In this paper, we assume that cause–effect relationships among variables can be represented by a Gaussian linear structural equation model and a corresponding directed acyclic graph. For a set of intermediate variables that satisfies the front-door criterion, we provide the variance formula of the estimated mean outcome under an external intervention in which a treatment variable is set to a specified

This paper presents a new methodology, called AFSSEN, to simultaneously select significant predictors and produce smooth estimates in a high-dimensional function-on-scalar linear model with sub-Gaussian errors. Outcomes are assumed to lie in a general real separable Hilbert space, H, while parameters lie in a subspace known as a Cameron-Martin space, K, which are closely related to Reproducing Kernel

For a high dimensional linear model with a finite number of covariates measured with errors, we study statistical inference on the parameters associated with the error-prone covariates, and propose a new corrected decorrelated score test and a corresponding score type estimator. This work was motivated by a real data example, where both low dimensional phenotypic variables and high dimensional genotypic

In this article, we consider a non-parametric Bayesian approach to multivariate quantile regression. The proposed approach involves modeling of related conditional distributions of a response vector given the covariates using a Dependent Dirichlet Process (DDP) prior. The DDP is used to introduce dependence across covariates. The flexible covariate-dependent mixture of multivariate Gaussian kernels

This paper focuses on semi-parametric estimation of multivariate expectiles for extreme levels of risk. Multivariate expectiles and their extremes have been the focus of plentiful research in recent years. In particular, it has been noted that due to the difficulty in estimating these values for elevated levels of risk, an alternative formulation of the underlying optimization problem would be necessary

Dimension reduction methods for matrix or tensor data have been an active research field in recent years. Li et al. (2010) introduced the notion of the Kronecker envelope and proposed dimension folding estimators for supervised dimension reduction. In a data analysis of cryogenic electron microscopy (cryo-EM) images (Chen et al., 2014), Kronecker envelope principal component analysis (PCA) was used

We derive a stochastic representation for the probability distribution on the positive orthant (0,∞)d whose association between components is minimal among all probability laws with ℓp-norm symmetric survival functions. It is given by a transformation of a uniform distribution on the standard unit simplex that is multiplied with an independent finite mixture of certain beta distributions and an additional

This work is concerned with testing the proportionality between two high dimensional covariance matrices. Several tests for proportional covariance matrices, based on modifying the classical likelihood ratio test and applicable in high dimension, have been proposed in the literature. Despite their usefulness, they tend to have unsatisfactory performance for nonnormal high dimensional multivariate data

Expectiles are the solution to an asymmetric least squares minimization problem for univariate data. They resemble the quantiles, and just like them, expectiles are indexed by a level α in the unit interval. In the present paper, we introduce and discuss the main properties of the (multivariate) expectile regions, a nested family of sets, whose instance with level 0<α≤1∕2 is built up by all points

This paper proposes a tool for dimension reduction where the dimension of the original space is reduced: the principal loading analysis. Principal loading analysis is a tool to reduce dimensions by discarding variables. The intuition is that variables are dropped which distort the covariance matrix only by a little. Our method is introduced and an algorithm for conducting principal loading analysis

This paper presents a variable selection method for the Euclidean distance-based classifier in high-dimensional settings. We are concerned that the expected probabilities of misclassification (EPMC) for the Euclidean distance-based classifier may be increasing with dimension when redundant variables are included in feature values. First, we show the Euclidean distance-based classifier with only non-redundant

We consider the problem of kernel classification with nonignorable missing data. Instead of imposing a fully parametric model for the selection probability, which can be quite sensitive to the violations of model assumptions, here we consider a semiparametric exponential tilting selection probability model in the spirit of Kim and Yu (2011). In addition to the existing parameter estimators, we also

Statistical shape analysis (SSA) may be understood like a field of Multivariate Analysis and has assumed a prominent position due to its applicability in various areas; such as in imagery processing, biology, anatomy, among others. Biological hypotheses have been often formulating in the SSA framework, which is based on the concept of “shape” whose related data are in a non Euclidian space. Thus, tailored

The Minimum Average Variance Estimation (MAVE) method and its variants have proven to be effective approaches to the dimension reduction problems. However, as far as we know, using MAVE to estimate the Central Mean Subspace (CMS) with multivariate response receives little attention. This paper proposes a weighted version of MAVE for the CMS with multivariate response. The proposed weighted MAVE method

The estimation of structured covariance matrix arises in many fields. An appropriate covariance structure not only improves the accuracy of covariance estimation but also increases the efficiency of mean parameter estimators in statistical models. In this paper, a novel statistical method is proposed, which selects the optimal Toeplitz covariance structure and estimates the covariance matrix, simultaneously

Consider the p×p matrix that is the product of a population covariance matrix and the inverse of another population covariance matrix. Suppose that their difference has a divergent rank with respect to p, when two samples of sizes n and T from the two populations are available, we construct its corresponding sample version. In the high-dimensional regime where both n and T are proportional to p, we

The paper introduces a new class of row-column (RC) association models for contingency tables by allowing the user to select both the scale on which interactions are measured as in Kateri and Papaioannou (1994) and the type of logit (local, global, continuation) suitable for the row and column classification variables as in Bartolucci and Forcina (2002). These choices determine the matrix of interactions

We consider the problem of testing for association between a functional variable belonging to a Hilbert space and a scalar variable. Particularly, we propose a distribution-free test statistic based on Kendall’s Tau, which is a popular method for determining the association between two random variables. The distribution of the test statistic under the null hypothesis of independence is established

The Heckman selection model is perhaps the most popular econometric model in the analysis of data with sample selection. The analyses of this model are based on the normality assumption for the error terms, however, in some applications, the distribution of the error term departs significantly from normality, for instance, in the presence of heavy tails and/or atypical observation. In this paper, we

Numerous problems remain in the construction of statistical depth for functional data. Issues stem largely from the absence of a well-conceived notion of symmetry. The present paper proposes a topologically valid notion of symmetry for distributions on functional metric spaces and a corresponding notion of depth. The latter is shown to satisfy the axiomatic definition of functional depth introduced

With the recent progress of data acquisition technology, classification of data exhibiting relational dependence, from online social interactions to multi-omics studies to linkage of electronic health records, continues to gain an ever increasing attention. By introducing a robust and inherently geometric concept of data depth we propose a new type of geometrically-enhanced classification method for

We extend projection theorems concerning Hellinger and Jones et al. divergences to the continuous case. These projection theorems reduce certain estimation problems on generalized exponential models to linear problems. We introduce the notion of regularity for generalized exponential models and show that the projection theorems in this case are similar to the ones in discrete and canonical case. We

Vine copulas are a type of multivariate dependence model, composed of a collection of bivariate copulas that are combined according to a specific underlying graphical structure. Their flexibility and practicality in moderate and high dimensions have contributed to the popularity of vine copulas, but relatively little attention has been paid to their extremal properties. To address this issue, we present

In this paper, we work on a general multivariate regression model under the regime that both p, the number of covariates, and n, the number of observations, are large with p∕n→κ(0<κ<∞). Unlike previous works that focus on a sparse regression vector β, we consider a more interesting situation in which β is composed of two groups: components in group I are large while components in group II are small

Most existing methods of variable selection in partially linear models (PLM) with ultrahigh dimensional covariates are based on partial residuals, which involve a two-step estimation procedure. While the estimation error produced in the first step may have an impact on the second step, multicollinearity among predictors adds additional challenges in the model selection procedure. In this paper, we

Spatio-temporal processes with a continuous index in space and time are useful for modeling spatio-temporal data in many scientific disciplines such as environmental and health sciences. However, approaches that enable simultaneous estimation of the mean and covariance functions for such spatio-temporal processes are limited. Here, we propose a flexible spatio-temporal model with partially linear regression

Lévy copulas are an important tool which can be used to build dependent Lévy processes. In a classical setting, they have been used to model financial applications. In a Bayesian framework they have been employed to introduce dependent nonparametric priors which allow to model heterogeneous data. This paper focuses on introducing a new class of Lévy copulas based on a class of subordinators recently

In the high dimensional setting, this article explores the problem of testing the complete independence of random variables having a multivariate normal distribution. A natural high-dimensional extension of the test in Schott (2005) is proposed for this purpose. The newly defined tests are asymptotically distribution-free as both the sample size and the number of variables go to infinity and hence

A unified formula for various moments of the multivariate normal distribution with sectional truncation is derived using a non-recursive method, where sectional truncation is given by several sections (regions) for selection including single and double truncation as special cases. The moments include raw, central, arbitrarily deviated, non-absolute, absolute and partially absolute moments with non-integer

The halfspace depth can be seen as a mapping that to a finite Borel measure μ on the Euclidean space Rd assigns its depth, being a function Rd→[0,∞):x↦Dx;μ. The depth of μ quantifies how much centrally positioned a point x is with respect to μ. This function is intended to serve as generalization of the quantile function to multivariate spaces. We consider the problem of finding the inverse mapping

The paper considers estimation of the multivariate normal mean under a multivariate normal prior with a singular precision matrix. Such a setup appears in the multi-task averaging, serial and spatial smoothing problems. The empirical and hierarchical Bayes estimators shrink the maximum likelihood estimator by projecting it to the null space of the precision matrix. Conditions for minimaxity are given

Inaccuracy measure is an important measure in information theory, which have been considered recently by many researchers, so that various generalizations have been introduced for this measure. In this paper, two new inaccuracy measures using co-copula and dual of a copula in copula theory are introduced and their properties under specific conditions are investigated. Including, under the establishment

In economic and business data, the correlation matrix is a stochastic process that fluctuates over time and exhibits seasonality. The most widely-used approaches for estimating and forecasting the correlation matrix (e.g., multivariate GARCH) often are hindered by computational difficulties and require strong assumptions. In this paper we propose a method for modeling and forecasting correlation matrices

Canonical correlation analysis (CCA) is a common method used to estimate the associations between two different sets of variables by maximizing the Pearson correlation between linear combinations of the two sets of variables. We propose a version of CCA for transelliptical distributions with an elliptical copula using pairwise Kendall’s tau to estimate a latent scatter matrix. Because Kendall’s tau

We propose a new two-stage procedure for detecting multiple outliers when the dimension of the data is much larger than the available sample size. In the first stage, the data are split into two disjoint sets, one containing non-outliers and the other containing the rest of the data that are considered as potential outliers. In the second stage, a series of hypothesis tests is conducted to test the

Seemingly unrelated regression is a natural framework for regressing multiple correlated responses on multiple predictors. The model is very flexible, with multiple linear regression and covariance selection models being special cases. However, its practical deployment in genomic data analysis under a Bayesian framework is limited due to both statistical and computational challenges. The statistical

This paper discusses certain properties of heterogeneous hypergeometric functions with two matrix arguments. These functions are newly defined but have already appeared in statistical literature and are useful when dealing with the derivation of certain distributions for the eigenvalues of singular beta-Wishart matrices. The joint density function of the eigenvalues and the distribution of the largest

This work is concerned with optimal designs for multivariate regression of responses of mixed variable types (continuous and binary) on quantitative and qualitative factors. New complete class results with respect to the Loewner ordering, and relevant Chebyshev systems are derived to identify a small class of designs, within which locally optimal designs can be found for a group of models and optimality

We propose a new nonparametric independence test for two functional random variables. The test is based on a new dependence metric, the so-called angle covariance, which fully characterizes the independence of the random variables and generalizes the projection covariance proposed for random vectors. The angle covariance has a number of desirable properties, including the equivalence of its zero value

A classical result of Slepian (1962) for the normal distribution and extended by Das Guptas et al. (1972) for elliptical distributions gives one-sided (lower orthant) comparison criteria for the distributions with respect to the (generalized) correlations. Müller and Scarsini (2000) established that the ordering conditions even characterize the stronger supermodular ordering in the normal case. In

In this paper the interest is to elaborate on the generalization of bivariate association measures, namely Spearman’s rho, Kendall’s tau, Blomqvist’s beta and Gini’s gamma, for a general dimension d≥2. Desirable properties and axioms for such generalizations are discussed, where special attention is given to the impact of the addition of: (i) an independent random variable to a random vector; (ii)

Variable screening is a commonly used procedure in high dimensional data analysis to reduce dimensionality and ensure the applicability of available statistical methods. Such a procedure is complicated and computationally burdensome because spurious correlations commonly exist among predictor variables, while important predictor variables may not have large marginal correlations with the response variable

The correlation ratio has been used to measure how much the behavior of one variable can be predicted by the other variable. In this paper, we derive a new expression of the correlation ratio based on copulas. We represent the copula correlation ratio in terms of Spearman’s rho of the ∗-product of two copulas. Our expression provides a new way to obtain the copula correlation ratio, which is especially

In this work, we develop the asymptotic theory of the Detrended Fluctuation Analysis (DFA) and Detrended Cross-Correlation Analysis (DCCA) for trend-stationary stochastic processes without any assumption on the specific form of the underlying distribution. All results are presented and derived under the general framework of potentially overlapping boxes for the polynomial fit. We prove the stationarity

We study the power of deep neural networks (DNNs) with sigmoid activation function. Recently, it was shown that DNNs approximate any d-dimensional, smooth function on a compact set with a rate of order W−p∕d, where W is the number of nonzero weights in the network and p is the smoothness of the function. Unfortunately, these rates only hold for a special class of sparsely connected DNNs. We ask ourselves

This article contributes to the current statistical theory of deep neural networks (DNNs). It was shown that DNNs are able to circumvent the so-called curse of dimensionality in case that suitable restrictions on the structure of the regression function hold. In most of those results the tuning parameter is the sparsity of the network, which describes the number of non-zero weights in the network.

We prove the almost sure convergence of Oja-type processes to eigenvectors of the expectation B of a random matrix while relaxing the i.i.d. assumption on the observed random matrices (Bn) and assuming either (Bn) converges to B or (E[Bn|Tn]) converges to B where Tn is the sigma-field generated by the events before time n. As an application of this generalization, the online PCA of a random vector

In this paper, a new method called mean-of-quantile is introduced to estimate multivariate quantiles. The consistency and asymptotic normality of mean-of-quantile estimators are investigated. Furthermore, we apply empirical likelihood to mean-of-quantile estimators. The effectiveness of our new method is illustrated by Monte Carlo simulations and an empirical example.

We construct a kernel density estimator on symmetric spaces of non-compact type and establish an upper bound for its convergence rate, analogous to the minimax rate for classical kernel density estimators on Euclidean space. Symmetric spaces of non-compact type include hyperboloids of constant curvature −1 and spaces of symmetric positive definite matrices. This paper obtains a simplified formula in

Independent Component Analysis (ICA) offers an effective data-driven approach for blind source extraction encountered in many signal and image processing problems. Although many ICA methods have been developed, they have received relatively little attention in the statistics literature, especially in terms of rigorous theoretical investigation for statistical inference. The current paper aims at narrowing

Estimation of a high dimensional precision matrix is a critical problem to many areas of statistics including Gaussian graphical models and inference on high dimensional data. Working under the structural assumption of sparsity, we propose a novel methodology for estimating such matrices while controlling the false positive rate, percentage of matrix entries incorrectly chosen to be non-zero. We specifically

The fastICA method is a popular dimension reduction technique used to reveal patterns in data. Here we show both theoretically and in practice that the approximations used in fastICA can result in patterns not being successfully recognised. We demonstrate this problem using a two-dimensional example where a clear structure is immediately visible to the naked eye, but where the projection chosen by

Subgroup identification serves as an important step towards precision medicine which has attracted great attention recently. On the other hand, longitudinal data with dropouts often arises in medical research. However there is little work in subgroup identification considering this data type. Therefore, in this paper we propose a new subgroup identification method based on concave fusion penalization

We investigate the class of splitting distributions as the composition of a singular multivariate distribution and a univariate distribution. It will be shown that most common parametric count distributions (multinomial, negative multinomial, multivariate hypergeometric, multivariate negative hypergeometric, …) can be written as splitting distributions with separate parameters for both components,

In this paper, a new mixture family of multivariate normal distributions, formed by mixing multivariate normal distribution and a skewed distribution, is constructed. Some properties of this family, such as characteristic function, moment generating function, and the first four moments are derived. The distributions of affine transformations and canonical forms of the model are also derived. An EM-type

The projection median as introduced by Durocher and Kirkpatrick (2005); Durocher and Kirkpatrick (2009) is a robust multivariate, nonparametric location estimator. It is a weighted average of points in a sample, where each point’s weight is proportional to the fraction of directions in which that point is a univariate median. The projection median has the highest possible asymptotic breakdown and is

The problem of estimating the scale matrix Σ in a multivariate additive model, with elliptical noise, is considered from a decision-theoretic point of view. As the natural estimators of the form Σˆa=aS (where S is the sample covariance matrix and a is a positive constant) perform poorly, we propose estimators of the general form Σˆa,G=a(S+SS+G(Z,S)), where S+ is the Moore–Penrose inverse of S and G(Z

A recursive max-linear vector models causal dependence between its components by expressing each node variable as a max-linear function of its parental nodes in a directed acyclic graph and some exogenous innovation. Motivated by extreme value theory, innovations are assumed to have regularly varying distribution tails. We propose a scaling technique in order to determine a causal order of the node

We introduce a nonlinear additive functional principal component analysis (NAFPCA) for vector-valued functional data. This is a generalization of functional principal component analysis and allows the relations among the random functions involved to be nonlinear. The method is constructed via two additively nested Hilbert spaces of functions, in which the first space characterizes the functional nature

With the advent of the era of big data, high dimensional covariance matrices are increasingly encountered and testing covariance structure has become an active area in contemporary statistical inference. Conventional testing methods fail when addressing high dimensional data due to the singularity of the sample covariance matrices. In this paper, we propose a novel test for the prominent identity test