Distance correlation is a new measure of dependence between random vectors. Distance covariance and distance correlation are analogous to product-moment covariance and correlation, but unlike the classical definition of correlation, distance correlation is zero only if the random vectors are independent. The empirical distance dependence measures are based on certain Euclidean distances between sample elements rather than sample moments, yet have a compact representation analogous to the classical covariance and correlation. Asymptotic properties and applications in testing independence are discussed. Implementation of the test and Monte Carlo results are also presented.
A new index based on empirical copulas, termed the Copula Statistic (CoS), is introduced for assessing the strength of multivariate dependence and for testing statistical independence. New properties of the copulas are proved. They allow us to define the CoS in terms of a relative distance function between the empirical copula, the Fréchet-Hoeffding bounds and the independence copula. Monte Carlo simulations reveal that for large sample sizes, the CoS is approximately normal. This property is utilised to develop a CoS-based statistical test of independence against various noisy functional depe…
To quantify the dependence between two random vectors of possibly different dimensions, we propose to rely on the properties of the 2-Wasserstein distance. We first propose two coefficients that are based on the Wasserstein distance between the actual distribution and a reference distribution with independent components. The coefficients are normalized to take values between 0 and 1, where 1 represents the maximal amount of dependence possible given the two multivariate margins. We then make a quasi-Gaussian assumption that yields two additional coefficients rooted in the same ideas as the fir…
We present an index of dependence that allows one to measure the joint or mutual dependence of a $d$-dimensional random vector with $d>2$. The index is based on a $d$-dimensional Kendall process. We further propose a standardized version of our index of dependence that is easy to interpret, and provide an algorithm for its computation. We discuss tests of total independence based on consistent estimates of the area under the Kendall curve. We evaluate the performance of our procedures via simulation, and apply our methods to a real data set.
The problem of testing mutually exclusive hypotheses with dependent test statistics is considered. Bayesian and frequentist approaches to multiplicity control are studied and compared to help gain understanding as to the effect of test statistic dependence on each approach. The Bayesian approach is shown to have excellent frequentist properties and is argued to be the most effective way of obtaining frequentist multiplicity control, without sacrificing power, when there is considerable test statistic dependence.
This paper discusses the statistical inference problem associated with testing for dependence between two continuous random variables using Kendall's $τ$ in the context of the missing data problem. We prove the worst-case identified set for this measure of association always includes zero. The consequence of this result is that robust inference for dependence using Kendall's $τ$, where robustness is with respect to the form of the missingness-generating process, is impossible.
Test statistics are often strongly dependent in large-scale multiple testing applications. Most corrections for multiplicity are unduly conservative for correlated test statistics, resulting in a loss of power to detect true positives. We show that the Westfall--Young permutation method has asymptotically optimal power for a broad class of testing problems with a block-dependence and sparsity structure among the tests, when the number of tests tends to infinity.
In this paper, change-point problems for long memory stochastic volatility models are considered. A general testing problem which includes various alternative hypotheses is discussed. Under the hypothesis of stationarity the limiting behavior of CUSUM- and Wilcoxon-type test statistics is derived. In this context, a limit theorem for the two-parameter empirical process of long memory stochastic volatility time series is proved. In particular, it is shown that the asymptotic distribution of CUSUM test statistics may not be affected by long memory, unlike Wilcoxon test statistics which are typic…
In this paper, we study the effect of dependence on detecting sparse signals. In particular, we focus on global testing against sparse alternatives for the means of binary outcomes following an Ising model, and establish how the interplay between the strength and sparsity of a signal determines its detectability under various notions of dependence. The profound impact of dependence is best illustrated under the Curie-Weiss model where we observe the effect of a "thermodynamic" phase transition. In particular, the critical state exhibits a subtle "blessing of dependence" phenomenon in that one …
The performance of multiple hypothesis testing is known to be affected by the statistical dependence among random variables involved. The mechanisms responsible for this, however, are not well understood. We study the effects of the dependence structure of a finite state hidden Markov model (HMM) on the likelihood ratios critical for optimal multiple testing on the hidden states. Various convergence results are obtained for the likelihood ratios as the observations of the HMM form an increasing long chain. Analytic expansions of the first and second order derivatives are obtained for the case …
In this article, we consider the problem of simultaneous testing of hypotheses when the individual test statistics are not necessarily independent. Specifically, we consider the problem of simultaneous testing of point null hypotheses against two-sided alternatives for the mean parameters of normally distributed random variables. We assume that conditionally given the vector of means, these random variables jointly follow a multivariate normal distribution with a known but arbitrary covariance matrix. We consider a Bayesian framework where each unknown mean parameter is modeled through a two-c…
Quantifying tail dependence is an important issue in insurance and risk management. The prevalent tail dependence coefficient (TDC), however, is known to underestimate the degree of tail dependence and it does not capture non-exchangeable tail dependence since it evaluates the limiting tail probability only along the main diagonal. To overcome these issues, two novel tail dependence measures called the maximal tail concordance measure (MTCM) and the average tail concordance measure (ATCM) are proposed. Both measures are constructed based on tail copulas and possess clear probabilistic interpre…
This paper proposes the cross-quantilogram to measure the quantile dependence between two time series. We apply it to test the hypothesis that one time series has no directional predictability to another time series. We establish the asymptotic distribution of the cross quantilogram and the corresponding test statistic. The limiting distributions depend on nuisance parameters. To construct consistent confidence intervals we employ the stationary bootstrap procedure; we show the consistency of this bootstrap. Also, we consider the self-normalized approach, which is shown to be asymptotically pi…
Motivated by applications in biological science, we propose a novel test to assess the conditional mean dependence of a response variable on a large number of covariates. Our procedure is built on the martingale difference divergence recently proposed in Shao and Zhang (2014), and it is able to detect a certain type of departure from the null hypothesis of conditional mean independence without making any specific model assumptions. Theoretically, we establish the asymptotic normality of the proposed test statistic under suitable assumption on the eigenvalues of a Hermitian operator, which is c…
The Kendall plot ($\K$-plot) is a plot measuring dependence between the components of a bivariate random variable. The $\K$-plot graphs the Kendall distribution function against the distribution function of $VU$, where $V$ and $U$ are independent uniform $[0,1]$ random variables. We associate $\K$-plots with the receiver operating characteristic ($\ROC$) curve, a well-accepted graphical tool in biostatistics for evaluating the ability of a biomarker to discriminate between two populations. The most commonly used global index of diagnostic accuracy of biomarkers is the area under the $\ROC$ cur…
We study asymptotic properties of Bayesian multiple testing procedures and provide sufficient conditions for strong consistency under general dependence structure. We also consider a novel Bayesian multiple testing procedure and associated error measures that coherently accounts for the dependence structure present in the model. We advocate posterior versions of FDR and FNR as appropriate error rates and show that their asymptotic convergence rates are directly associated with the Kullback-Leibler divergence from the true model. Our results hold even when the class of postulated models is miss…
In the context of multiple hypotheses testing, the proportion $π_0$ of true null hypotheses in the pool of hypotheses to test often plays a crucial role, although it is generally unknown a priori. A testing procedure using an implicit or explicit estimate of this quantity in order to improve its efficency is called adaptive. In this paper, we focus on the issue of False Discovery Rate (FDR) control and we present new adaptive multiple testing procedures with control of the FDR. First, in the context of assuming independent $p$-values, we present two new procedures and give a unified review of …
We propose a series-based nonparametric specification test for a regression function when data are spatially dependent, the `space' being of a general economic or social nature. Dependence can be parametric, parametric with increasing dimension, semiparametric or any combination thereof, thus covering a vast variety of settings. These include spatial error models of varying types and levels of complexity. Under a new smooth spatial dependence condition, our test statistic is asymptotically standard normal. To prove the latter property, we establish a central limit theorem for quadratic forms i…
Statistical analysis of multimodal imaging data is a challenging task, since the data involves high-dimensionality, strong spatial correlations and complex data structures. In this paper, we propose rigorous statistical testing procedures for making inferences on the complex dependence of multimodal imaging data. Motivated by the analysis of multi-task fMRI data in the Human Connectome Project (HCP) study, we particularly address three hypothesis testing problems: (a) testing independence among imaging modalities over brain regions, (b) testing independence between brain regions within imaging…
Relational data are often represented as a square matrix, the entries of which record the relationships between pairs of objects. Many statistical methods for the analysis of such data assume some degree of similarity or dependence between objects in terms of the way they relate to each other. However, formal tests for such dependence have not been developed. We provide a test for such dependence using the framework of the matrix normal model, a type of multivariate normal distribution parameterized in terms of row- and column-specific covariance matrices. We develop a likelihood ratio test (L…
Complex large-scale studies, such as those related to microarray data and fMRI studies, often involve testing multiple hierarchically ordered hypotheses. However, most existing false discovery rate (FDR) controlling procedures do not exploit the inherent hierarchical structure among the tested hypotheses. In this paper, we first present a generalized stepwise procedure which generalizes the usual stepwise procedure to the case where each hypothesis is tested with a different set of critical constants. This procedure is helpful in creating a general framework under which our hierarchical testin…
We propose new concepts in order to analyze and model the dependence structure between two time series. Our methods rely exclusively on the order structure of the data points. Hence, the methods are stable under monotone transformations of the time series and robust against small perturbations or measurement errors. Ordinal pattern dependence can be characterized by four parameters. We propose estimators for these parameters, and we calculate their asymptotic distributions. Furthermore, we derive a test for structural breaks within the dependence structure. All results are supplemented by simu…
Testing for white noise has been well studied in the literature of econometrics and statistics. For most of the proposed test statistics, such as the well-known Box-Pierce's test statistic with fixed lag truncation number, the asymptotic null distributions are obtained under independent and identically distributed assumptions and may not be valid for the dependent white noise. Due to recent popularity of conditional heteroscedastic models (e.g., GARCH models), which imply nonlinear dependence with zero autocorrelation, there is a need to understand the asymptotic properties of the existing tes…
Heavy-tailed errors impair the accuracy of the least squares estimate, which can be spoiled by a single grossly outlying observation. As argued in the seminal work of Peter Huber in 1973 [{\it Ann. Statist.} {\bf 1} (1973) 799--821], robust alternatives to the method of least squares are sorely needed. To achieve robustness against heavy-tailed sampling distributions, we revisit the Huber estimator from a new perspective by letting the tuning parameter involved diverge with the sample size. In this paper, we develop nonasymptotic concentration results for such an adaptive Huber estimator, name…
