Mohsen Maleki, Arezo Hajrajabi, Reinaldo B. Arellano-Valle.

Source: Brazilian Journal of Probability and Statistics, Volume 34, Number 2, 273--290.

Abstract:
In this paper, we study the finite mixtures of autoregressive processes assuming that the distribution of innovations (errors) belongs to the class of scale mixture of skew-normal (SMSN) distributions. The SMSN distributions allow a simultaneous modeling of the existence of outliers, heavy tails and asymmetries in the distribution of innovations. Therefore, a statistical methodology based on the SMSN family allows us to use a robust modeling on some non-linear time series with great flexibility, to accommodate skewness, heavy tails and heterogeneity simultaneously. The existence of convenient hierarchical representations of the SMSN distributions facilitates also the implementation of an ECME-type of algorithm to perform the likelihood inference in the considered model. Simulation studies and the application to a real data set are finally presented to illustrate the usefulness of the proposed model.

Zhengwei Liu, Qi Li, Fukang Zhu.

Source: Brazilian Journal of Probability and Statistics, Volume 34, Number 2, 251--272.

Abstract:
To predict time series of counts with small values and remarkable fluctuations, an available model is the $r$ states random environment process based on the negative binomial thinning operator and the geometric marginal. However, we argue that the aforementioned model may suffer from the following two drawbacks. First, under the condition of no prior information, the overdispersed property of the geometric distribution may cause the predictions fluctuate greatly. Second, because of the constraints on the model parameters, some estimated parameters are close to zero in real-data examples, which may not objectively reveal the correlation relationship. For the first drawback, an $r$ states random environment process based on the binomial thinning operator and the Poisson marginal is introduced. For the second drawback, we propose a generalized $r$ states random environment integer-valued autoregressive model based on the binomial thinning operator to model fluctuations of data. Yule–Walker and conditional maximum likelihood estimates are considered and their performances are assessed via simulation studies. Two real-data sets are conducted to illustrate the better performances of the proposed models compared with some existing models.

Saralees Nadarajah, Yuancheng Si.

Source: Brazilian Journal of Probability and Statistics, Volume 34, Number 1, 183--187.

Abstract:
Da Paz, Balakrishnan and Bazan [Braz. J. Probab. Stat. 33 (2019), 455–479] introduced the L-logistic distribution, studied its properties including estimation issues and illustrated a data application. This note derives a closed form expression for moment properties of the distribution. Some computational issues are discussed.

Flávio B. Gonçalves, Marcos O. Prates, Victor Hugo Lachos.

Source: Brazilian Journal of Probability and Statistics, Volume 34, Number 1, 51--70.

Abstract:
In this paper, we present a novel methodology to perform Bayesian model selection in linear models with heavy-tailed distributions. We consider a finite mixture of distributions to model a latent variable where each component of the mixture corresponds to one possible model within the symmetrical class of normal independent distributions. Naturally, the Gaussian model is one of the possibilities. This allows for a simultaneous analysis based on the posterior probability of each model. Inference is performed via Markov chain Monte Carlo—a Gibbs sampler with Metropolis–Hastings steps for a class of parameters. Simulated examples highlight the advantages of this approach compared to a segregated analysis based on arbitrarily chosen model selection criteria. Examples with real data are presented and an extension to censored linear regression is introduced and discussed.

Fábio P. Lima, Francisco Cribari-Neto.

Source: Brazilian Journal of Probability and Statistics, Volume 34, Number 1, 18--34.

Abstract:
We address the issue of performing testing inference in small samples in the class of beta regression models. We consider the likelihood ratio test and its standard bootstrap version. We also consider two alternative resampling-based tests. One of them uses the bootstrap test statistic replicates to numerically estimate a Bartlett correction factor that can be applied to the likelihood ratio test statistic. By doing so, we avoid estimation of quantities located in the tail of the likelihood ratio test statistic null distribution. The second alternative resampling-based test uses a fast double bootstrap scheme in which a single second level bootstrapping resample is performed for each first level bootstrap replication. It delivers accurate testing inferences at a computational cost that is considerably smaller than that of a standard double bootstrapping scheme. The Monte Carlo results we provide show that the standard likelihood ratio test tends to be quite liberal in small samples. They also show that the bootstrap tests deliver accurate testing inferences even when the sample size is quite small. An empirical application is also presented and discussed.

Wesley Bertoli, Katiane S. Conceição, Marinho G. Andrade, Francisco Louzada.

Source: Brazilian Journal of Probability and Statistics, Volume 33, Number 4, 826--860.

Abstract:
The primary goal of this paper is to introduce the zero-modified Poisson–Lindley regression model as an alternative to model overdispersed count data exhibiting inflation or deflation of zeros in the presence of covariates. The zero-modification is incorporated by considering that a zero-truncated process produces positive observations and consequently, the proposed model can be fitted without any previous information about the zero-modification present in a given dataset. A fully Bayesian approach based on the g-prior method has been considered for inference concerns. An intensive Monte Carlo simulation study has been conducted to evaluate the performance of the developed methodology and the maximum likelihood estimators. The proposed model was considered for the analysis of a real dataset on the number of bids received by $126$ U.S. firms between 1978–1985, and the impact of choosing different prior distributions for the regression coefficients has been studied. A sensitivity analysis to detect influential points has been performed based on the Kullback–Leibler divergence. A general comparison with some well-known regression models for discrete data has been presented.

Flávio B. Gonçalves, Bárbara C. C. Dias.

Source: Brazilian Journal of Probability and Statistics, Volume 33, Number 4, 782--800.

Abstract:
Educational assessment usually considers a contextual questionnaire to extract relevant information from the applicants. This may include items related to socio-economical profile as well as items to extract other characteristics potentially related to applicant’s performance in the test. A careful analysis of the questionnaires jointly with the test’s results may evidence important relations between profiles and test performance. The most coherent way to perform this task in a statistical context is to use the information from the questionnaire to help explain the variability of the abilities in a joint model-based approach. Nevertheless, the responses to the questionnaire typically present missing values which, in some cases, may be missing not at random. This paper proposes a statistical methodology to model the abilities in dichotomous IRT models using the information of the contextual questionnaires via linear regression. The proposed methodology models the missing data jointly with the all the observed data, which allows for the estimation of the former. The missing data modelling is flexible enough to allow the specification of missing not at random structures. Furthermore, even if those structures are not assumed a priori, they can be estimated from the posterior results when assuming missing (completely) at random structures a priori. Statistical inference is performed under the Bayesian paradigm via an efficient MCMC algorithm. Simulated and real examples are presented to investigate the efficiency and applicability of the proposed methodology.

Marco A. R. Ferreira.

Source: Brazilian Journal of Probability and Statistics, Volume 33, Number 4, 734--744.

Abstract:
We study the limiting behavior of the one-at-a-time Gibbs sampler for the intrinsic conditional autoregressive model with centering on the fly. The intrinsic conditional autoregressive model is widely used as a prior for random effects in hierarchical models for spatial modeling. This model is defined by full conditional distributions that imply an improper joint “density” with a multivariate Gaussian kernel and a singular precision matrix. To guarantee propriety of the posterior distribution, usually at the end of each iteration of the Gibbs sampler the random effects are centered to sum to zero in what is widely known as centering on the fly. While this works well in practice, this informal computational way to recenter the random effects obscures their implied prior distribution and prevents the development of formal Bayesian procedures. Here we show that the implied prior distribution, that is, the limiting distribution of the one-at-a-time Gibbs sampler for the intrinsic conditional autoregressive model with centering on the fly is a singular Gaussian distribution with a covariance matrix that is the Moore–Penrose inverse of the precision matrix. This result has important implications for the development of formal Bayesian procedures such as reference priors and Bayes-factor-based model selection for spatial models.

Dani Gamerman.

Source: Brazilian Journal of Probability and Statistics, Volume 33, Number 4, 686--705.

Abstract:
Point processes are one of the most commonly encountered observation processes in Spatial Statistics. Model-based inference for them depends on the likelihood function. In the most standard setting of Poisson processes, the likelihood depends on the intensity function, and can not be computed analytically. A number of approximating techniques have been proposed to handle this difficulty. In this paper, we review recent work on exact solutions that solve this problem without resorting to approximations. The presentation concentrates more heavily on discrete time but also considers continuous time. The solutions are based on model specifications that impose smoothness constraints on the intensity function. We also review approaches to include a regression component and different ways to accommodate it while accounting for additional heterogeneity. Applications are provided to illustrate the results. Finally, we discuss possible extensions to account for discontinuities and/or jumps in the intensity function.

Rosineide F. da Paz, Narayanaswamy Balakrishnan, Jorge Luis Bazán.

Source: Brazilian Journal of Probability and Statistics, Volume 33, Number 3, 455--479.

Abstract:
Tadikamalla and Johnson [ Biometrika 69 (1982) 461–465] developed the $L_{B}$ distribution to variables with bounded support by considering a transformation of the standard Logistic distribution. In this manuscript, a convenient parametrization of this distribution is proposed in order to develop regression models. This distribution, referred to here as L-Logistic distribution, provides great flexibility and includes the uniform distribution as a particular case. Several properties of this distribution are studied, and a Bayesian approach is adopted for the parameter estimation. Simulation studies, considering prior sensitivity analysis, recovery of parameters and comparison of algorithms, and robustness to outliers are all discussed showing that the results are insensitive to the choice of priors, efficiency of the algorithm MCMC adopted, and robustness of the model when compared with the beta distribution. Applications to estimate the vulnerability to poverty and to explain the anxiety are performed. The results to applications show that the L-Logistic regression models provide a better fit than the corresponding beta regression models.

Luisa Rivas, Manuel Galea.

Source: Brazilian Journal of Probability and Statistics, Volume 33, Number 2, 402--424.

Abstract:
In this paper, we present a regression model where the response variable is a count data that follows a Waring distribution. The Waring regression model allows for analysis of phenomena where the Geometric regression model is inadequate, because the probability of success on each trial, $p$, is different for each individual and $p$ has an associated distribution. Estimation is performed by maximum likelihood, through the maximization of the $Q$-function using EM algorithm. Diagnostic measures are calculated for this model. To illustrate the results, an application to real data is presented. Some specific details are given in the Appendix of the paper.

Francisco Cribari-Neto, Rodney V. Fonseca.

Source: Brazilian Journal of Probability and Statistics, Volume 33, Number 2, 329--355.

Abstract:
The log-linear Birnbaum–Saunders model has been widely used in empirical applications. We introduce an extension of this model based on a recently proposed version of the Birnbaum–Saunders distribution which is more flexible than the standard Birnbaum–Saunders law since its density may assume both unimodal and bimodal shapes. We show how to perform point estimation, interval estimation and hypothesis testing inferences on the parameters that index the regression model we propose. We also present a number of diagnostic tools, such as residual analysis, local influence, generalized leverage, generalized Cook’s distance and model misspecification tests. We investigate the usefulness of model selection criteria and the accuracy of prediction intervals for the proposed model. Results of Monte Carlo simulations are presented. Finally, we also present and discuss an empirical application.

Alain Desgagné, Philippe Gagnon.

Source: Brazilian Journal of Probability and Statistics, Volume 33, Number 2, 205--221.

Abstract:
Whole robustness is a nice property to have for statistical models. It implies that the impact of outliers gradually vanishes as they approach plus or minus infinity. So far, the Bayesian literature provides results that ensure whole robustness for the location-scale model. In this paper, we make two contributions. First, we generalise the results to attain whole robustness in simple linear regression through the origin, which is a necessary step towards results for general linear regression models. We allow the variance of the error term to depend on the explanatory variable. This flexibility leads to the second contribution: we provide a simple Bayesian approach to robustly estimate finite population means and ratios. The strategy to attain whole robustness is simple since it lies in replacing the traditional normal assumption on the error term by a super heavy-tailed distribution assumption. As a result, users can estimate the parameters as usual, using the posterior distribution.

John J. Dziak, Donna L. Coffman, Matthew Reimherr, Justin Petrovich, Runze Li, Saul Shiffman, Mariya P. Shiyko.

Source: Statistics Surveys, Volume 13, 150--180.

Abstract:
Researchers are sometimes interested in predicting a distal or external outcome (such as smoking cessation at follow-up) from the trajectory of an intensively recorded longitudinal variable (such as urge to smoke). This can be done in a semiparametric way via scalar-on-function regression. However, the resulting fitted coefficient regression function requires special care for correct interpretation, as it represents the joint relationship of time points to the outcome, rather than a marginal or cross-sectional relationship. We provide practical guidelines, based on experience with scientific applications, for helping practitioners interpret their results and illustrate these ideas using data from a smoking cessation study.

Solveig Engebretsen, Ingrid K. Glad.

Source: Statistics Surveys, Volume 13, 1--51.

Abstract:
In numerous problems where the aim is to estimate the effect of a predictor variable on a response, one can assume a monotone relationship. For example, dose-effect models in medicine are of this type. In a multiple regression setting, additive monotone regression models assume that each predictor has a monotone effect on the response. In this paper, we present an overview and comparison of very recent frequentist methods for fitting additive monotone regression models. Three of the methods we present can be used both in the high dimensional setting, where the number of parameters $p$ exceeds the number of observations $n$, and in the classical multiple setting where $1<pleq n$. However, many of the most recent methods only apply to the classical setting. The methods are compared through simulation experiments in terms of efficiency, prediction error and variable selection properties in both settings, and they are applied to the Boston housing data. We conclude with some recommendations on when the various methods perform best.

Phillip S. Kott.

Source: Statistics Surveys, Volume 12, 1--17.

Abstract:
Fitting complex survey data to regression equations is explored under a design-sensitive model-based framework. A robust version of the standard model assumes that the expected value of the difference between the dependent variable and its model-based prediction is zero no matter what the values of the explanatory variables. The extended model assumes only that the difference is uncorrelated with the covariates. Little is assumed about the error structure of this difference under either model other than independence across primary sampling units. The standard model often fails in practice, but the extended model very rarely does. Under this framework some of the methods developed in the conventional design-based, pseudo-maximum-likelihood framework, such as fitting weighted estimating equations and sandwich mean-squared-error estimation, are retained but their interpretations change. Few of the ideas here are new to the refereed literature. The goal instead is to collect those ideas and put them into a unified conceptual framework.

Mariella Dimiccoli.

Source: Statistics Surveys, Volume 10, 53--99.

Abstract:
Cone regression is a particular case of quadratic programming that minimizes a weighted sum of squared residuals under a set of linear inequality constraints. Several important statistical problems such as isotonic, concave regression or ANOVA under partial orderings, just to name a few, can be considered as particular instances of the cone regression problem. Given its relevance in Statistics, this paper aims to address the fundamentals of cone regression from a theoretical and practical point of view. Several formulations of the cone regression problem are considered and, focusing on the particular case of concave regression as an example, several algorithms are analyzed and compared both qualitatively and quantitatively through numerical simulations. Several improvements to enhance numerical stability and bound the computational cost are proposed. For each analyzed algorithm, the pseudo-code and its corresponding code in Matlab are provided. The results from this study demonstrate that the choice of the optimization approach strongly impacts the numerical performances. It is also shown that methods are not currently available to solve efficiently cone regression problems with large dimension (more than many thousands of points). We suggest further research to fill this gap by exploiting and adapting classical multi-scale strategy to compute an approximate solution.

Gery Geenens

Source: Statist. Surv., Volume 5, 30--43.

Abstract:
Recently, some nonparametric regression ideas have been extended to the case of functional regression. Within that framework, the main concern arises from the infinite dimensional nature of the explanatory objects. Specifically, in the classical multivariate regression context, it is well-known that any nonparametric method is affected by the so-called “curse of dimensionality”, caused by the sparsity of data in high-dimensional spaces, resulting in a decrease in fastest achievable rates of convergence of regression function estimators toward their target curve as the dimension of the regressor vector increases. Therefore, it is not surprising to find dramatically bad theoretical properties for the nonparametric functional regression estimators, leading many authors to condemn the methodology. Nevertheless, a closer look at the meaning of the functional data under study and on the conclusions that the statistician would like to draw from it allows to consider the problem from another point-of-view, and to justify the use of slightly modified estimators. In most cases, it can be entirely legitimate to measure the proximity between two elements of the infinite dimensional functional space via a semi-metric, which could prevent those estimators suffering from what we will call the “curse of infinite dimensionality”.

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Arctic sea ice extent (SIE) in September 2019 ranked second-to-lowest in history and is trending downward. The understanding of how internal variability amplifies the effects of external $ ext{CO}_2$ forcing is still limited. We propose the VARCTIC, which is a Vector Autoregression (VAR) designed to capture and extrapolate Arctic feedback loops. VARs are dynamic simultaneous systems of equations, routinely estimated to predict and understand the interactions of multiple macroeconomic time series. Hence, the VARCTIC is a parsimonious compromise between fullblown climate models and purely statistical approaches that usually offer little explanation of the underlying mechanism. Our "business as usual" completely unconditional forecast has SIE hitting 0 in September by the 2060s. Impulse response functions reveal that anthropogenic $ ext{CO}_2$ emission shocks have a permanent effect on SIE - a property shared by no other shock. Further, we find Albedo- and Thickness-based feedbacks to be the main amplification channels through which $ ext{CO}_2$ anomalies impact SIE in the short/medium run. Conditional forecast analyses reveal that the future path of SIE crucially depends on the evolution of $ ext{CO}_2$ emissions, with outcomes ranging from recovering SIE to it reaching 0 in the 2050s. Finally, Albedo and Thickness feedbacks are shown to play an important role in accelerating the speed at which predicted SIE is heading towards 0.

In this paper we propose a bimodal gamma distribution using a quadratic transformation based on the alpha-skew-normal model. We discuss several properties of this distribution such as mean, variance, moments, hazard rate and entropy measures. Further, we propose a new regression model with censored data based on the bimodal gamma distribution. This regression model can be very useful to the analysis of real data and could give more realistic fits than other special regression models. Monte Carlo simulations were performed to check the bias in the maximum likelihood estimation. The proposed models are applied to two real data sets found in literature.

In the Gaussian sequence model $Y= heta_0 + varepsilon$ in $mathbb{R}^n$, we study the fundamental limit of approximating the signal $ heta_0$ by a class $Theta(d,d_0,k)$ of (generalized) splines with free knots. Here $d$ is the degree of the spline, $d_0$ is the order of differentiability at each inner knot, and $k$ is the maximal number of pieces. We show that, given any integer $dgeq 0$ and $d_0in{-1,0,ldots,d-1}$, the minimax rate of estimation over $Theta(d,d_0,k)$ exhibits the following phase transition: egin{equation*} egin{aligned} inf_{widetilde{ heta}}sup_{ hetainTheta(d,d_0, k)}mathbb{E}_ heta|widetilde{ heta} - heta|^2 asymp_d egin{cases} kloglog(16n/k), & 2leq kleq k_0,\ klog(en/k), & k geq k_0+1. end{cases} end{aligned} end{equation*} The transition boundary $k_0$, which takes the form $lfloor{(d+1)/(d-d_0) floor} + 1$, demonstrates the critical role of the regularity parameter $d_0$ in the separation between a faster $log log(16n)$ and a slower $log(en)$ rate. We further show that, once encouraging an additional '$d$-monotonicity' shape constraint (including monotonicity for $d = 0$ and convexity for $d=1$), the above phase transition is eliminated and the faster $kloglog(16n/k)$ rate can be achieved for all $k$. These results provide theoretical support for developing $ell_0$-penalized (shape-constrained) spline regression procedures as useful alternatives to $ell_1$- and $ell_2$-penalized ones.

Disaggregation regression has become an important tool in spatial disease mapping for making fine-scale predictions of disease risk from aggregated response data. By including high resolution covariate information and modelling the data generating process on a fine scale, it is hoped that these models can accurately learn the relationships between covariates and response at a fine spatial scale. However, validating these high resolution predictions can be a challenge, as often there is no data observed at this spatial scale. In this study, disaggregation regression was performed on simulated data in various settings and the resulting fine-scale predictions are compared to the simulated ground truth. Performance was investigated with varying numbers of data points, sizes of aggregated areas and levels of model misspecification. The effectiveness of cross validation on the aggregate level as a measure of fine-scale predictive performance was also investigated. Predictive performance improved as the number of observations increased and as the size of the aggregated areas decreased. When the model was well-specified, fine-scale predictions were accurate even with small numbers of observations and large aggregated areas. Under model misspecification predictive performance was significantly worse for large aggregated areas but remained high when response data was aggregated over smaller regions. Cross-validation correlation on the aggregate level was a moderately good predictor of fine-scale predictive performance. While the simulations are unlikely to capture the nuances of real-life response data, this study gives insight into the effectiveness of disaggregation regression in different contexts.

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.

Machine learning models with both good predictability and high interpretability are crucial for decision support systems. Linear regression is one of the most interpretable prediction models. However, the linearity in a simple linear regression worsens its predictability. In this work, we introduce a locally adaptive interpretable regression (LoAIR). In LoAIR, a metamodel parameterized by neural networks predicts percentile of a Gaussian distribution for the regression coefficients for a rapid adaptation. Our experimental results on public benchmark datasets show that our model not only achieves comparable or better predictive performance than the other state-of-the-art baselines but also discovers some interesting relationships between input and target variables such as a parabolic relationship between CO2 emissions and Gross National Product (GNP). Therefore, LoAIR is a step towards bridging the gap between econometrics, statistics, and machine learning by improving the predictive ability of linear regression without depreciating its interpretability.

An accurate and prompt diagnosis of pediatric pneumonia is imperative for successful treatment intervention. One approach to diagnose pneumonia cases is using radiographic data. In this article, we propose a novel parsimonious scalar-on-image classification model adopting the ideas of functional data analysis. Our main idea is to treat images as functional measurements and exploit underlying covariance structures to select basis functions; these bases are then used in approximating both image profiles and corresponding regression coefficient. We re-express the regression model into a standard generalized linear model where the functional principal component scores are treated as covariates. We apply the method to (1) classify pneumonia against healthy and viral against bacterial pneumonia patients, and (2) test the null effect about the association between images and responses. Extensive simulation studies show excellent numerical performance in terms of classification, hypothesis testing, and efficient computation.

Ridge regression (RR) is a regularization technique that penalizes the L2-norm of the coefficients in linear regression. One of the challenges of using RR is the need to set a hyperparameter ($alpha$) that controls the amount of regularization. Cross-validation is typically used to select the best $alpha$ from a set of candidates. However, efficient and appropriate selection of $alpha$ can be challenging, particularly where large amounts of data are analyzed. Because the selected $alpha$ depends on the scale of the data and predictors, it is not straightforwardly interpretable. Here, we propose to reparameterize RR in terms of the ratio $gamma$ between the L2-norms of the regularized and unregularized coefficients. This approach, called fractional RR (FRR), has several benefits: the solutions obtained for different $gamma$ are guaranteed to vary, guarding against wasted calculations, and automatically span the relevant range of regularization, avoiding the need for arduous manual exploration. We provide an algorithm to solve FRR, as well as open-source software implementations in Python and MATLAB (https://github.com/nrdg/fracridge). We show that the proposed method is fast and scalable for large-scale data problems, and delivers results that are straightforward to interpret and compare across models and datasets.

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.

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.

Emmanuel J. Candès, Pragya Sur.

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

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

Xi Chen, Weidong Liu, Yichen Zhang.

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

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

Ke Zhu.

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

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

Xin Bing, Marten H. Wegkamp.

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

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

Long Feng, Cun-Hui Zhang.

Source: The Annals of Statistics, Volume 47, Number 6, 3069--3098.

Abstract:
The Lasso is biased. Concave penalized least squares estimation (PLSE) takes advantage of signal strength to reduce this bias, leading to sharper error bounds in prediction, coefficient estimation and variable selection. For prediction and estimation, the bias of the Lasso can be also reduced by taking a smaller penalty level than what selection consistency requires, but such smaller penalty level depends on the sparsity of the true coefficient vector. The sorted $ell_{1}$ penalized estimation (Slope) was proposed for adaptation to such smaller penalty levels. However, the advantages of concave PLSE and Slope do not subsume each other. We propose sorted concave penalized estimation to combine the advantages of concave and sorted penalizations. We prove that sorted concave penalties adaptively choose the smaller penalty level and at the same time benefits from signal strength, especially when a significant proportion of signals are stronger than the corresponding adaptively selected penalty levels. A local convex approximation for sorted concave penalties, which extends the local linear and quadratic approximations for separable concave penalties, is developed to facilitate the computation of sorted concave PLSE and proven to possess desired prediction and estimation error bounds. Our analysis of prediction and estimation errors requires the restricted eigenvalue condition on the design, not beyond, and provides selection consistency under a required minimum signal strength condition in addition. Thus, our results also sharpens existing results on concave PLSE by removing the upper sparse eigenvalue component of the sparse Riesz condition.

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.

Qiyang Han, Tengyao Wang, Sabyasachi Chatterjee, Richard J. Samworth.

Source: The Annals of Statistics, Volume 47, Number 5, 2440--2471.

Abstract:
We study the least squares regression function estimator over the class of real-valued functions on $[0,1]^{d}$ that are increasing in each coordinate. For uniformly bounded signals and with a fixed, cubic lattice design, we establish that the estimator achieves the minimax rate of order $n^{-min{2/(d+2),1/d}}$ in the empirical $L_{2}$ loss, up to polylogarithmic factors. Further, we prove a sharp oracle inequality, which reveals in particular that when the true regression function is piecewise constant on $k$ hyperrectangles, the least squares estimator enjoys a faster, adaptive rate of convergence of $(k/n)^{min(1,2/d)}$, again up to polylogarithmic factors. Previous results are confined to the case $dleq2$. Finally, we establish corresponding bounds (which are new even in the case $d=2$) in the more challenging random design setting. There are two surprising features of these results: first, they demonstrate that it is possible for a global empirical risk minimisation procedure to be rate optimal up to polylogarithmic factors even when the corresponding entropy integral for the function class diverges rapidly; second, they indicate that the adaptation rate for shape-constrained estimators can be strictly worse than the parametric rate.

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.

Qiyang Han, Jon A. Wellner.

Source: The Annals of Statistics, Volume 47, Number 4, 2286--2319.

Abstract:
We study the performance of the least squares estimator (LSE) in a general nonparametric regression model, when the errors are independent of the covariates but may only have a $p$th moment ($pgeq1$). In such a heavy-tailed regression setting, we show that if the model satisfies a standard “entropy condition” with exponent $alphain(0,2)$, then the $L_{2}$ loss of the LSE converges at a rate [mathcal{O}_{mathbf{P}}igl(n^{-frac{1}{2+alpha}}vee n^{-frac{1}{2}+frac{1}{2p}}igr).] Such a rate cannot be improved under the entropy condition alone. This rate quantifies both some positive and negative aspects of the LSE in a heavy-tailed regression setting. On the positive side, as long as the errors have $pgeq1+2/alpha$ moments, the $L_{2}$ loss of the LSE converges at the same rate as if the errors are Gaussian. On the negative side, if $p<1+2/alpha$, there are (many) hard models at any entropy level $alpha$ for which the $L_{2}$ loss of the LSE converges at a strictly slower rate than other robust estimators. The validity of the above rate relies crucially on the independence of the covariates and the errors. In fact, the $L_{2}$ loss of the LSE can converge arbitrarily slowly when the independence fails. The key technical ingredient is a new multiplier inequality that gives sharp bounds for the “multiplier empirical process” associated with the LSE. We further give an application to the sparse linear regression model with heavy-tailed covariates and errors to demonstrate the scope of this new inequality.

Benedikt Bauer, Michael Kohler.

Source: The Annals of Statistics, Volume 47, Number 4, 2261--2285.

Abstract:
Assuming that a smoothness condition and a suitable restriction on the structure of the regression function hold, it is shown that least squares estimates based on multilayer feedforward neural networks are able to circumvent the curse of dimensionality in nonparametric regression. The proof is based on new approximation results concerning multilayer feedforward neural networks with bounded weights and a bounded number of hidden neurons. The estimates are compared with various other approaches by using simulated data.

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.

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.

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.

Chih-Yuan Hsu, Yi-Hau Chen, Ruoh-Rong Yu, Tsung-Wei Hung.

Source: The Annals of Applied Statistics, Volume 14, Number 1, 160--177.

Abstract:
Workers in Taiwan overall have been suffering from long-lasting wage stagnation since the mid-1990s. In particular, there seems to be little mobility for the wages of Taiwanese workers to transit across wage quantile groups. It is of interest to see if certain groups of workers, such as female, lower educated and younger generation workers, suffer from the problem more seriously than the others. This work tries to apply a systematic statistical approach to study this issue, based on the longitudinal data from the Panel Study of Family Dynamics (PSFD) survey conducted in Taiwan since 1999. We propose the quantile transition regression model, generalizing recent methodology for quantile association, to assess the wage status transition with respect to the marginal wage quantiles over time as well as the effects of certain demographic and job factors on the wage status transition. Estimation of the model can be based on the composite likelihoods utilizing the binary, or ordinal-data information regarding the quantile transition, with the associated asymptotic theory established. A goodness-of-fit procedure for the proposed model is developed. The performances of the estimation and the goodness-of-fit procedures for the quantile transition model are illustrated through simulations. The application of the proposed methodology to the PSFD survey data suggests that female, private-sector workers with higher age and education below postgraduate level suffer from more severe wage status stagnation than the others.

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.

Emily Berg, Danhyang Lee.

Source: The Annals of Applied Statistics, Volume 13, Number 4, 2158--2188.

Abstract:
Quantiles of the distributions of several measures of erosion are important parameters in the Conservation Effects Assessment Project, a survey intended to quantify soil and nutrient loss on crop fields. Because sample sizes for domains of interest are too small to support reliable direct estimators, model based methods are needed. Quantile regression is appealing for CEAP because finding a single family of parametric models that adequately describes the distributions of all variables is difficult and small area quantiles are parameters of interest. We construct empirical Bayes predictors and bootstrap mean squared error estimators based on the linearly interpolated generalized Pareto distribution (LIGPD). We apply the procedures to predict county-level quantiles for four types of erosion in Wisconsin and validate the procedures through simulation.

Bret Zeldow, Vincent Lo Re III, Jason Roy.

Source: The Annals of Applied Statistics, Volume 13, Number 3, 1989--2010.

Abstract:
Bayesian Additive Regression Trees (BART) is a flexible machine learning algorithm capable of capturing nonlinearities between an outcome and covariates and interactions among covariates. We extend BART to a semiparametric regression framework in which the conditional expectation of an outcome is a function of treatment, its effect modifiers, and confounders. The confounders are allowed to have unspecified functional form, while treatment and effect modifiers that are directly related to the research question are given a linear form. The result is a Bayesian semiparametric linear regression model where the posterior distribution of the parameters of the linear part can be interpreted as in parametric Bayesian regression. This is useful in situations where a subset of the variables are of substantive interest and the others are nuisance variables that we would like to control for. An example of this occurs in causal modeling with the structural mean model (SMM). Under certain causal assumptions, our method can be used as a Bayesian SMM. Our methods are demonstrated with simulation studies and an application to dataset involving adults with HIV/Hepatitis C coinfection who newly initiate antiretroviral therapy. The methods are available in an R package called semibart.

Gaoxiang Jia, Xinlei Wang, Qiwei Li, Wei Lu, Ximing Tang, Ignacio Wistuba, Yang Xie.

Source: The Annals of Applied Statistics, Volume 13, Number 3, 1617--1647.

Abstract:
Formalin-fixed paraffin-embedded (FFPE) samples have great potential for biomarker discovery, retrospective studies and diagnosis or prognosis of diseases. Their application, however, is hindered by the unsatisfactory performance of traditional gene expression profiling techniques on damaged RNAs. NanoString nCounter platform is well suited for profiling of FFPE samples and measures gene expression with high sensitivity which may greatly facilitate realization of scientific and clinical values of FFPE samples. However, methodological development for normalization, a critical step when analyzing this type of data, is far behind. Existing methods designed for the platform use information from different types of internal controls separately and rely on an overly-simplified assumption that expression of housekeeping genes is constant across samples for global scaling. Thus, these methods are not optimized for the nCounter system, not mentioning that they were not developed for FFPE samples. We construct an integrated system of random-coefficient hierarchical regression models to capture main patterns and characteristics observed from NanoString data of FFPE samples and develop a Bayesian approach to estimate parameters and normalize gene expression across samples. Our method, labeled RCRnorm, incorporates information from all aspects of the experimental design and simultaneously removes biases from various sources. It eliminates the unrealistic assumption on housekeeping genes and offers great interpretability. Furthermore, it is applicable to freshly frozen or like samples that can be generally viewed as a reduced case of FFPE samples. Simulation and applications showed the superior performance of RCRnorm.

Ying Chen, J. S. Marron, Jiejie Zhang.

Source: The Annals of Applied Statistics, Volume 13, Number 3, 1590--1616.

Abstract:
Electricity prices are high dimensional, serially dependent and have seasonal variations. We propose a Warping Functional AutoRegressive (WFAR) model that simultaneously accounts for the cross time-dependence and seasonal variations of the large dimensional data. In particular, electricity price curves are obtained by smoothing over the $24$ discrete hourly prices on each day. In the functional domain, seasonal phase variations are separated from level amplitude changes in a warping process with the Fisher–Rao distance metric, and the aligned (season-adjusted) electricity price curves are modeled in the functional autoregression framework. In a real application, the WFAR model provides superior out-of-sample forecast accuracy in both a normal functioning market, Nord Pool, and an extreme situation, the California market. The forecast performance as well as the relative accuracy improvement are stable for different markets and different time periods.

Lisa Schlosser, Torsten Hothorn, Reto Stauffer, Achim Zeileis.

Source: The Annals of Applied Statistics, Volume 13, Number 3, 1564--1589.

Abstract:
To obtain a probabilistic model for a dependent variable based on some set of explanatory variables, a distributional approach is often adopted where the parameters of the distribution are linked to regressors. In many classical models this only captures the location of the distribution but over the last decade there has been increasing interest in distributional regression approaches modeling all parameters including location, scale and shape. Notably, so-called nonhomogeneous Gaussian regression (NGR) models both mean and variance of a Gaussian response and is particularly popular in weather forecasting. Moreover, generalized additive models for location, scale and shape (GAMLSS) provide a framework where each distribution parameter is modeled separately capturing smooth linear or nonlinear effects. However, when variable selection is required and/or there are nonsmooth dependencies or interactions (especially unknown or of high-order), it is challenging to establish a good GAMLSS. A natural alternative in these situations would be the application of regression trees or random forests but, so far, no general distributional framework is available for these. Therefore, a framework for distributional regression trees and forests is proposed that blends regression trees and random forests with classical distributions from the GAMLSS framework as well as their censored or truncated counterparts. To illustrate these novel approaches in practice, they are employed to obtain probabilistic precipitation forecasts at numerous sites in a mountainous region (Tyrol, Austria) based on a large number of numerical weather prediction quantities. It is shown that the novel distributional regression forests automatically select variables and interactions, performing on par or often even better than GAMLSS specified either through prior meteorological knowledge or a computationally more demanding boosting approach.

Bo Ning, Seonghyun Jeong, Subhashis Ghosal.

Source: Bernoulli, Volume 26, Number 3, 2353--2382.

Abstract:
We study frequentist properties of a Bayesian high-dimensional multivariate linear regression model with correlated responses. The predictors are separated into many groups and the group structure is pre-determined. Two features of the model are unique: (i) group sparsity is imposed on the predictors; (ii) the covariance matrix is unknown and its dimensions can also be high. We choose a product of independent spike-and-slab priors on the regression coefficients and a new prior on the covariance matrix based on its eigendecomposition. Each spike-and-slab prior is a mixture of a point mass at zero and a multivariate density involving the $ell_{2,1}$-norm. We first obtain the posterior contraction rate, the bounds on the effective dimension of the model with high posterior probabilities. We then show that the multivariate regression coefficients can be recovered under certain compatibility conditions. Finally, we quantify the uncertainty for the regression coefficients with frequentist validity through a Bernstein–von Mises type theorem. The result leads to selection consistency for the Bayesian method. We derive the posterior contraction rate using the general theory by constructing a suitable test from the first principle using moment bounds for certain likelihood ratios. This leads to posterior concentration around the truth with respect to the average Rényi divergence of order $1/2$. This technique of obtaining the required tests for posterior contraction rate could be useful in many other problems.