Measuring the (causal) direction and strength of dependence between two variables (events), Xi and Xj , is fundamental for all science. Our survey of decades-long literature on statistical dependence reveals that most assume symmetry in the sense that the strength of dependence of Xi on Xj exactly equals the strength of dependence of Xj on Xi. However, we show that such symmetry is often untrue in many real-world examples, being neither necessary nor sufficient. Vinod's (2014) asymmetric matrix R* in [-1, 1] of generalized correlation coefficients provides intuitively appealing, readily interpretable, and superior measures of dependence. This paper proposes statistical inference for R* using Taraldsen's (2021) exact sampling distribution of correlation coefficients and the bootstrap. When the direction is known, proposed asymmetric (one-tail) tests have greater power.
Positivity is one of the three conditions for causal inference from observational data. The standard way to validate positivity is to analyze the distribution of propensity. However, to democratize the ability to do causal inference by non-experts, it is required to design an algorithm to (i) test positivity and (ii) explain where in the covariate space positivity is lacking. The latter could be used to either suggest the limitation of further causal analysis and/or encourage experimentation where positivity is violated. The contribution of this paper is first present the problem of automatic …
Neyman and Pearson's theory of testing hypotheses does not warrant minimal epistemic reliability: the feature of driving to true conclusions more often than to false ones. The theory does not protect from the possible negative effects of the pragmatic value-laden unequal setting of error probabilities on the theory's epistemic reliability. Most importantly, in the case of a negative impact, no methodological adjustment is available to neutralize it, so in such cases, the discussed pragmatic-value-ladenness of the theory inevitably compromises the goal of attaining truth.
In this paper we introduce a robust to outliers Wilcoxon change-point testing procedure, for distinguishing between short-range dependent time series with a change in mean at unknown time and stationary long-range dependent time series. We establish the asymptotic distribution of the test statistic under the null hypothesis for $L_1$ near epoch dependent processes and show its consistency under the alternative. The Wilcoxon-type testing procedure similarly as the CUSUM-type testing procedure of Berkes, Horváth, Kokoszka and Shao (2006), requires estimation of the location of a possible change-…
We consider multiple testing means of many dependent Normal random variables that do not necessarily follow a joint Normal distribution. Under weak dependence, we show the uniform consistency of proportion estimators that are constructed as solutions to Lebesgue-Stieltjes equations for the setting of a point, bounded and one-sided null, respectively, and characterize via the index of weak dependence the sparsest proportion these estimators can consistently estimate. On the other hand, under a principal correlation structure and employing a suitable definition of p-value for composite null hypo…
Modelling multivariate tail dependence is one of the key challenges in extreme-value theory. Multivariate extremes are usually characterized using parametric models, some of which have simpler submodels at the boundary of their parameter space. Hypothesis tests are proposed for tail dependence parameters that, under the null hypothesis, are on the boundary of the alternative hypothesis. The asymptotic distribution of the weighted least squares estimator (Einmahl, Kiriliouk and Segers, Extremes 21, pages 205-233, 2018) is given when the true parameter vector is on the boundary of the parameter …
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…
Selecting skilled mutual funds through the multiple testing framework has received increasing attention from finance researchers and statisticians. The intercept $α$ of Carhart four-factor model is commonly used to measure the true performance of mutual funds, and positive $α$'s are considered as skilled. We observe that the standardized OLS estimates of $α$'s across the funds possess strong dependence and nonnormality structures, indicating that the conventional multiple testing methods are inadequate for selecting the skilled funds. We start from a decision theoretic perspective, and propose…
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.
This paper is concerned with testing and dating structural breaks in the dependence structure of multivariate time series. We consider a cumulative sum (CUSUM) type test for constant copula-based dependence measures, such as Spearman's rank correlation and quantile dependencies. The asymptotic null distribution is not known in closed form and critical values are estimated by an i.i.d. bootstrap procedure. We analyze size and power properties in a simulation study under different dependence measure settings, such as skewed and fat-tailed distributions. To date break points and to decide whether…
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…
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…
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…
Extremal dependence describes the strength of correlation between the largest observations of two variables. It is usually measured with symmetric dependence coefficients that do not depend on the order of the variables. In many cases, there is a natural asymmetry between extreme observations that can not be captured by such coefficients. An example for such asymmetry are large discharges at an upstream and a downstream stations on a river network: an extreme discharge at the upstream station will directly influence the discharge at the downstream station, but not vice versa. Simple measures f…
在这篇文章中，我们提出了一种新的方法来测试两组变量之间的依赖关系的基本任务。独立性的原假设和相关性的备择假设下的响应密度由非参数贝叶斯模型确定。在原假设下，联合分布由两个独立的Dirichlet过程混合（Dirichlet Process Mixture，DIFR）先验的乘积建模;在另一种情况下，全联合密度由多变量DIFR先验建模。然后，检验基于支持备择假设的后验概率。该检验不仅在线性相关性检验方面具有良好的性能，而且在非线性相关性检验方面也优于其他方法。在基因表达数据的分析中，我们比较了不同的方法来测试基因之间的成对依赖性。结果表明，该测试确定了一些依赖结构，没有检测到的其他测试。
We investigate testing of the hypothesis of independence between a covariate and the marks in a marked point process. It would be rather straightforward if the (unmarked) point process were independent of the covariate and the marks. In practice, however, such an assumption is questionable and possible dependence between the point process and the covariate or the marks may lead to incorrect conclusions. Therefore, we propose to investigate the complete dependence structure in the triangle points--marks--covariates together. We take advantage of the recent development of the nonparametric rando…
We study a broad class of asymmetric copulas introduced by Liebscher (2008) as a combination of multiple - usually symmetric - copulas. The main thrust of the paper is to provide new theoretical properties including exact tail dependence expressions and stability properties. A subclass of Liebscher copulas obtained by combining Fréchet copulas is studied in more details. We establish further dependence properties for copulas of this class and show that they are characterized by an arbitrary number of singular components. Furthermore, we introduce a novel iterative representation for general Li…
We propose a specification test for conditional location--scale models based on extremal dependence properties of the standardized residuals. We do so comparing the left-over serial extremal dependence -- as measured by the pre-asymptotic tail copula -- with that arising under serial independence at different lags. Our main theoretical results show that the proposed Portmanteau-type test statistics have nuisance parameter-free asymptotic limits. The test statistics are easy to compute, as they only depend on the standardized residuals, and critical values are likewise easily obtained from the …
Since the inception of Bitcoin in 2008, cryptocurrencies have played an increasing role in the world of e-commerce, but the recent turbulence in the cryptocurrency market in 2018 has raised some concerns about their stability and associated risks. For investors, it is crucial to uncover the dependence relationships between cryptocurrencies for a more resilient portfolio diversification. Moreover, the stochastic behavior in both tails is important, as long positions are sensitive to a decrease in prices (lower tail), while short positions are sensitive to an increase in prices (upper tail). In …
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.
