In spatial statistics, the screening effect historically refers to the situation when the observations located far from the predictand receive a small (ideally, zero) kriging weight. Several factors play a crucial role in this phenomenon: among them, the spatial design, the dimension of the spatial domain where the observations are defined, the mean-square properties of the underlying random field and its covariance function or, equivalently, its spectral density.

The tour de force by Michael L. Stein provides a formal definition of the screening effect and puts emphasis on the Matérn covariance function, advocated as a good covariance function to yield such an effect. Yet, it is often recommended not to use covariance functions with a compact support. This paper shows that some classes of covariance functions being compactly supported allow for a screening effect according to Stein’s definition, in both regular and irregular settings of the spatial design. Further, numerical experiments suggest that the screening effect under a class of compactly supported covariance functions is even stronger than the screening effect under a Matérn model.

With the large amounts of modern financial and econometric data available from disparate informational sources, it becomes increasingly critical to develop inferential tools for the impact of exogenous factors on volatility of financial time series. We develop a new Local Covariate Trend test (LOCOT) for the significance of an exogenous covariate in the autoregressive conditional heteroscedastic volatility model, where the covariate effect can be nonlinear. The new LOCOT statistic is based on an artificial high-dimensional one-way ANOVA where the number of factor levels increases with the sample size. We derive asymptotic properties of the new LOCOT statistic and show its competitive finite sample performance in a broad range of simulation studies. We illustrate utility of the new testing approach in application to volatility analysis of three major cryptoassets and their relationship with the prices of gold and the S&P500 index.

Robust methods, though ubiquitous in practice, are yet to be fully understood in the context of regularized estimation and high dimensions. Even simple questions become challenging very quickly. For example, classical statistical theory identifies equivalence between model-averaged and composite quantile estimation. However, little to nothing is known about such equivalence between methods that encourage sparsity. This paper provides a toolbox to further study robustness in these settings and focuses on prediction. In particular, we study optimally weighted model-averaged as well as composite $l_{1}$-regularized estimation. Optimal weights are determined by minimizing the asymptotic mean squared error. This approach incorporates the effects of regularization, without the assumption of perfect selection, as is often used in practice. Such weights are then optimal for prediction quality. Through an extensive simulation study, we show that no single method systematically outperforms others. We find, however, that model-averaged and composite quantile estimators often outperform least-squares methods, even in the case of Gaussian model noise. Real data application witnesses the method’s practical use through the reconstruction of compressed audio signals.

We study minimax estimation of two-dimensional totally positive distributions. Such distributions pertain to pairs of strongly positively dependent random variables and appear frequently in statistics and probability. In particular, for distributions with $\beta$-Hölder smooth densities where $\beta\in(0,2)$, we observe polynomially faster minimax rates of estimation when, additionally, the total positivity condition is imposed. Moreover, we demonstrate fast algorithms to compute the proposed estimators and corroborate the theoretical rates of estimation by simulation studies.

We consider the problem of estimating the mean of a distribution supported by the $k$-dimensional probability simplex in the setting where an $\varepsilon$ fraction of observations are subject to adversarial corruption. A simple particular example is the problem of estimating the distribution of a discrete random variable. Assuming that the discrete variable takes $k$ values, the unknown parameter $\boldsymbol{\theta}$ is a $k$-dimensional vector belonging to the probability simplex. We first describe various settings of contamination and discuss the relation between these settings. We then establish minimax rates when the quality of estimation is measured by the total-variation distance, the Hellinger distance, or the $\mathbb{L}^{2}$-distance between two probability measures. We also provide confidence regions for the unknown mean that shrink at the minimax rate. Our analysis reveals that the minimax rates associated to these three distances are all different, but they are all attained by the sample average. Furthermore, we show that the latter is adaptive to the possible sparsity of the unknown vector. Some numerical experiments illustrating our theoretical findings are reported.

Respondent-Driven sampling (RDS) is a sampling method devised to overcome challenges with sampling hard-to-reach human populations. The sampling starts with a limited number of individuals who are asked to recruit a small number of their contacts. Every surveyed individual is subsequently given the same opportunity to recruit additional members of the target population until a pre-established sample size is achieved. The recruitment process consequently implies that the survey respondents are responsible for deciding who enters the study. Most RDS prevalence estimators assume that participants select among their contacts completely at random. The main objective of this work is to correct the inference for departure from this assumption, such as systematic recruitment based on the characteristics of the individuals or based on the nature of relationships. To accomplish this, we introduce three forms of non-random recruitment, provide estimators for these recruitment behaviors and extend three estimators and their associated variance procedures. The proposed methodology is assessed through a simulation study capturing various sampling and network features. Finally, the proposed methods are applied to a public health setting.

Shrinkage priors are becoming more and more popular in Bayesian modeling for high dimensional sparse problems due to its computational efficiency. Recent works show that a polynomially decaying prior leads to satisfactory posterior asymptotics under regression models. In the literature, statisticians have investigated how the global shrinkage parameter, i.e., the scale parameter, in a heavy tailed prior affects the posterior contraction. In this work, we explore how the shape of the prior, or more specifically, the polynomial order of the prior tail affects the posterior. We discover that, under the sparse normal means model, the polynomial order does affect the multiplicative constant of the posterior contraction rate. More importantly, if the polynomial order is sufficiently close to 1, it will induce the optimal Bayesian posterior convergence, in the sense that the Bayesian contraction rate is sharply minimax, i.e., not only the order, but also the multiplicative constant of the posterior contraction rate are optimal. The above Bayesian sharp minimaxity holds when the global shrinkage parameter follows a deterministic choice which depends on the unknown sparsity $s$. Therefore, a Beta-prior modeling is further proposed, such that our sharply minimax Bayesian procedure is adaptive to unknown $s$. Our theoretical discoveries are justified by simulation studies.

In this work, we propose a way to construct Gaussian processes indexed by multidimensional distributions. More precisely, we tackle the problem of defining positive definite kernels between multivariate distributions via notions of optimal transport and appealing to Hilbert space embeddings. Besides presenting a characterization of radial positive definite and strictly positive definite kernels on general Hilbert spaces, we investigate the statistical properties of our theoretical and empirical kernels, focusing in particular on consistency as well as the special case of Gaussian distributions. A wide set of applications is presented, both using simulations and implementation with real data.

We study the independence structure of finitely exchangeable distributions over random vectors and random networks. In particular, we provide necessary and sufficient conditions for an exchangeable vector so that its elements are completely independent or completely dependent. We also provide a sufficient condition for an exchangeable vector so that its elements are marginally independent. We then generalize these results and conditions for exchangeable random networks. In this case, it is demonstrated that the situation is more complex. We show that the independence structure of exchangeable random networks lies in one of six regimes that are two-fold dual to one another, represented by undirected and bidirected independence graphs in graphical model sense with graphs that are complement of each other. In addition, under certain additional assumptions, we provide necessary and sufficient conditions for the exchangeable network distributions to be faithful to each of these graphs.

We study a non-linear statistical inverse problem, where we observe the noisy image of a quantity through a non-linear operator at some random design points. We consider the widely used Tikhonov regularization (or method of regularization) approach to estimate the quantity for the non-linear ill-posed inverse problem. The estimator is defined as the minimizer of a Tikhonov functional, which is the sum of a data misfit term and a quadratic penalty term. We develop a theoretical analysis for the minimizer of the Tikhonov regularization scheme using the concept of reproducing kernel Hilbert spaces. We discuss optimal rates of convergence for the proposed scheme, uniformly over classes of admissible solutions, defined through appropriate source conditions.

Performing numerical integration when the integrand itself cannot be evaluated point-wise is a challenging task that arises in statistical analysis, notably in Bayesian inference for models with intractable likelihood functions. Markov chain Monte Carlo (MCMC) algorithms have been proposed for this setting, such as the pseudo-marginal method for latent variable models and the exchange algorithm for a class of undirected graphical models. As with any MCMC algorithm, the resulting estimators are justified asymptotically in the limit of the number of iterations, but exhibit a bias for any fixed number of iterations due to the Markov chains starting outside of stationarity. This “burn-in” bias is known to complicate the use of parallel processors for MCMC computations. We show how to use coupling techniques to generate unbiased estimators in finite time, building on recent advances for generic MCMC algorithms. We establish the theoretical validity of some of these procedures, by extending existing results to cover the case of polynomially ergodic Markov chains. The efficiency of the proposed estimators is compared with that of standard MCMC estimators, with theoretical arguments and numerical experiments including state space models and Ising models.

A high-dimensional $r$-factor model for an $n$-dimensional vector time series is characterised by the presence of a large eigengap (increasing with $n$) between the $r$-th and the $(r+1)$-th largest eigenvalues of the covariance matrix. Consequently, Principal Component (PC) analysis is the most popular estimation method for factor models and its consistency, when $r$ is correctly estimated, is well-established in the literature. However, popular factor number estimators often suffer from the lack of an obvious eigengap in empirical eigenvalues and tend to over-estimate $r$ due, for example, to the existence of non-pervasive factors affecting only a subset of the series. We show that the errors in the PC estimators resulting from the over-estimation of $r$ are non-negligible, which in turn lead to the violation of the conditions required for factor-based large covariance estimation. To remedy this, we propose new estimators of the factor model based on scaling the entries of the sample eigenvectors. We show both theoretically and numerically that the proposed estimators successfully control for the over-estimation error, and investigate their performance when applied to risk minimisation of a portfolio of financial time series.

Considering a class of stochastic differential equations driven by a locally stable process, we address the joint parametric estimation, based on high frequency observations of the process on a fixed time interval, of the drift coefficient, the scale coefficient and the jump activity of the process. Extending the methodology proposed in [6], where the jump activity was assumed to be known, we obtain two different rates of convergence in estimating simultaneously the scale parameter and the jump activity, depending on the scale coefficient. If the scale coefficient is multiplicative: $a(x,\sigma )=\sigma \overline{a}(x)$, the joint estimation of the scale coefficient and the jump activity behaves as for the translated stable process studied in [5] and the rate of convergence of our estimators is non diagonal. In the non multiplicative case, the results are different and we obtain a diagonal and faster rate of convergence which coincides with the one obtained in estimating marginally each parameter. In both cases, the estimation method is illustrated by numerical simulations showing that our estimators are rather easy to implement.

The Rasch model is widely used in the field of psychometrics when $n$ persons under test answer $m$ questions and the score, which describes the correctness of the answers, is given by a binary $n\times m$-matrix. We consider the Mixed-Effect Rasch Model, in which the persons are chosen randomly from a huge population. The goal is to estimate the ability density of this population under nonparametric constraints, which turns out to be a statistical linear inverse problem with an unknown but estimable operator. Based on our previous result on asymptotic equivalence to a two-layer Gaussian model, we construct an estimation procedure and study its asymptotic optimality properties as $n$ tends to infinity, as does $m$, but moderately with respect to $n$. Moreover numerical simulations are provided.

The Gaussian width is a fundamental quantity in probability, statistics and geometry, known to underlie the intrinsic difficulty of estimation and hypothesis testing. In this work, we show how the Gaussian width, when localized to any given point of an ellipse, can be controlled by the Kolmogorov width of a set similarly localized. Among other consequences, this connection, when coupled with a previous result due to Chatterjee, leads to a tight characterization of the estimation error of least-squares regression as a function of the true regression vector within the ellipse. This characterization reveals that the rate of error decay varies substantially as a function of location: as a concrete example, in Sobolev ellipses of smoothness $\alpha $, we exhibit rates that vary from $(\sigma ^{2})^{\frac{2\alpha }{2\alpha +1}}$, corresponding to the classical global rate, to the faster rate $(\sigma ^{2})^{\frac{4\alpha }{4\alpha +1}}$. We also show how the local Kolmogorov width can be related to local metric entropy.

In many problems, a sensible estimator of a possibly multivariate monotone function may fail to be monotone. We study the correction of such an estimator obtained via projection onto the space of functions monotone over a finite grid in the domain. We demonstrate that this corrected estimator has no worse supremal estimation error than the initial estimator, and that analogously corrected confidence bands contain the true function whenever the initial bands do, at no loss to band width. Additionally, we demonstrate that the corrected estimator is asymptotically equivalent to the initial estimator if the initial estimator satisfies a stochastic equicontinuity condition and the true function is Lipschitz and strictly monotone. We provide simple sufficient conditions in the special case that the initial estimator is asymptotically linear, and illustrate the use of these results for estimation of a G-computed distribution function. Our stochastic equicontinuity condition is weaker than standard uniform stochastic equicontinuity, which has been required for alternative correction procedures. This allows us to apply our results to the bivariate correction of the local linear estimator of a conditional distribution function known to be monotone in its conditioning argument. Our experiments suggest that the projection step can yield significant practical improvements.

This paper focuses on the distributed learning in nonparametric regression framework. With sufficient computational resources, the efficiency of distributed algorithms improves as the number of machines increases. We aim to analyze how the number of machines affects statistical optimality. We establish an upper bound for the number of machines to achieve statistical minimax in two settings: nonparametric estimation and hypothesis testing. Our framework is general compared with existing work. We build a unified frame in distributed inference for various regression problems, including thin-plate splines and additive regression under random design: univariate, multivariate, and diverging-dimensional designs. The main tool to achieve this goal is a tight bound of an empirical process by introducing the Green function for equivalent kernels. Thorough numerical studies back theoretical findings.

The extremal index $\theta $, a number in the interval $[0,1]$, is known to be a measure of primal importance for analyzing the extremes of a stationary time series. New rank-based estimators for $\theta $ are proposed which rely on the construction of approximate samples from the exponential distribution with parameter $\theta $ that is then to be fitted via the method of moments. The new estimators are analyzed both theoretically as well as empirically through a large-scale simulation study. In specific scenarios, in particular for time series models with $\theta \approx 1$, they are found to be superior to recent competitors from the literature.

Discrete time trawl processes constitute a large class of time series parameterized by a trawl sequence $(a_{j})_{j\in \mathbb{N}}$ and defined though a sequence of independent and identically distributed (i.i.d.) copies of a continuous time process $(\gamma (t))_{t\in \mathbb{R}}$ called the seed process. They provide a general framework for modeling linear or non-linear long range dependent time series. We investigate the spectral estimation, either pointwise or broadband, of long range dependent discrete-time trawl processes. The difficulty arising from the variety of seed processes and of trawl sequences is twofold. First, the spectral density may take different forms, often including smooth additive correction terms. Second, trawl processes with similar spectral densities may exhibit very different statistical behaviors. We prove the consistency of our estimators under very general conditions and we show that a wide class of trawl processes satisfy them. This is done in particular by introducing a weighted weak dependence index that can be of independent interest. The broadband spectral estimator includes an estimator of the long memory parameter. We complete this work with numerical experiments to evaluate the finite sample size performance of this estimator for various integer valued discrete time trawl processes.

In this paper, under discrete observations, we study Cramér-type moderate deviations (extended central limit theorem) for parameter estimation in Ornstein-Uhlenbeck process. Our results contain both stationary and explosive cases. For applications, we propose test statistics which can be used to construct rejection regions in the hypothesis testing for the drift coefficient, and the corresponding probability of type II error tends to zero exponentially. Simulation study shows that our test statistics have good finite-sample performances both in size and power. The main methods include the deviation inequalities for multiple Wiener-Itô integrals, as well as the asymptotic analysis techniques.

Given a pair of graphs $G_{1}$ and $G_{2}$ and a vertex set of interest in $G_{1}$, the vertex nomination (VN) problem seeks to find the corresponding vertices of interest in $G_{2}$ (if they exist) and produce a rank list of the vertices in $G_{2}$, with the corresponding vertices of interest in $G_{2}$ concentrating, ideally, at the top of the rank list. In this paper, we define and derive the analogue of Bayes optimality for VN with multiple vertices of interest, and we define the notion of maximal consistency classes in vertex nomination. This theory forms the foundation for a novel VN adversarial contamination model, and we demonstrate with real and simulated data that there are VN schemes that perform effectively in the uncontaminated setting, and adversarial network contamination adversely impacts the performance of our VN scheme. We further define a network regularization method for mitigating the impact of the adversarial contamination, and we demonstrate the effectiveness of regularization in both real and synthetic data.

The classical regular and partial autocorrelation functions are powerful tools for stationary time series modelling and analysis. However, it is increasingly recognized that many time series are not stationary and the use of classical global autocorrelations can give misleading answers. This article introduces two estimators of the local partial autocorrelation function and establishes their asymptotic properties. The article then illustrates the use of these new estimators on both simulated and real time series. The examples clearly demonstrate the strong practical benefits of local estimators for time series that exhibit nonstationarities.

We consider the problem of estimating sparse approximate factor models. In a first step, we jointly estimate the factor loading parameters and the error - or idiosyncratic - covariance matrix based on the Gaussian quasi-maximum likelihood method. Conditionally on these first step estimators, using the SCAD, MCP and Lasso regularisers, we obtain a sparse error covariance matrix based on a Gaussian QML and, as an alternative criterion, a least squares loss function. Under suitable regularity conditions, we derive error bounds for the regularised idiosyncratic factor model matrix for both Gaussian QML and least squares losses. Moreover, we establish the support recovery property, including the case when the regulariser is non-convex. These theoretical results are supported by empirical studies.

Random measures are the key ingredient for effective nonparametric Bayesian modeling of time-to-event data. This paper focuses on priors for the hazard rate function, a popular choice being the kernel mixture with respect to a gamma random measure. Sampling schemes are usually based on approximations of the underlying random measure, both a priori and conditionally on the data. Our main goal is the quantification of approximation errors through the Wasserstein distance. Though easy to simulate, the Wasserstein distance is generally difficult to evaluate, making tractable and informative bounds essential. Here we accomplish this task on the wider class of completely random measures, yielding a measure of discrepancy between many noteworthy random measures, including the gamma, generalized gamma and beta families. By specializing these results to gamma kernel mixtures, we achieve upper and lower bounds for the Wasserstein distance between hazard rates, cumulative hazard rates and survival functions.

This work examines under what circumstances adaptivity for truncated SVD estimation can be achieved by an early stopping rule based on the smoothed residuals $\|(AA^{\top })^{\alpha /2}(Y-A\widehat{\mu }^{(m)})\|^{2}$. Lower and upper bounds for the risk are derived, which show that moderate smoothing of the residuals can be used to adapt over classes of signals with varying smoothness, while oversmoothing yields suboptimal convergence rates. The range of smoothness classes for which adaptation is possible can be controlled via $\alpha $. The theoretical results are illustrated by Monte-Carlo simulations.

Discriminating patients with Alzheimer’s disease (AD) from healthy subjects is a crucial task in the research of Alzheimer’s disease. The task can be potentially achieved by linear discriminant analysis (LDA), which is one of the most classical and popular classification techniques. However, the classification problem becomes challenging for LDA because of the high-dimensionality and the spatial dependency of the brain imaging data. To address the challenges, researchers have proposed various ways to generalize LDA into high-dimensional context in recent years. However, these existing methods did not reach any consensus on how to incorporate spatially dependent structure. In light of the current needs and limitations, we propose a new classification method, named as Penalized Maximum Likelihood Estimation LDA (PMLE-LDA). The proposed method uses Matérn covariance function to describe the spatial correlation of brain regions. Additionally, PMLE is designed to model the sparsity of high-dimensional features. The spatial location information is used to address the singularity of the covariance. Tapering technique is introduced to reduce computational burden. We show in theory that the proposed method can not only provide consistent results of parameter estimation and feature selection, but also generate an asymptotically optimal classifier driven by high dimensional data with specific spatially dependent structure. Finally, the method is validated through simulations and an application into ADNI data for classifying Alzheimer’s patients.

For a regression problem with a binary label response, we examine the problem of constructing confidence intervals for the label probability conditional on the features. In a setting where we do not have any information about the underlying distribution, we would ideally like to provide confidence intervals that are distribution-free—that is, valid with no assumptions on the distribution of the data. Our results establish an explicit lower bound on the length of any distribution-free confidence interval, and construct a procedure that can approximately achieve this length. In particular, this lower bound is independent of the sample size and holds for all distributions with no point masses, meaning that it is not possible for any distribution-free procedure to be adaptive with respect to any type of special structure in the distribution.

This article studies nonparametric methods to estimate the co-integrated volatility of multi-dimensional Lévy processes with high frequency data. We construct a spectral estimator for the co-integrated volatility and prove minimax rates for an appropriate bounded nonparametric class of Lévy processes. Given $n$ observations of increments over intervals of length $1/n$, the rates of convergence are $1/\sqrt{n}$ if $r\leq 1$ and $(n\log n)^{(r-2)/2}$ if $r>1$, where $r$ is the co-jump activity index and corresponds to the intensity of dependent jumps. These rates are optimal in a minimax sense. We bound the co-jump activity index from below by the harmonic mean of the jump activity indices of the components. Finally, we assess the efficiency of our estimator by comparing it with estimators in the existing literature.

We study Empirical Risk Minimizers (ERM) and Regularized Empirical Risk Minimizers (RERM) for regression problems with convex and $L$-Lipschitz loss functions. We consider a setting where $|{\mathcal{O}}|$ malicious outliers contaminate the labels. In that case, under a local Bernstein condition, we show that the $L_{2}$-error rate is bounded by $r_{N}+AL|{\mathcal{O}}|/N$, where $N$ is the total number of observations, $r_{N}$ is the $L_{2}$-error rate in the non-contaminated setting and $A$ is a parameter coming from the local Bernstein condition. When $r_{N}$ is minimax-rate-optimal in a non-contaminated setting, the rate $r_{N}+AL|{\mathcal{O}}|/N$ is also minimax-rate-optimal when $|{\mathcal{O}}|$ outliers contaminate the label. The main results of the paper can be used for many non-regularized and regularized procedures under weak assumptions on the noise. We present results for Huber’s M-estimators (without penalization or regularized by the $\ell _{1}$-norm) and for general regularized learning problems in reproducible kernel Hilbert spaces when the noise can be heavy-tailed.

This paper is devoted to the offline multiple changes detection for long-range dependent processes. The observations are supposed to satisfy a semi-parametric long-range dependent assumption with distinct memory parameters on each stage. A penalized local Whittle contrast is considered for estimating all the parameters, notably the number of changes. Consistency as well as convergence rates are obtained. Monte-Carlo experiments exhibit the accuracy of the estimators. They also show that the estimation of the number of breaks is improved by using a data-driven slope heuristic procedure of choice of the penalization parameter.

In this paper we study asymptotic properties of random forests within the framework of nonlinear time series modeling. While random forests have been successfully applied in various fields, the theoretical justification has not been considered for their use in a time series setting. Under mild conditions, we prove a uniform concentration inequality for regression trees built on nonlinear autoregressive processes and, subsequently, we use this result to prove consistency for a large class of random forests. The results are supported by various simulations.

Statistical analysis of network is an active research area and the literature counts a lot of papers concerned with network models and statistical analysis of networks. However, very few papers deal with missing data in network analysis and we reckon that, in practice, networks are often observed with missing values. In this paper we focus on the Stochastic Block Model with valued edges and consider a MCAR setting by assuming that every dyad (pair of nodes) is sampled identically and independently of the others with probability $\rho >0$. We prove that maximum likelihood estimators and its variational approximations are consistent and asymptotically normal in the presence of missing data as soon as the sampling probability $\rho $ satisfies $\rho \gg \log (n)/n$.

A new approach to detect change points based on differential smoothing and multiple testing is presented for long data sequences modeled as piecewise constant functions plus stationary ergodic Gaussian noise. As an application of the STEM algorithm for peak detection developed in Schwartzman et al. [27] and Cheng and Schwartzman [5], the method detects change points as significant local maxima and minima after smoothing and differentiating the observed sequence. The algorithm, combined with the Benjamini-Hochberg procedure for thresholding p-values, provides asymptotic strong control of the False Discovery Rate (FDR) and power consistency, as the length of the sequence and the size of the jumps get large. Simulations show that FDR levels are maintained in non-asymptotic conditions and guide the choice of smoothing bandwidth. The methods are illustrated in magnetometer sensor data and genomic array-CGH data. An R package named “dSTEM” is available in R Cran.

We consider the nonparametric estimation of the isotropic pair correlation function (PCF) of inhomogeneous point processes when replicates are available. Based on carefully designed estimating equations, two types of nonparametric estimators, i.e., the local polynomial estimator and the orthogonal series estimator, are proposed and studied. The proposed estimators circumvent the problems caused by the need for estimating the unknown intensity function for kernel smoothed PCF estimators and they are free of edge correction terms. Asymptotic properties are investigated for both estimators and valid point-wise confidence bands are derived. Finite sample performances of the proposed estimators are demonstrated by simulation as well as an application to the Sina Weibo posting data.

Missing responses is a common type of data where the interested outcomes are not always observed. In this paper, we develop two new kernel machines to handle such a case, which can be used for both regression and classification. The first proposed kernel machine uses $\text{only}$ the complete cases where both response and covariates are observed. It is, however, subject to some assumption limitations. Our second proposed doubly-robust kernel machine overcomes such limitations regardless of the misspecification of either the missing mechanism or the conditional distribution of the response. Theoretical properties, including the oracle inequalities for the excess risk, universal consistency, and learning rates are established. We demonstrate the superiority of the proposed methods to some existing methods by simulation and illustrate their application to a real data set concerning a survey about homeless people.

We consider the problem of predicting several response variables using the same set of explanatory variables. This setting naturally induces a group structure over the coefficient matrix, in which every explanatory variable corresponds to a set of related coefficients. Most of the existing methods that utilize this group formation assume that the similarities between related coefficients arise solely through a joint sparsity structure. In this paper, we propose a procedure for constructing multivariate regression models, that directly capture and model the within-group similarities, by employing a multivariate linear mixed model formulation, with a joint estimation of covariance matrices for coefficients and errors via penalized likelihood. Our approach, which we term MrRCE for Multivariate random Regression with Covariance Estimation, encourages structured similarity in parameters, in which coefficients for the same variable in related tasks share the same sign and similar magnitude. We illustrate the benefits of our approach in synthetic and real examples, and show that the proposed method outperforms natural competitors and alternative estimators under several model settings.

In this paper we propose a new nonparametric regression technique. Our proposal has common ground with existing two-step procedures in that it starts with a parametric model. However, our approach differs from others in the choice of parametric start within the parametric family. Our proposal chooses a function that is the projection of the unknown regression function onto the parametric family in a certain metric, while the existing methods select the best approximation in the usual $L_{2}$ metric. We find that the difference leads to substantial improvement in the performance of regression estimators in comparison with direct one-step estimation, irrespective of the choice of a parametric model. This is in contrast with the existing two-step methods, which fail if the chosen parametric model is largely misspecified. We demonstrate this with sound theory and numerical experiment.

Finite mixtures are a flexible modeling tool for irregularly shaped densities and samples from heterogeneous populations. When modeling with mixtures using an exchangeable prior on the component features, the component labels are arbitrary and are indistinguishable in posterior analysis. This makes it impossible to attribute any meaningful interpretation to the marginal posterior distributions of the component features. We propose a model in which a small number of observations are assumed to arise from some of the labeled component densities. The resulting model is not exchangeable, allowing inference on the component features without post-processing. Our method assigns meaning to the component labels at the modeling stage and can be justified as a data-dependent informative prior on the labelings. We show that our method produces interpretable results, often (but not always) similar to those resulting from relabeling algorithms, with the added benefit that the marginal inferences originate directly from a well specified probability model rather than a post hoc manipulation. We provide asymptotic results leading to practical guidelines for model selection that are motivated by maximizing prior information about the class labels and demonstrate our method on real and simulated data.

We discuss goodness-of-fit tests for stationary ergodic processes based on limit theorems for marked empirical processes viewed as elements of an $L^{2}$ space. Our limit theorems cover a weighted process that enables an Anderson–Darling-type test statistic for the goodness-of-fit tests to be proposed. The procedures presented for these goodness-of-fit tests are novel.

The computational complexity of some depths that satisfy the projection property, such as the halfspace depth or the projection depth, is known to be high, especially for data of higher dimensionality. In such scenarios, the exact depth is frequently approximated using a randomized approach: The data are projected into a finite number of directions uniformly distributed on the unit sphere, and the minimal depth of these univariate projections is used to approximate the true depth. We provide a theoretical background for this approximation procedure. Several uniform consistency results are established, and the corresponding uniform convergence rates are provided. For elliptically symmetric distributions and the halfspace depth it is shown that the obtained uniform convergence rates are sharp. In particular, guidelines for the choice of the number of random projections in order to achieve a given precision of the depths are stated.

In this communication we examine the relationship between maxi–min, Bayes and Nash designs for some hypothesis testing problems. In particular we consider the problem of sample allocation in the standard analysis of variance framework and show that the maxi–min design is also a Bayes solution with respect to the least favourable prior, as well as a solution to a game theoretic problem, which we refer to as a Nash design. In addition, an extension to tests for order is provided.

Motivated by customer loyalty plans and scholarship programs, we study tie-breaker designs which are hybrids of randomized controlled trials (RCTs) and regression discontinuity designs (RDDs). We quantify the statistical efficiency of a tie-breaker design in which a proportion $\Delta $ of observed subjects are in the RCT. In a two line regression, statistical efficiency increases monotonically with $\Delta $, so efficiency is maximized by an RCT. We point to additional advantages of tie-breakers versus RDD: for a nonparametric regression the boundary bias is much less severe and for quadratic regression, the variance is greatly reduced. For a two line model we can quantify the short term value of the treatment allocation and this comparison favors smaller $\Delta $ with the RDD being best. We solve for the optimal tradeoff between these exploration and exploitation goals. The usual tie-breaker design applies an RCT on the middle $\Delta $ subjects as ranked by the assignment variable. We quantify the efficiency of other designs such as experimenting only in the second decile from the top. We also show that in some general parametric models a Monte Carlo evaluation can be replaced by matrix algebra.

Motivated by recent works on the high-dimensional logistic regression, we establish that the existence of the maximum likelihood estimate exhibits a phase transition for a wide range of generalized linear models with binary outcome and elliptical covariates. This extends a previous result of Candès and Sur who proved the phase transition for the logistic regression with Gaussian covariates. Our result reveals a rich structure in the phase transition phenomenon, which is simply overlooked by Gaussianity. The main tools for deriving the result are data separation, convex geometry and stochastic approximation. We also conduct simulation studies to corroborate our theoretical findings, and explore other features of the problem.

Model selection plays a critical role in statistical inference and a large literature has been devoted to this topic. Despite extensive research attention on model selection, research gaps still remain. An important but relatively unexplored problem concerns truncated and censored data with measurement error. Although analysis of left-truncated and right-censored (LTRC) data has received extensive interests in survival analysis, there has been no research on model selection for LTRC data with measurement error. In this paper, we take up this important problem and develop inferential procedures to handle model selection for LTRC data with measurement error in covariates. Our development employs the local model misspecification framework ([6]; [10]) and emphasizes the use of the focus information criterion (FIC). We develop valid estimators using the model averaging scheme and establish theoretical results to justify the validity of our methods. Numerical studies are conducted to assess the performance of the proposed methods.

Graphical models represent a powerful framework to incorporate conditional independence structure for the statistical analysis of high-dimensional data. In this paper we focus on Directed Acyclic Graphs (DAGs). In the Gaussian setting, a prior recently introduced for the parameters associated to the (modified) Cholesky decomposition of the precision matrix is the DAG-Wishart. The flexibility introduced through a rich choice of shape hyperparameters coupled with conjugacy are two desirable assets of this prior which are especially welcome for estimation and prediction. In this paper we look at the DAG-Wishart prior from the perspective of model selection, with special reference to its consistency properties in high dimensional settings. We show that Bayes factor consistency only holds when comparing two DAGs which do not belong to the same Markov equivalence class, equivalently they encode distinct conditional independencies; a similar result holds for posterior ratio consistency. We also prove that DAG-Wishart distributions with arbitrarily chosen hyperparameters will lead to incompatible priors for model selection, because they assign different marginal likelihoods to Markov equivalent graphs. To overcome this difficulty, we propose a constructive method to specify DAG-Wishart priors whose suitably constrained shape hyperparameters ensure compatibility for DAG model selection.

Identification of causal relations among variables is central to many scientific investigations, as in regulatory network analysis of gene interactions and brain network analysis of effective connectivity of causal relations between regions of interest. Statistically, causal relations are often modeled by a directed acyclic graph (DAG), and hence that reconstruction of a DAG’s structure leads to the discovery of causal relations. Yet, reconstruction of a DAG’s structure from observational data is impossible because a DAG Gaussian model is usually not identifiable with unequal error variances. In this article, we reconstruct a DAG’s structure with the help of interventional data. Particularly, we construct a constrained likelihood to regularize intervention in addition to adjacency matrices to identify a DAG’s structure, subject to an error variance constraint to further reinforce the model identifiability. Theoretically, we show that the proposed constrained likelihood leads to identifiable models, thus correct reconstruction of a DAG’s structure through parameter estimation even with unequal error variances. Computationally, we design efficient algorithms for the proposed method. In simulations, we show that the proposed method enables to produce a higher accuracy of reconstruction with the help of interventional observations.

The Blumenthal–Getoor (BG) index characterizes the jump measure of an infinitely active Lévy process. It determines sample path properties and affects the behavior of various econometric procedures. If the process contains a diffusion term, existing estimators of the BG index based on high-frequency observations achieve rates of convergence which are suboptimal by a polynomial factor. In this paper, a novel estimator for the BG index and the successive BG indices is presented, attaining the optimal rate of convergence. If an additional proportionality factor needs to be inferred, the proposed estimator is rate-optimal up to logarithmic factors. Furthermore, our method yields a new efficient volatility estimator which accounts for jumps of infinite variation. All parameters are estimated jointly by the generalized method of moments. A simulation study compares the finite sample behavior of the proposed estimators with competing methods from the financial econometrics literature.

Random coefficient regression (RCR) models are the regression versions of random effects models in analysis of variance and panel data analysis. Optimal detection of the presence of random coefficients (equivalently, optimal testing of the hypothesis of constant regression coefficients) has been an open problem for many years. The simple regression case has been solved recently and the multiple regression case is considered here. The latter poses several theoretical challenges: (a) a nonstandard ULAN structure, with log-likelihood gradients vanishing at the null; (b) cone-shaped alternatives under which traditional optimality concepts are no longer adequate; (c) nuisance parameters that are not identified under the null but have a significant impact on local powers. We propose a new (local and asymptotic) concept of optimality for this problem and, for specified error densities, derive parametrically optimal procedures. A suitable modification of the Gaussian version of the latter is shown to qualify as a pseudo-Gaussian test. The asymptotic performances of those pseudo-Gaussian tests, however, are quite poor under skewed and heavy-tailed densities. We therefore also construct rank-based tests, possibly based on data-driven scores, the asymptotic relative efficiencies of which are remarkably high with respect to their pseudo-Gaussian counterparts.

Several classical methods exist for controlling the false discovery exceedance (FDX) for large-scale multiple testing problems, among them the Lehmann-Romano procedure (Lehmann and Romano 2005) ($[\mathrm{LR}]$ below) and the Guo-Romano procedure (Guo and Romano 2007) ($[\mathrm{GR}]$ below). While these two procedures are the most prominent, they were originally designed for homogeneous test statistics, that is, when the null distribution functions of the $p$-values $F_{i}$, $1\leq i\leq m$, are all equal. In many applications, however, the data are heterogeneous which leads to heterogeneous null distribution functions. Ignoring this heterogeneity induces a lack of power. In this paper, we develop three new procedures that incorporate the $F_{i}$’s, while maintaining rigorous FDX control. The heterogeneous version of $[\mathrm{LR}]$, denoted $[\mathrm{HLR}]$, is based on the arithmetic average of the $F_{i}$’s, while the heterogeneous version of $[\mathrm{GR}]$, denoted $[\mathrm{HGR}]$, is based on the geometric average of the $F_{i}$’s. We also introduce a procedure $[\mathrm{PB}]$, that is based on the Poisson-binomial distribution and that uniformly improves $[\mathrm{HLR}]$ and $[\mathrm{HGR}]$, at the price of a higher computational complexity. Perhaps surprisingly, this shows that, contrary to the known theory of false discovery rate (FDR) control under heterogeneity, the way to incorporate the $F_{i}$’s can be particularly simple in the case of FDX control, and does not require any further correction term. The performances of the new proposed procedures are illustrated by real and simulated data in two important heterogeneous settings: first, when the test statistics are continuous but the $p$-values are weighted by some known independent weight vector, e.g., coming from co-data sets; second, when the test statistics are discretely distributed, as is the case for data representing frequencies or counts. Our new procedures are implemented in the R package $\mathtt{FDX}$, see Junge and Döhler (2020).

We present new families of goodness-of-fit tests of uniformity on a full-dimensional set $W\subset \mathbb{R}^{d}$ based on statistics related to edge lengths of random geometric graphs. Asymptotic normality of these statistics is proven under the null hypothesis as well as under fixed alternatives. The derived tests are consistent and their behaviour for some contiguous alternatives can be controlled. A simulation study suggests that the procedures can compete with or are better than established goodness-of-fit tests. We show with a real data example that the new tests can detect non-uniformity of a small sample data set, where most of the competitors fail.

Conditional heteroskedastic financial time series are commonly modelled by (G)ARCH processes. ARCH$(1)$ and GARCH were recently established in $C[0,1]$ and $L^{2}[0,1]$. This article provides sufficient conditions for the existence of strictly stationary solutions, weak dependence and finite moments of (G)ARCH processes for any order in $C[0,1]$ and $L^{p}[0,1]$. It deduces explicit asymptotic upper bounds of estimation errors for the shift term, the complete (G)ARCH operators and the projections of ARCH operators on finite-dimensional subspaces. The operator estimaton is based on Yule-Walker equations, and estimating the GARCH operators also involves a result estimating operators in invertible linear processes being valid beyond the scope of (G)ARCH. Moreover, our results regarding (G)ARCH can be transferred to functional AR(MA).

For any finite point set in $D$-dimensional space equipped with the 1-norm, we present random linear embeddings to $k$-dimensional space, with a new metric, having the following properties. For any pair of points from the point set that are not too close, the distance between their images is a strictly concave increasing function of their original distance, up to multiplicative error. The target dimension $k$ need only be quadratic in the logarithm of the size of the point set to ensure the result holds with high probability. The linear embeddings are random matrices composed of standard Cauchy random variables, and the proofs rely on Chernoff bounds for sums of iid random variables. The new metric is translation invariant, but is not induced by a norm.

In this paper we present new theoretical results for the Dantzig and Lasso estimators of the drift in a high dimensional Ornstein-Uhlenbeck model under sparsity constraints. Our focus is on oracle inequalities for both estimators and error bounds with respect to several norms. In the context of the Lasso estimator our paper is strongly related to [11], where the same problem was investigated under row sparsity. We improve their rates and also prove the restricted eigenvalue property solely under ergodicity assumption on the model. Finally, we demonstrate a numerical analysis to uncover the finite sample performance of the Dantzig and Lasso estimators.

We consider a location-dispersion regression model for heavy-tailed distributions when the multidimensional covariate is deterministic. In a first step, nonparametric estimators of the regression and dispersion functions are introduced. This permits, in a second step, to derive an estimator of the conditional extreme-value index computed on the residuals. Finally, a plug-in estimator of extreme conditional quantiles is built using these two preliminary steps. It is shown that the resulting semi-parametric estimator is asymptotically Gaussian and may benefit from the same rate of convergence as in the unconditional situation. Its finite sample properties are illustrated both on simulated and real tsunami data.

Dependent data with hierarchical structures arises commonly from a variety of application, and analysis of such data is often challenging due to the complexity in modeling dependence structures and the computation intensity. In this paper, we propose a Bayesian hierarchical copula model (BHCM) to accommodate hierarchical structures of dependent data, where the subject-level dependence is modeled by the copula-based model and the hierarchical structure is described using random dependence parameters. We introduce a layer-by-layer sampling scheme for conducting Bayesian inferences. Our proposed BHCM enjoys the flexibility of modeling various complex association structures, while retaining manageable computation. Extensive simulation studies show that our proposed estimators outperform conventional likelihood-based estimators in a variety of finite sample settings. We apply the BHCM to analyze the Vertebral Column dataset arising from UCI Machine Learning Repository.

Global null testing is a classical problem going back about a century to Fisher’s and Stouffer’s combination tests. In this work, we present simple martingale analogs of these classical tests, which are applicable in two distinct settings: (a) the online setting in which there is a possibly infinite sequence of $p$-values, and (b) the batch setting, where one uses prior knowledge to preorder the hypotheses. Through theory and simulations, we demonstrate that our martingale variants have higher power than their classical counterparts even when the preordering is only weakly informative. Finally, using a recent idea of “masking” $p$-values, we develop a novel interactive test for the global null that can take advantage of covariates and repeated user guidance to create a data-adaptive ordering that achieves higher detection power against structured alternatives.