Temporal data are increasingly prevalent in modern data science. A fundamental question is whether two time series are related or not. Existing approaches often have limitations, such as relying on parametric assumptions, detecting only linear associations, and requiring multiple tests and corrections. While many non-parametric and universally consistent dependence measures have recently been proposed, directly applying them to temporal data can inflate the p-value and result in an invalid test. To address these challenges, this paper introduces the temporal dependence statistic with block permutation to test independence between temporal data. Under proper assumptions, the proposed procedure is asymptotically valid and universally consistent for testing independence between stationary time series, and capable of estimating the optimal dependence lag that maximizes the dependence. Moreover, it is compatible with a rich family of distance and kernel based dependence measures, eliminates the need for multiple testing, and exhibits excellent testing power in various simulation settings.
Testing (conditional) independence of multivariate random variables is a task central to statistical inference and modelling in general - though unfortunately one for which to date there does not exist a practicable workflow. State-of-art workflows suffer from the need for heuristic or subjective manual choices, high computational complexity, or strong parametric assumptions. We address these problems by establishing a theoretical link between multivariate/conditional independence testing, and model comparison in the multivariate predictive modelling aka supervised learning task. This link all…
This paper develops a model-free sequential test for conditional independence. The proposed test allows researchers to analyze an incoming i.i.d. data stream with any arbitrary dependency structure, and safely conclude whether a feature is conditionally associated with the response under study. We allow the processing of data points online, as soon as they arrive, and stop data acquisition once significant results are detected, rigorously controlling the type-I error rate. Our test can work with any sophisticated machine learning algorithm to enhance data efficiency to the extent possible. The…
Testing Conditional Mean Independence Using Generative Neural Networks
University of Illinois at Urbana-Champaign（伊利诺伊大学厄巴纳-香槟分校）；University of Maryland, College Park（马里兰大学帕克分校）；Washington University in St Louis（圣路易斯华盛顿大学）
Department of Statistical Science University College London（伦敦大学学院统计科学系）；Leibniz Institute for Prevention Research and Epidemiology – BIPS & Faculty of Mathematics and Computer Science University of Bremen（不来梅大学莱布尼茨预防研究与流行病学研究所- BIPS及数学与计算机科学学院）
We propose the conditional predictive impact (CPI), a consistent and unbiased estimator of the association between one or several features and a given outcome, conditional on a reduced feature set. Building on the knockoff framework of Candès et al. (2018), we develop a novel testing procedure that works in conjunction with any valid knockoff sampler, supervised learning algorithm, and loss function. The CPI can be efficiently computed for high-dimensional data without any sparsity constraints. We demonstrate convergence criteria for the CPI and develop statistical inference procedures for eva…
Competing risks data with discrete lifetime comes up in practice. However, only limited literature exists for such data. In this paper, we propose a non-parametric test based on U-statistics for testing independence of time to failure and cause of failure of competing risks data when the lifetime is a discrete random variable. Asymptotic distribution of the proposed test statistic is derived. An extensive Monte Carlo simulation study is conducted to assess the finite sample performance of the proposed test. The flexibility of the testing procedure is illustrated using real data sets on oral ca…
We propose an estimator of the Hilbert-Schmidt Independence Criterion obtained from an appropriate modification of the usual estimator. We then get asymptotic normality of this estimator both under independence hypothesis and under the alternative hypothesis. A new test for independence of random variables valued into metric spaces is then introduced, and a simulation study that allows to compare the proposed test to an existing one is provided
University of Cambridge（剑桥大学）；The Alan Turing Institute（艾伦·图灵研究所）；University of California Los Angeles（加州大学洛杉矶分校）
We consider the hypothesis testing problem of detecting conditional dependence, with a focus on high-dimensional feature spaces. Our contribution is a new test statistic based on samples from a generative adversarial network designed to approximate directly a conditional distribution that encodes the null hypothesis, in a manner that maximizes power (the rate of true negatives). We show that such an approach requires only that density approximation be viable in order to ensure that we control type I error (the rate of false positives); in particular, no assumptions need to be made on the form …
We study the problem of independence testing given independent and identically distributed pairs taking values in a $σ$-finite, separable measure space. Defining a natural measure of dependence $D(f)$ as the squared $L^2$-distance between a joint density $f$ and the product of its marginals, we first show that there is no valid test of independence that is uniformly consistent against alternatives of the form $\{f: D(f) \geq ρ^2 \}$. We therefore restrict attention to alternatives that impose additional Sobolev-type smoothness constraints, and define a permutation test based on a basis expansi…
Goodness-of-fit (GoF) testing is ubiquitous in statistics, with direct ties to model selection, confidence interval construction, conditional independence testing, and multiple testing, just to name a few applications. While testing the GoF of a simple (point) null hypothesis provides an analyst great flexibility in the choice of test statistic while still ensuring validity, most GoF tests for composite null hypotheses are far more constrained, as the test statistic must have a tractable distribution over the entire null model space. A notable exception is co-sufficient sampling (CSS): resampl…
We study a class of distributed hypothesis testing against conditional independence problems. Under the criterion that stipulates minimization of the Type II error rate subject to a (constant) upper bound $ε$ on the Type I error rate, we characterize the set of encoding rates and exponent for both discrete memoryless and memoryless vector Gaussian settings.
As a crucial problem in statistics is to decide whether additional variables are needed in a regression model. We propose a new multivariate test to investigate the conditional mean independence of Y given X conditioning on some known effect Z, i.e., E(Y|X, Z) = E(Y|Z). Assuming that E(Y|Z) and Z are linearly related, we reformulate an equivalent notion of conditional mean independence through transformation, which is approximated in practice. We apply the martingale difference divergence (Shao and Zhang, 2014) to measure conditional mean dependence, and show that the estimation error from app…
We consider testing marginal independence versus conditional independence in a trivariate Gaussian setting. The two models are non-nested and their intersection is a union of two marginal independences. We consider two sequences of such models, one from each type of independence, that are closest to each other in the Kullback-Leibler sense as they approach the intersection. They become indistinguishable if the signal strength, as measured by the product of two correlation parameters, decreases faster than the standard parametric rate. Under local alternatives at such rate, we show that the asy…
A distributed binary hypothesis testing (HT) problem over a noisy (discrete and memoryless) channel studied previously by the authors is investigated from the perspective of the strong converse property. It was shown by Ahlswede and Csiszár that a strong converse holds in the above setting when the channel is rate-limited and noiseless. Motivated by this observation, we show that the strong converse continues to hold in the noisy channel setting for a special case of HT known as testing against independence (TAI). The proof utilizes the blowing up lemma and the recent change of measure techniq…
We propose a test of independence of two multivariate random vectors, given a sample from the underlying population. Our approach, which we call MINT, is based on the estimation of mutual information, whose decomposition into joint and marginal entropies facilitates the use of recently-developed efficient entropy estimators derived from nearest neighbour distances. The proposed critical values, which may be obtained from simulation (in the case where one marginal is known) or resampling, guarantee that the test has nominal size, and we provide local power analyses, uniformly over classes of de…
Hypothesis testing is a useful statistical tool in determining whether a given model should be rejected based on a sample from the population. Sample data may contain sensitive information about individuals, such as medical information. Thus it is important to design statistical tests that guarantee the privacy of subjects in the data. In this work, we study hypothesis testing subject to differential privacy, specifically chi-squared tests for goodness of fit for multinomial data and independence between two categorical variables. We propose new tests for goodness of fit and independence testi…
We consider the problem of conditional independence testing of $X$ and $Y$ given $Z$ where $X,Y$ and $Z$ are three real random variables and $Z$ is continuous. We focus on two main cases - when $X$ and $Y$ are both discrete, and when $X$ and $Y$ are both continuous. In view of recent results on conditional independence testing (Shah and Peters, 2018), one cannot hope to design non-trivial tests, which control the type I error for all absolutely continuous conditionally independent distributions, while still ensuring power against interesting alternatives. Consequently, we identify various, nat…
In this paper, we introduce a ${\mathcal L}_2$ type test for testing mutual independence and banded dependence structure for high dimensional data. The test is constructed based on the pairwise distance covariance and it accounts for the non-linear and non-monotone dependences among the data, which cannot be fully captured by the existing tests based on either Pearson correlation or rank correlation. Our test can be conveniently implemented in practice as the limiting null distribution of the test statistic is shown to be standard normal. It exhibits excellent finite sample performance in our …
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…
The Multinomial Logit (MNL) model and the axiom it satisfies, the Independence of Irrelevant Alternatives (IIA), are together the most widely used tools of discrete choice. The MNL model serves as the workhorse model for a variety of fields, but is also widely criticized, with a large body of experimental literature claiming to document real-world settings where IIA fails to hold. Statistical tests of IIA as a modelling assumption have been the subject of many practical tests focusing on specific deviations from IIA over the past several decades, but the formal size properties of hypothesis te…
The partial copula provides a method for describing the dependence between two random variables $X$ and $Y$ conditional on a third random vector $Z$ in terms of nonparametric residuals $U_1$ and $U_2$. This paper develops a nonparametric test for conditional independence by combining the partial copula with a quantile regression based method for estimating the nonparametric residuals. We consider a test statistic based on generalized correlation between $U_1$ and $U_2$ and derive its large sample properties under consistency assumptions on the quantile regression procedure. We demonstrate thro…
Model selection for Gaussian concentration graph is based on multiple testing of pairwise conditional independence. In practical applications partial correlation tests are widely used. However it is not known whether partial correlation test is uniformly most powerful for pairwise conditional independence testing. This question is answered in the paper. Uniformly most powerful unbiased test of Neymann structure is obtained. It turns out, that this test can be reduced to usual partial correlation test. It implies that partial correlation test is uniformly most powerful unbiased one.
In this paper the nonparametric quantile regression model is considered in a location-scale context. The asymptotic properties of the empirical independence process based on covariates and estimated residuals are investigated. In particular an asymptotic expansion and weak convergence to a Gaussian process are proved. The results can, on the one hand, be applied to test for validity of the location-scale model. On the other hand, they allow to derive various specification tests in conditional quantile location-scale models. In detail a test for monotonicity of the conditional quantile curve is…
For two or more classes (or types) of points, nearest neighbor contingency tables (NNCTs) are constructed using nearest neighbor (NN) frequencies and are used in testing spatial segregation of the classes. Pielou's test of independence, Dixon's cell-specific, class-specific, and overall tests are the tests based on NNCTs (i.e., they are NNCT-tests). These tests are designed and intended for use under the null pattern of random labeling (RL) of completely mapped data. However, it has been shown that Pielou's test is not appropriate for testing segregation against the RL pattern while Dixon's te…
We treat the problem of testing independence between m continuous variables when m can be larger than the available sample size n. We consider three types of test statistics that are constructed as sums or sums of squares of pairwise rank correlations. In the asymptotic regime where both m and n tend to infinity, a martingale central limit theorem is applied to show that the null distributions of these statistics converge to Gaussian limits, which are valid with no specific distributional or moment assumptions on the data. Using the framework of U-statistics, our result covers a variety of ran…
In this paper, the problem of error control of stepwise multiple testing procedures is considered. For two-sided hypotheses, control of both type 1 and type 3 (or directional) errors is required, and thus mixed directional familywise error rate control and mixed directional false discovery rate control are each considered by incorporating both types of errors in the error rate. Mixed directional familywise error rate control of stepwise methods in multiple testing has proven to be a challenging problem, as demonstrated in Shaffer (1980). By an appropriate formulation of the problem, some new s…
We find the asymptotic distribution of the multi-dimensional multi-scale and kernel estimators for high-frequency financial data with microstructure. Sampling times are allowed to be asynchronous and endogenous. In the process, we show that the classes of multi-scale and kernel estimators for smoothing noise perturbation are asymptotically equivalent in the sense of having the same asymptotic distribution for corresponding kernel and weight functions. The theory leads to multi-dimensional stable central limit theorems and feasible versions. Hence they allow to draw statistical inference for a …
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 …
In this paper, we investigate local permutation tests for testing conditional independence between two random vectors $X$ and $Y$ given $Z$. The local permutation test determines the significance of a test statistic by locally shuffling samples which share similar values of the conditioning variables $Z$, and it forms a natural extension of the usual permutation approach for unconditional independence testing. Despite its simplicity and empirical support, the theoretical underpinnings of the local permutation test remain unclear. Motivated by this gap, this paper aims to establish theoretical …
In this article, we propose a class of $L_q$-norm based U-statistics for a family of global testing problems related to high-dimensional data. This includes testing of mean vector and its spatial sign, simultaneous testing of linear model coefficients, and testing of component-wise independence for high-dimensional observations, among others. Under the null hypothesis, we derive asymptotic normality and independence between $L_q$-norm based U-statistics for several $q$s under mild moment and cumulant conditions. A simple combination of two studentized $L_q$-based test statistics via their $p$-…
In the classical two-sample problem, the conventional approach for testing distributions equality is based on the difference between the two marginal empirical distribution functions, whereas a test for independence is based on the contrast between the bivariate and the product of the marginal empirical distribution functions. In this article we consider the problem of testing independence and distributions equality when the observer is "colour blind" so he cannot distinguish the distribution which has generated each of the two measurements. Within a nonparametric framework, we propose an empi…
In this paper an autoregressive time series model with conditional heteroscedasticity is considered, where both conditional mean and conditional variance function are modeled nonparametrically. A test for the model assumption of independence of innovations from past time series values is suggested. The test is based on an weighted $L^2$-distance of empirical characteristic functions. The asymptotic distribution under the null hypothesis of independence is derived and consistency against fixed alternatives is shown. A smooth autoregressive residual bootstrap procedure is suggested and its perfo…
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…
This work investigates the problem of testing whether $d$ functional random variables are jointly independent using a modified estimator of the $d$-variable Hilbert Schmidt Indepedence Criterion ($d$HSIC) which generalizes HSIC for the case where $d \geq 2$. We then get asymptotic normality of this estimator both under joint independence hypothesis and under the alternative hypothesis. A simulation study shows good performance of the proposed test on finite sample.
We study a class of $K$-encoder hypothesis testing against conditional independence problems. Under the criterion that stipulates minimization of the Type II error subject to a (constant) upper bound $ε$ on the Type I error, we characterize the set of encoding rates and exponent for both discrete memoryless and memoryless vector Gaussian settings. For the DM setting, we provide a converse proof and show that it is achieved using the Quantize-Bin-Test scheme of Rahman and Wagner. For the memoryless vector Gaussian setting, we develop a tight outer bound by means of a technique that relies on th…
From Conditional to Unconditional Independence: Testing Conditional Independence via Transport Maps
Institute of Statistics and Big Data, Renmin University of China（中国人民大学统计与大数据研究院）；Department of Data Science and Artificial Intelligence, The Hong Kong Polytechnic University（香港理工大学数据科学与人工智能学院）；Department of Applied Mathematics, The Hong Kong Polytechnic University（香港理工大学应用数学系）
Sequential Kernel-based Conditional Independence Testing via Adaptive Betting
Department of Computer Science, University of British Columbia, Vancouver, Canada（不列颠哥伦比亚大学计算机科学系，温哥华，加拿大）；Alberta Machine Intelligence Institute, Edmonton, Canada（阿尔伯塔机器智能研究所，爱德蒙顿，加拿大）
This paper introduces the \textit{weighted partial copula} function for testing conditional independence. The proposed test procedure results from these two ingredients: (i) the test statistic is an explicit Cramer-von Mises transformation of the \textit{weighted partial copula}, (ii) the regions of rejection are computed using a bootstrap procedure which mimics conditional independence by generating samples from the product measure of the estimated conditional marginals. Under conditional independence, the weak convergence of the \textit{weighted partial copula proces}s is established when th…
High-Dimensional Independence Testing via Maximum and Average Distance Correlations
Department of Applied Economics and Statistics, University of Delaware（应用经济学与统计学系，德克萨斯大学）；Department of Statistics, Operations, and Data Science, Temple University（统计学、运营与数据科学系，泰勒大学）
Federal University of São Carlos（圣卡洛斯联邦大学）；University of São Paulo（圣保罗大学）
Conditional independence testing is a key problem required by many machine learning and statistics tools. In particular, it is one way of evaluating the usefulness of some features on a supervised prediction problem. We propose a novel conditional independence test in a predictive setting, and show that it achieves better power than competing approaches in several settings. Our approach consists in deriving a p-value using a permutation test where the predictive power using the unpermuted dataset is compared with the predictive power of using dataset where the feature(s) of interest are permut…
Constraint based causal structure learning for point processes require empirical tests of local independence. Existing tests require strong model assumptions, e.g. that the true data generating model is a Hawkes process with no latent confounders. Even when restricting attention to Hawkes processes, latent confounders are a major technical difficulty because a marginalized process will generally not be a Hawkes process itself. We introduce an expansion similar to Volterra expansions as a tool to represent marginalized intensities. Our main theoretical result is that such expansions can approxi…
