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

Joint models for a wide class of response variables and longitudinal measurements consist on a mixed-effects model to fit longitudinal trajectories whose random effects enter as covariates in a generalized linear model for the primary response. They provide a useful way to assess association between these two kinds of data, which in clinical studies are often collected jointly on a series of individuals

In this paper, test statistics for repeated measures design are introduced when the dimension is large. By large dimension is meant the number of repeated measures and the total sample size grow together but either one could be larger than the other. Asymptotic distribution of the statistics are derived for the equal as well as unequal covariance cases in the balanced as well as unbalanced cases. The

We consider semiparametric analysis of competing risks data subject to mixed case interval censoring. The Fine-Gray model (Fine & Gray, 1999) is used to model the cumulative incidence function and is coupled with sieve semiparametric maximum likelihood estimation based on univariate or multivariate likelihood. The univariate likelihood of cause-specific data enables separate estimation of cumulative

Motivated by differential co-expression analysis in genomics, we consider in this paper estimation and testing of high-dimensional differential correlation matrices. An adaptive thresholding procedure is introduced and theoretical guarantees are given. Minimax rate of convergence is established and the proposed estimator is shown to be adaptively rate-optimal over collections of paired correlation

In biomedical studies on HIV RNA dynamics, viral loads generate repeated measures that are often subjected to upper and lower detection limits, and hence these responses are either left- or right-censored. Linear and non-linear mixed-effects censored (LMEC/NLMEC) models are routinely used to analyse these longitudinal data, with normality assumptions for the random effects and residual errors. However

Inference about dependencies in a multiway data array can be made using the array normal model, which corresponds to the class of multivariate normal distributions with separable covariance matrices. Maximum likelihood and Bayesian methods for inference in the array normal model have appeared in the literature, but there have not been any results concerning the optimality properties of such estimators

This paper proposes a new method for estimating sparse precision matrices in the high dimensional setting. It has been popular to study fast computation and adaptive procedures for this problem. We propose a novel approach, called Sparse Column-wise Inverse Operator, to address these two issues. We analyze an adaptive procedure based on cross validation, and establish its convergence rate under the

T-optimum designs for model discrimination are notoriously difficult to find because of the computational difficulty involved in solving an optimization problem that involves two layers of optimization. Only a handful of analytical T-optimal designs are available for the simplest problems; the rest in the literature are found using specialized numerical procedures for a specific problem. We propose

This paper is concerned with the inference of nonparametric mean function in a time series context. The commonly used kernel smoothing estimate is asymptotically normal and the traditional inference procedure then consistently estimates the asymptotic variance function and relies upon normal approximation. Consistent estimation of the asymptotic variance function involves another level of nonparametric

The effective degrees of freedom is a useful concept for describing model complexity. Recently the number of effective degrees of freedom has been shown to relate to the concept of conditional Akaike information (cAI) in the mixed effects models. This relationship was made explicit under linear mixed-effects models with i.i.d. errors, and later also extended to the generalized linear and the proportional

Most existing works on specification testing assume that we have direct observations from the model of interest. We study specification testing for Markov models based on contaminated observations. The evolving model dynamics of the unobservable Markov chain is implicitly coded into the conditional distribution of the observed process. To test whether the underlying Markov chain follows a parametric

This paper is motivated by a wide range of background correction problems in gene array data analysis, where the raw gene expression intensities are measured with error. Estimating a conditional density function from the contaminated expression data is a key aspect of statistical inference and visualization in these studies. We propose re-weighted deconvolution kernel methods to estimate the conditional

When an n × 1 random vector X = (X1, …, Xn ) T has a sign-invariant distribution, Strait [J. Multivariate Anal. 4 (1974) 494-496] proved that the expectations of max(0, X1, X1 + X2, …, X1 + Xn ) and max(0, X1, …, Xn ) are equal. In this note we assume a weaker condition that (X1, X2, …, Xn ) and (-X1, X2, …, Xn ) are equal in distribution and prove a more general result that the expectations of Lr

Linear mixed models (LMMs) are widely used for regression analysis of data that are assumed to be clustered or correlated. Assessing model fit is important for valid inference but to date no confirmatory tests are available to assess the adequacy of the fixed effects part of LMMs against general alternatives. We therefore propose a class of goodness-of-fit tests for the mean structure of LMMs. Our

In this article, we propose a computationally efficient approach to estimate (large) p-dimensional covariance matrices of ordered (or longitudinal) data based on an independent sample of size n. To do this, we construct the estimator based on a k-band partial autocorrelation matrix with the number of bands chosen using an exact multiple hypothesis testing procedure. This approach is considerably faster

Semi-competing risks data frequently arise in biomedical studies when time to a disease landmark event is subject to dependent censoring by death, the observation of which however is not precluded by the occurrence of the landmark event. In observational studies, the analysis of such data can be further complicated by left truncation. In this work, we study a varying co-efficient subdistribution regression