The aim of this paper is to develop a new framework of surface functional models (SFM) for surface functional data which contains repeated observations in two domains (typically, time-location). The primary problem of interest is to investigate the relationship between a response and the two domains, where the numbers of observations in both domains within a subject may be diverging. The SFMs are far

Functional canonical correlation analysis (FCCA) is a tool for exploring the associations between a pair of functional data. However, when the association is nonlinear or even nonmonotone, FCCA can fail to discover any meaningful relationship between the pair. In this paper, nonlinear FCCA estimators are constructed based on some popular measures of dependence – distance covariance and distance correlation

The existing literature of copula-based regression assumes that complete data are available, but this assumption is violated in many real applications. The present paper allows the regressand and regressors to be missing at random (MAR). We formulate a generalized regression model which unifies many prominent cases such as the conditional mean and quantile regressions. A semiparametric copula and the

Graphical modelling techniques based on sparse estimation have been applied to infer complex networks in many fields, including biology and medicine, engineering, finance and social sciences. One structural feature of some of these networks that poses a challenge for statistical inference is the presence of a small number of strongly interconnected nodes, which are called hubs. For example, in microbiome

The negative multinomial distribution is a multivariate generalization of the negative binomial distribution. In this paper, we consider the problem of estimating an unknown matrix of probabilities on the basis of observations of negative multinomial variables under the standardized squared error loss. First, a general sufficient condition for a shrinkage estimator to dominate the UMVU estimator is

We consider the multivariate independence testing problem between pairs of random vectors for high-dimensional data and develop three high-dimensional nonparametric independence tests based on spatial sign and spatial rank, which have greater power than many existing popular tests, especially for heavy-tailed distributions. Under the elliptically symmetric distributions, which are much more general

Categorical data are often observed as counts resulting from a fixed number of trials in which each trial consists of making one selection from a prespecified set of categories. The multinomial distribution serves as a standard model for such data but assumes that trials are independent and identically distributed. Extensions such as the Dirichlet-multinomial and random-clumped multinomial distribution

In this paper, we propose a matrix extension of the scale and shape mixtures of multivariate skew normal distributions and present some particular cases of this new class. We also present several formal properties of this class, such as the marginal distributions, the moment generating function, the distribution of linear and quadratic forms, and the selection and stochastic representations. In addition

We derive an expression for the Spearman rank correlation of bivariate scale mixtures of normals (SMN) and we show that within this class, for any value of the correlation parameter, the Spearman rank correlation of the normal is the greatest in absolute value. We then provide expressions for the symmetric generalized hyperbolic, the Bessel, and the Laplace distributions. We further derive an expression

We consider the problem of testing for the nullity of conditional expectations of Hilbert space-valued random variables. We allow for conditioning variables taking values in finite or infinite Hilbert spaces. This testing problem occurs, for instance, when checking the goodness-of-fit or the effect of some infinite-dimensional covariates in regression models for functional data. Testing the independence

Hidden Markov models (HMMs) describe the relationship between two stochastic processes, namely, an observed outcome process and an unobservable finite-state transition process. Given their ability to model dynamic heterogeneity, HMMs are extensively used to analyze heterogeneous longitudinal data. A majority of early developments in HMMs assume that observation times are discrete and regular. This

The problem of characterizing a multivariate distribution of a random vector using examination of univariate combinations of vector components is an essential issue of multivariate analysis. The likelihood principle plays a prominent role in developing powerful statistical inference tools. In this context, we raise the question: can the univariate likelihood function based on a random vector be used

We give an approximate formula for the distribution of the largest eigenvalue of real Wishart matrices by the expected Euler characteristic method for general dimension. The formula is expressed in terms of a definite integral with parameters. We derive a differential equation satisfied by the integral for the 2×2 matrix case and perform a numerical analysis of it.

In this paper, (positive) linear stochastic orderings of random vectors X and Y having scale mixtures of the multivariate skew-normal distribution are studied. Necessary and sufficient convenient conditions for a⊤X to be less than a⊤Y, when a is a vector of positive values, in the sense of usual, convex and increasing convex stochastic orders are grasped. The results are potentially applied to conduct

Identifying large-scale conditional dependence structures through graphical models is a challenging yet practical problem. Under ultra-high dimensional settings, a screening procedure is generally suggested before variable selection to reduce computational costs. However, most existing screening methods examine the marginal correlations, thus not suitable to discover the conditional dependence in graphical

Corrupted data sets containing noisy or missing observations are prevalent in various contemporary applications such as economics, finance and bioinformatics. Despite the recent methodological and algorithmic advances in high-dimensional multi-response regression, how to achieve scalable and interpretable estimation under contaminated covariates is unclear. In this paper, we develop a new methodology

In this article, we consider discrimination analyses in high-dimensional cases where the dimension of the predictor vector diverges with the sample size in a theoretical setting. The emphasis is on the case where the number of classes is bigger than two. We first deal with the asymptotic misclassification rates of linear discrimination rules under various conditions. In practical high-dimensional classification

We propose a significance test to determine if data on a regular d-dimensional grid can be assumed to be a realization of Gaussian process. By accounting for the spatial dependence of the observations, we derive statistics analogous to sample skewness and kurtosis. We show that the sum of squares of these two statistics converges to a chi-square distribution with two degrees of freedom. This leads

The multivariate skew normal (MSN) and multivariate skew t (MST) distributions have received considerable attention in the past two decades because of their appealing mathematical properties and their usefulness for modeling skewed data. We develop sparse regularization methodology for estimating the skewness parameters of these two distributions. This methodology facilitates skewness selection, i

Hypergeometric functions and zonal polynomials are the tools usually addressed in the literature to deal with the expected value of the elementary symmetric functions in non-central Wishart latent roots. The method here proposed recovers the expected value of these symmetric functions by using the umbral operator applied to the trace of suitable polynomial matrices and their cumulants. The employment

Estimation of the covariance matrix for high-dimensional multivariate datasets is a challenging and important problem in modern statistics. In this paper, we focus on high-dimensional Gaussian DAG models where sparsity is induced on the Cholesky factor L of the inverse covariance matrix. In recent work, (Cao et al., 2019), we established high-dimensional sparsity selection consistency for a hierarchical

We treat the problem of testing mutual independence of k high-dimensional random vectors when the data are multivariate normal and k≥2 is a fixed integer. For this purpose, we focus on the vector correlation coefficient, ρV and propose an extension of its classical estimator which is constructed to correct potential sources of inconsistency related to the high dimensionality. Building on the proposed

In this paper, we discuss a two-way multivariate analysis of variance in high-dimensional settings. With a high-dimensional setting, we propose new approximate tests that work well under the following conditions: 1. The error vectors do not necessarily follow a multivariate normal distribution, 2. The cell sizes are unequal, 3. The cell covariance matrices are unequal, and 4. The dimension p is much

Factor models have been proposed for a broad range of observed variables such as binary, Gaussian, and variables of mixed types. They typically model a pairwise interaction parameter matrix with a low-rank and a diagonal component. The low-rank component can be interpreted as the effect of a few latent quantitative variables which are common in social science applications. Another line of research

A test of modality of rotationally symmetric distributions on hyperspheres is proposed. The test is based on a modified multivariate kurtosis defined for directional data on Sd. We first reveal a relationship between the multivariate kurtosis and the types of modality for Euclidean data. In particular, the kurtosis of a rotationally symmetric distribution with decreasing sectional density is greater

This paper develops a test of goodness-of-fit for elliptical distributions. The test is invariant to affine-linear transformations and has a convenient expression that can be broken into components containing diagnostic information to be used to identify possible departures when the test rejects. The test is developed for the bivariate Laplace, logistic and Pearson type II distributions, as well as

This article is concerned with random objects in the complex projective space ℂPk−2. It is shown that the Veronese–Whitney (VW) antimean, which is the extrinsic antimean of a random point on ℂPk−2 relative to the VW-embedding, is given by the point on ℂPk−2 represented by the eigenvector corresponding to the smallest eigenvalue of the expected mean of the VW-embedding of the random point, provided

We consider robust and nonparametric estimation of the coefficient of tail dependence in presence of random covariates. The estimator is obtained by fitting the extended Pareto distribution locally to properly transformed bivariate observations using the minimum density power divergence criterion. We establish convergence in probability and asymptotic normality of the proposed estimator under some

The estimation of the mean matrix of the multivariate normal distribution is addressed in the high dimensional setting. Efron–Morris-type linear shrinkage estimators with ridge modification for the precision matrix instead of the Moore–Penrose generalized inverse are considered, and the weights in the ridge-type linear shrinkage estimators are estimated in terms of minimizing the Stein unbiased risk

In this paper, we investigate the asymptotic distribution of likelihood ratio tests in models with several groups, when the number of groups converges with the dimension and sample size to infinity. We derive central limit theorems for the logarithm of various test statistics and compare our results with the approximations obtained from a central limit theorem where the number of groups is fixed.

In this paper, a new model-free feature screening method based on classification accuracy of marginal classifiers is proposed for ultrahigh dimensional classification. Different from existing methods, which use the differences of means or differences of conditional cumulative distribution functions between classes as the screening indexes, we propose a new feature screening method to rank the importance

Causal inference on multiple non-independent outcomes raises serious challenges, because multivariate techniques that properly account for the outcome’s dependence structure need to be considered. We focus on the case of binary outcomes framing our discussion in the potential outcome approach to causal inference. We define causal effects of treatment on joint outcomes introducing the notion of product

This paper considers conditional moment models where the parameters of interest include both finite-dimensional parameters and unknown functions. We propose sup-Wald, sup-quasi-likelihood ratio and sup-Lagrange multiplier statistics for testing functionals restrictions uniformly over the support for both finite and infinite dimensional components of the parameters. The trinity of three statistics holds

Hierarchical models are the workhorse of much of Bayesian analysis, yet there is uncertainty as to which priors to use for hyperparameters. Formal approaches to objective Bayesian analysis, such as the Jeffreys-rule approach or reference prior approach, are only implementable in simple hierarchical settings. It is thus common to use less formal approaches, such as utilizing formal priors from non-hierarchical

Conditional Kendall’s tau is a measure of dependence between two random variables, conditionally on some covariates. We assume a regression-type relationship between conditional Kendall’s tau and some covariates, in a parametric setting with a large number of transformations of a small number of regressors. This model may be sparse, and the underlying parameter is estimated through a penalized criterion

Updating of the Gaussian graphical model via shrinkage estimation is studied. This shrinkage is towards a nonzero parameter value representing prior quantitative information. Once new data become available, the previously estimated parameter needs updating. Shrinkage provides the means to this end, using the latter as a shrinkage target to acquire an updated estimate. The process of iteratively updating

The comparison of two risky portfolios has always been of interest in insurance and finance. Classically, it is often assumed that the portfolio claims are independent, but in practice, this assumption is not usually true and we need to study portfolios with dependent claims. In this paper, we consider two risky portfolios with dependent claims whose dependencies are modeled using Archimedean copulas

Suppose X is an N×n complex matrix whose entries are centered, independent, and identically distributed random variables with variance 1∕n and whose fourth moment is of order O(n−2). Suppose A is a deterministic matrix whose smallest and largest singular values are bounded below and above respectively, and z≠0 is a complex number. First we consider the matrix XAX∗−z, and obtain asymptotic probability

Correlation matrices are the sub-class of positive definite real matrices with all entries on the diagonal equal to unity. Earlier work has exhibited a parametrisation of the corresponding Cholesky factorisation in terms of partial correlations, and also in terms of hyperspherical co-ordinates. We show how the two are related, starting from the definition of the partial correlations in terms of the

This paper explores likelihood-based tests for parameter constancy in I(2) cointegrated vector autoregressive (CVAR) models. A new class of test statistics for parameter stability is introduced in the I(2) CVAR framework. This study proves that their asymptotic distributions take non-standard forms involving the integrals of Brownian motions, but they are free of any nuisance parameters. It is thus

A new test of independence between random elements is presented in this article. The test is based on a functional of the Cramér–von Mises type, which is applied to a U-process that is defined from the recurrence rates. Theorems of asymptotic distribution under H0, and consistency under a wide class of alternatives are obtained. The results under contiguous alternatives are also shown. The test has

This paper studies the problem of functional data clustering. Functional data have their own characteristics and contain rich information that cannot be obtained when regarding the data as multivariate data. Functional data are inherently infinite-dimensional, so classical clustering techniques for finite-dimensional data may not be suitable for functional data. There are several clustering methods

Case-controls studies are popular epidemiological designs for detecting gene-environment interactions in the etiology of complex diseases, where the genetic susceptibility and environmental exposures may often be reasonably assumed independent in the source population. Various papers have presented analytical methods exploiting gene-environment independence to achieve better efficiency, all of which

The largest eigenvalue of a single or a double Wishart matrix, both known as Roy's largest root, plays an important role in a variety of applications. Recently, via a small noise perturbation approach with fixed dimension and degrees of freedom, Johnstone and Nadler derived simple yet accurate approximations to its distribution in the real valued case, under a rank-one alternative. In this paper, we

The partially linear model (PLM) is a useful semiparametric extension of the linear model that has been well studied in the statistical literature. This paper proposes a variable selection procedure for the PLM with ultrahigh dimensional predictors. The proposed method is different from the existing penalized least squares procedure in that it relies on partial correlation between the partial residuals

Forward regression, a classical variable screening method, has been widely used for model building when the number of covariates is relatively low. However, forward regression is seldom used in high-dimensional settings because of the cumbersome computation and unknown theoretical properties. Some recent works have shown that forward regression, coupled with an extended Bayesian information criterion

With its important biological implications, modeling the associations of gene expression (GE) and copy number variation (CNV) has been extensively conducted. Such analysis is challenging because of the high data dimensionality, lack of knowledge regulating CNVs for a specific GE, different behaviors of the cis-acting and trans-acting CNVs, possible long-tailed distributions and contamination of GE

This paper studies the asymptotic properties of a sparse linear regression estimator, referred to as broken adaptive ridge (BAR) estimator, resulting from an L 0-based iteratively reweighted L 2 penalization algorithm using the ridge estimator as its initial value. We show that the BAR estimator is consistent for variable selection and has an oracle property for parameter estimation. Moreover, we show

The estimation of continuous treatment effect functions using observational data often requires parametric specification of the effect curves, the conditional distributions of outcomes and treatment assignments given multi-dimensional covariates. While nonparametric extensions are possible, they typically suffer from the curse of dimensionality. Dimension reduction is often inevitable and we propose

Principal Component Analysis (PCA) is a classical method for reducing the dimensionality of data by projecting them onto a subspace that captures most of their variation. Effective use of PCA in modern applications requires understanding its performance for data that are both high-dimensional and heteroscedastic. This paper analyzes the statistical performance of PCA in this setting, i.e., for high-dimensional

By optimizing index functions against different outcomes, we propose a multivariate single-index model (SIM) for development of medical indices that simultaneously work with multiple outcomes. Fitting of a multivariate SIM is not fundamentally different from fitting a univariate SIM, as the former can be written as a sum of multiple univariate SIMs with appropriate indicator functions. What have not

Variable selection plays a fundamental role in high-dimensional data analysis. Various methods have been developed for variable selection in recent years. Well-known examples are forward stepwise regression (FSR) and least angle regression (LARS), among others. These methods typically add variables into the model one by one. For such selection procedures, it is crucial to find a stopping criterion

Covariate measurement error is a common problem. Improper treatment of measurement errors may affect the quality of estimation and the accuracy of inference. Extensive literature exists on homoscedastic measurement error models, but little research exists on heteroscedastic measurement. In this paper, we consider a general parametric regression model allowing for a covariate measured with heteroscedastic

Large-margin classifiers are powerful techniques for classification problems. Although binary large-margin classifiers are heavily studied, multicategory problems are more complicated and challenging. A common approach is to construct k different decision functions for a k-class problem with a sum-to-zero constraint. However, such a constraint can be inefficient. Moreover, many large-margin classifiers

We consider the problem of high-dimensional classification between two groups with unequal covariance matrices. Rather than estimating the full quadratic discriminant rule, we propose to perform simultaneous variable selection and linear dimension reduction on the original data, with the subsequent application of quadratic discriminant analysis on the reduced space. In contrast to quadratic discriminant

It is frequently of interest to jointly analyze two paired sequences of multiple tests. This paper studies the problem of detecting whether there are more pairs of tests that are significant in both sequences than would be expected by chance. The asymptotic detection boundary is derived in terms of parameters such as the sparsity of non-null cases in each sequence, the effect sizes of the signals,

Many modern statistical problems can be cast in the framework of multivariate regression, where the main task is to make statistical inference for a possibly sparse and low-rank coefficient matrix. The low-rank structure in the coefficient matrix is of intrinsic multivariate nature, which, when combined with sparsity, can further lift dimension reduction, conduct variable selection, and facilitate

In the era of big data, integrative analyses that pool data from different sources are now extensively conducted in order to improve performance. Among many interesting applications, genomics research is an area where integrative methods become popular tools to identify prognostic biomarkers for various diseases. In this paper, we propose such a framework for pathway and gene identification. Our method

Exploring resting-state brain functional connectivity of autism spectrum disorders (ASD) using functional magnetic resonance imaging (fMRI) data has become a popular topic over the past few years. The data in a standard brain template consist of over 170,000 voxel specific points in time for each human subject. Such an ultra-high dimensionality makes the voxel-level functional connectivity analysis

Missing data occur frequently in a wide range of applications. In this paper, we consider estimation of high-dimensional covariance matrices in the presence of missing observations under a general missing completely at random model in the sense that the missingness is not dependent on the values of the data. Based on incomplete data, estimators for bandable and sparse covariance matrices are proposed