihood penalties via extensive simulations under misspecification and censoring. We present two applications concerning the effect of gene expression on colon and breast cancer.

We present LCE, a Local Cascade Ensemble for traditional (tabular) multivariate data classification, and its extension LCEM for Multivariate Time Series (MTS) classification. LCE is a new hybrid ensemble method that combines an explicit boosting-bagging approach to handle the usual bias-variance tradeoff faced by machine learning models and an implicit divide-and-conquer approach to individualize classifier errors on different parts of the training data. Our evaluation firstly shows that the hybrid ensemble method LCE outperforms the state-of-the-art classifiers on the UCI datasets and that LCEM outperforms the state-of-the-art MTS classifiers on the UEA datasets. Furthermore, LCEM provides explainability by design and manifests robust performance when faced with challenges arising from continuous data collection (different MTS length, missing data and noise).

This paper deals with robust marginal estimation under a general regression model when missing data occur in the response and also in some of covariates. The target is a marginal location parameter which is given through an $M-$functional. To obtain robust Fisher--consistent estimators, properly defined marginal distribution function estimators are considered. These estimators avoid the bias due to missing values by assuming a missing at random condition. Three methods are considered to estimate the marginal distribution function which allows to obtain the $M-$location of interest: the well-known inverse probability weighting, a convolution--based method that makes use of the regression model and an augmented inverse probability weighting procedure that prevents against misspecification. The robust proposed estimators and the classical ones are compared through a numerical study under different missing models including clean and contaminated samples. We illustrate the estimators behaviour under a nonlinear model. A real data set is also analysed.

Text summarization refers to the process that generates a shorter form of text from the source document preserving salient information. Recently, many models for text summarization have been proposed. Most of those models were evaluated using recall-oriented understudy for gisting evaluation (ROUGE) scores. However, as ROUGE scores are computed based on n-gram overlap, they do not reflect semantic meaning correspondences between generated and reference summaries. Because Korean is an agglutinative language that combines various morphemes into a word that express several meanings, ROUGE is not suitable for Korean summarization. In this paper, we propose evaluation metrics that reflect semantic meanings of a reference summary and the original document, Reference and Document Aware Semantic Score (RDASS). We then propose a method for improving the correlation of the metrics with human judgment. Evaluation results show that the correlation with human judgment is significantly higher for our evaluation metrics than for ROUGE scores.

This paper revisits the problem of computing empirical cumulative distribution functions (ECDF) efficiently on large, multivariate datasets. Computing an ECDF at one evaluation point requires $mathcal{O}(N)$ operations on a dataset composed of $N$ data points. Therefore, a direct evaluation of ECDFs at $N$ evaluation points requires a quadratic $mathcal{O}(N^2)$ operations, which is prohibitive for large-scale problems. Two fast and exact methods are proposed and compared. The first one is based on fast summation in lexicographical order, with a $mathcal{O}(N{log}N)$ complexity and requires the evaluation points to lie on a regular grid. The second one is based on the divide-and-conquer principle, with a $mathcal{O}(Nlog(N)^{(d-1){vee}1})$ complexity and requires the evaluation points to coincide with the input points. The two fast algorithms are described and detailed in the general $d$-dimensional case, and numerical experiments validate their speed and accuracy. Secondly, the paper establishes a direct connection between cumulative distribution functions and kernel density estimation (KDE) for a large class of kernels. This connection paves the way for fast exact algorithms for multivariate kernel density estimation and kernel regression. Numerical tests with the Laplacian kernel validate the speed and accuracy of the proposed algorithms. A broad range of large-scale multivariate density estimation, cumulative distribution estimation, survival function estimation and regression problems can benefit from the proposed numerical methods.

Uncertainties in a structure is inevitable, which generally lead to variation in dynamic response predictions. For a complex structure, brute force Monte Carlo simulation for response variation analysis is infeasible since one single run may already be computationally costly. Data driven meta-modeling approaches have thus been explored to facilitate efficient emulation and statistical inference. The performance of a meta-model hinges upon both the quality and quantity of training dataset. In actual practice, however, high-fidelity data acquired from high-dimensional finite element simulation or experiment are generally scarce, which poses significant challenge to meta-model establishment. In this research, we take advantage of the multi-level response prediction opportunity in structural dynamic analysis, i.e., acquiring rapidly a large amount of low-fidelity data from reduced-order modeling, and acquiring accurately a small amount of high-fidelity data from full-scale finite element analysis. Specifically, we formulate a composite neural network fusion approach that can fully utilize the multi-level, heterogeneous datasets obtained. It implicitly identifies the correlation of the low- and high-fidelity datasets, which yields improved accuracy when compared with the state-of-the-art. Comprehensive investigations using frequency response variation characterization as case example are carried out to demonstrate the performance.

I study the minimax-optimal design for a two-arm controlled experiment where conditional mean outcomes may vary in a given set. When this set is permutation symmetric, the optimal design is complete randomization, and using a single partition (i.e., the design that only randomizes the treatment labels for each side of the partition) has minimax risk larger by a factor of $n-1$. More generally, the optimal design is shown to be the mixed-strategy optimal design (MSOD) of Kallus (2018). Notably, even when the set of conditional mean outcomes has structure (i.e., is not permutation symmetric), being minimax-optimal for variance still requires randomization beyond a single partition. Nonetheless, since this targets precision, it may still not ensure sufficient uniformity in randomization to enable randomization (i.e., design-based) inference by Fisher's exact test to appropriately detect violations of null. I therefore propose the inference-constrained MSOD, which is minimax-optimal among all designs subject to such uniformity constraints. On the way, I discuss Johansson et al. (2020) who recently compared rerandomization of Morgan and Rubin (2012) and the pure-strategy optimal design (PSOD) of Kallus (2018). I point out some errors therein and set straight that randomization is minimax-optimal and that the "no free lunch" theorem and example in Kallus (2018) are correct.

An important component of every country's COVID-19 response is fast and efficient testing -- to identify and isolate cases, as well as for early detection of local hotspots. For many countries, producing a sufficient number of tests has been a serious limiting factor in their efforts to control COVID-19 infections. Group testing is a well-established mathematical tool, which can provide a serious and rapid improvement to this situation. In this note, we compare several well-established group testing schemes in the context of qPCR testing for COVID-19. We include example calculations, where we indicate which testing architectures yield the greatest efficiency gains in various settings. We find that for identification of individuals with COVID-19, array testing is usually the best choice, while for estimation of COVID-19 prevalence rates in the total population, Gibbs-Gower testing usually provides the most accurate estimates given a fixed and relatively small number of tests. This note is intended as a helpful handbook for labs implementing group testing methods.

Effective property prediction methods can help accelerate the search for COVID-19 antivirals either through accurate in-silico screens or by effectively guiding on-going at-scale experimental efforts. However, existing prediction tools have limited ability to accommodate scarce or fragmented training data currently available. In this paper, we introduce a novel approach to learn predictors that can generalize or extrapolate beyond the heterogeneous data. Our method builds on and extends recently proposed invariant risk minimization, adaptively forcing the predictor to avoid nuisance variation. We achieve this by continually exercising and manipulating latent representations of molecules to highlight undesirable variation to the predictor. To test the method we use a combination of three data sources: SARS-CoV-2 antiviral screening data, molecular fragments that bind to SARS-CoV-2 main protease and large screening data for SARS-CoV-1. Our predictor outperforms state-of-the-art transfer learning methods by significant margin. We also report the top 20 predictions of our model on Broad drug repurposing hub.

The R package mvord implements composite likelihood estimation in the class of multivariate ordinal regression models with a multivariate probit and a multivariate logit link. A flexible modeling framework for multiple ordinal measurements on the same subject is set up, which takes into consideration the dependence among the multiple observations by employing different error structures. Heterogeneity in the error structure across the subjects can be accounted for by the package, which allows for covariate dependent error structures. In addition, different regression coefficients and threshold parameters for each response are supported. If a reduction of the parameter space is desired, constraints on the threshold as well as on the regression coefficients can be specified by the user. The proposed multivariate framework is illustrated by means of a credit risk application.

An R package for computing the all-subsets regression problem is presented. The proposed algorithms are based on computational strategies recently developed. A novel algorithm for the best-subset regression problem selects subset models based on a predetermined criterion. The package user can choose from exact and from approximation algorithms. The core of the package is written in C++ and provides an efficient implementation of all the underlying numerical computations. A case study and benchmark results illustrate the usage and the computational efficiency of the package.

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.

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.

Monika Bhattacharjee, Arup Bose.

Source: The Annals of Statistics, Volume 47, Number 6, 3470--3503.

Abstract:
Consider a high-dimensional linear time series model where the dimension $p$ and the sample size $n$ grow in such a way that $p/n o 0$. Let $hat{Gamma }_{u}$ be the $u$th order sample autocovariance matrix. We first show that the LSD of any symmetric polynomial in ${hat{Gamma }_{u},hat{Gamma }_{u}^{*},ugeq 0}$ exists under independence and moment assumptions on the driving sequence together with weak assumptions on the coefficient matrices. This LSD result, with some additional effort, implies the asymptotic normality of the trace of any polynomial in ${hat{Gamma }_{u},hat{Gamma }_{u}^{*},ugeq 0}$. We also study similar results for several independent MA processes. We show applications of the above results to statistical inference problems such as in estimation of the unknown order of a high-dimensional MA process and in graphical and significance tests for hypotheses on coefficient matrices of one or several such independent processes.

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.

Reinhard Heckel, Nihar B. Shah, Kannan Ramchandran, Martin J. Wainwright.

Source: The Annals of Statistics, Volume 47, Number 6, 3099--3126.

Abstract:
We consider sequential or active ranking of a set of $n$ items based on noisy pairwise comparisons. Items are ranked according to the probability that a given item beats a randomly chosen item, and ranking refers to partitioning the items into sets of prespecified sizes according to their scores. This notion of ranking includes as special cases the identification of the top-$k$ items and the total ordering of the items. We first analyze a sequential ranking algorithm that counts the number of comparisons won, and uses these counts to decide whether to stop, or to compare another pair of items, chosen based on confidence intervals specified by the data collected up to that point. We prove that this algorithm succeeds in recovering the ranking using a number of comparisons that is optimal up to logarithmic factors. This guarantee does depend on whether or not the underlying pairwise probability matrix, satisfies a particular structural property, unlike a significant body of past work on pairwise ranking based on parametric models such as the Thurstone or Bradley–Terry–Luce models. It has been a long-standing open question as to whether or not imposing these parametric assumptions allows for improved ranking algorithms. For stochastic comparison models, in which the pairwise probabilities are bounded away from zero, our second contribution is to resolve this issue by proving a lower bound for parametric models. This shows, perhaps surprisingly, that these popular parametric modeling choices offer at most logarithmic gains for stochastic comparisons.

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.

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

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

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

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.

How small the pieces are. When a system is split into components, it's important to get the right degree of componentization. Small, fine-grained components give much greater flexibility in assembling precisely the right combination of functionality, but they are more difficult to co-ordinate. Much larger, coarse-grained components are easier to manage but may become too unwieldy. Performance and management considerations tend to favor the use of more coarsely grained messages in a service oriented architecture, whereas earlier generations of distributed computing have preferred a much finer level of granularity.

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.

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.

Zhenguo Gao, Pang Du, Ran Jin, John L. Robertson.

Source: The Annals of Applied Statistics, Volume 14, Number 1, 143--159.

Abstract:
Liver procurement experiments with surface-temperature monitoring motivated Gao et al. ( J. Amer. Statist. Assoc. 114 (2019) 773–781) to develop a variance change-point detection method under a smoothly-changing mean trend. However, the spotwise change points yielded from their method do not offer immediate information to surgeons since an organ is often transplanted as a whole or in part. We develop a new practical method that can analyze a defined portion of the organ surface at a time. It also provides a novel addition to the developing field of functional data monitoring. Furthermore, numerical challenge emerges for simultaneously modeling the variance functions of 2D locations and the mean function of location and time. The respective sample sizes in the scales of 10,000 and 1,000,000 for modeling these functions make standard spline estimation too costly to be useful. We introduce a multistage subsampling strategy with steps educated by quickly-computable preliminary statistical measures. Extensive simulations show that the new method can efficiently reduce the computational cost and provide reasonable parameter estimates. Application of the new method to our liver surface temperature monitoring data shows its effectiveness in providing accurate status change information for a selected portion of the organ in the experiment.

Li Zhu, Zhiguang Huo, Tianzhou Ma, Steffi Oesterreich, George C. Tseng.

Source: The Annals of Applied Statistics, Volume 13, Number 4, 2611--2636.

Abstract:
Variable selection is a pervasive problem in modern high-dimensional data analysis where the number of features often exceeds the sample size (a.k.a. small-n-large-p problem). Incorporation of group structure knowledge to improve variable selection has been widely studied. Here, we consider prior knowledge of a hierarchical overlapping group structure to improve variable selection in regression setting. In genomics applications, for instance, a biological pathway contains tens to hundreds of genes and a gene can be mapped to multiple experimentally measured features (such as its mRNA expression, copy number variation and methylation levels of possibly multiple sites). In addition to the hierarchical structure, the groups at the same level may overlap (e.g., two pathways can share common genes). Incorporating such hierarchical overlapping groups in traditional penalized regression setting remains a difficult optimization problem. Alternatively, we propose a Bayesian indicator model that can elegantly serve the purpose. We evaluate the model in simulations and two breast cancer examples, and demonstrate its superior performance over existing models. The result not only enhances prediction accuracy but also improves variable selection and model interpretation that lead to deeper biological insight of the disease.

John R. Tipton, Mevin B. Hooten, Connor Nolan, Robert K. Booth, Jason McLachlan.

Source: The Annals of Applied Statistics, Volume 13, Number 4, 2363--2388.

Abstract:
Multivariate compositional count data arise in many applications including ecology, microbiology, genetics and paleoclimate. A frequent question in the analysis of multivariate compositional count data is what underlying values of a covariate(s) give rise to the observed composition. Learning the relationship between covariates and the compositional count allows for inverse prediction of unobserved covariates given compositional count observations. Gaussian processes provide a flexible framework for modeling functional responses with respect to a covariate without assuming a functional form. Many scientific disciplines use Gaussian process approximations to improve prediction and make inference on latent processes and parameters. When prediction is desired on unobserved covariates given realizations of the response variable, this is called inverse prediction. Because inverse prediction is often mathematically and computationally challenging, predicting unobserved covariates often requires fitting models that are different from the hypothesized generative model. We present a novel computational framework that allows for efficient inverse prediction using a Gaussian process approximation to generative models. Our framework enables scientific learning about how the latent processes co-vary with respect to covariates while simultaneously providing predictions of missing covariates. The proposed framework is capable of efficiently exploring the high dimensional, multi-modal latent spaces that arise in the inverse problem. To demonstrate flexibility, we apply our method in a generalized linear model framework to predict latent climate states given multivariate count data. Based on cross-validation, our model has predictive skill competitive with current methods while simultaneously providing formal, statistical inference on the underlying community dynamics of the biological system previously not available.

Stephen Berg, Jun Zhu, Murray K. Clayton, Monika E. Shea, David J. Mladenoff.

Source: The Annals of Applied Statistics, Volume 13, Number 4, 2312--2340.

Abstract:
The Wisconsin Public Land Survey database describes historical forest composition at high spatial resolution and is of interest in ecological studies of forest composition in Wisconsin just prior to significant Euro-American settlement. For such studies it is useful to identify recurring subpopulations of tree species known as communities, but standard clustering approaches for subpopulation identification do not account for dependence between spatially nearby observations. Here, we develop and fit a latent discrete Markov random field model for the purpose of identifying and classifying historical forest communities based on spatially referenced multivariate tree species counts across Wisconsin. We show empirically for the actual dataset and through simulation that our latent Markov random field modeling approach improves prediction and parameter estimation performance. For model fitting we introduce a new stochastic approximation algorithm which enables computationally efficient estimation and classification of large amounts of spatial multivariate count data.

Hongbin Zhang, Lang Wu.

Source: The Annals of Applied Statistics, Volume 13, Number 4, 2140--2157.

Abstract:
For a time-to-event outcome with censored time-varying covariates, a joint Cox model with a linear mixed effects model is the standard modeling approach. In some applications such as AIDS studies, mechanistic nonlinear models are available for some covariate process such as viral load during anti-HIV treatments, derived from the underlying data-generation mechanisms and disease progression. Such a mechanistic nonlinear covariate model may provide better-predicted values when the covariates are left censored or mismeasured. When the focus is on the impact of the time-varying covariate process on the survival outcome, an accelerated failure time (AFT) model provides an excellent alternative to the Cox proportional hazard model since an AFT model is formulated to allow the influence of the outcome by the entire covariate process. In this article, we consider a nonlinear mixed effects model for the censored covariates in an AFT model, implemented using a Monte Carlo EM algorithm, under the framework of a joint model for simultaneous inference. We apply the joint model to an HIV/AIDS data to gain insights for assessing the association between viral load and immunological restoration during antiretroviral therapy. Simulation is conducted to compare model performance when the covariate model and the survival model are misspecified.

Gabriela V. Cohen Freue, David Kepplinger, Matías Salibián-Barrera, Ezequiel Smucler.

Source: The Annals of Applied Statistics, Volume 13, Number 4, 2065--2090.

Abstract:
In large-scale quantitative proteomic studies, scientists measure the abundance of thousands of proteins from the human proteome in search of novel biomarkers for a given disease. Penalized regression estimators can be used to identify potential biomarkers among a large set of molecular features measured. Yet, the performance and statistical properties of these estimators depend on the loss and penalty functions used to define them. Motivated by a real plasma proteomic biomarkers study, we propose a new class of penalized robust estimators based on the elastic net penalty, which can be tuned to keep groups of correlated variables together in the selected model and maintain robustness against possible outliers. We also propose an efficient algorithm to compute our robust penalized estimators and derive a data-driven method to select the penalty term. Our robust penalized estimators have very good robustness properties and are also consistent under certain regularity conditions. Numerical results show that our robust estimators compare favorably to other robust penalized estimators. Using our proposed methodology for the analysis of the proteomics data, we identify new potentially relevant biomarkers of cardiac allograft vasculopathy that are not found with nonrobust alternatives. The selected model is validated in a new set of 52 test samples and achieves an area under the receiver operating characteristic (AUC) of 0.85.

Ye Zhang, Zhigang Yao, Patrik Forssén, Torgny Fornstedt.

Source: The Annals of Applied Statistics, Volume 13, Number 4, 2011--2042.

Abstract:
The means to obtain the rate constants of a chemical reaction is a fundamental open problem in both science and the industry. Traditional techniques for finding rate constants require either chemical modifications of the reactants or indirect measurements. The rate constant map method is a modern technique to study binding equilibrium and kinetics in chemical reactions. Finding a rate constant map from biosensor data is an ill-posed inverse problem that is usually solved by regularization. In this work, rather than finding a deterministic regularized rate constant map that does not provide uncertainty quantification of the solution, we develop an adaptive variational Bayesian approach to estimate the distribution of the rate constant map, from which some intrinsic properties of a chemical reaction can be explored, including information about rate constants. Our new approach is more realistic than the existing approaches used for biosensors and allows us to estimate the dynamics of the interactions, which are usually hidden in a deterministic approximate solution. We verify the performance of the new proposed method by numerical simulations, and compare it with the Markov chain Monte Carlo algorithm. The results illustrate that the variational method can reliably capture the posterior distribution in a computationally efficient way. Finally, the developed method is also tested on the real biosensor data (parathyroid hormone), where we provide two novel analysis tools—the thresholding contour map and the high order moment map—to estimate the number of interactions as well as their rate constants.

Johann Gagnon-Bartsch, Yotam Shem-Tov.

Source: The Annals of Applied Statistics, Volume 13, Number 3, 1464--1483.

Abstract:
The gold standard for identifying causal relationships is a randomized controlled experiment. In many applications in the social sciences and medicine, the researcher does not control the assignment mechanism and instead may rely upon natural experiments or matching methods as a substitute to experimental randomization. The standard testable implication of random assignment is covariate balance between the treated and control units. Covariate balance is commonly used to validate the claim of as good as random assignment. We propose a new nonparametric test of covariate balance. Our Classification Permutation Test (CPT) is based on a combination of classification methods (e.g., random forests) with Fisherian permutation inference. We revisit four real data examples and present Monte Carlo power simulations to demonstrate the applicability of the CPT relative to other nonparametric tests of equality of multivariate distributions.