Distribution-Free Tests of Independence in High Dimensions

We consider the testing of mutual independence among all entries in a $d$-dimensional random vector based on $n$ independent observations. We study two families of distribution-free test statistics, which include Kendall's tau and Spearman's rho as important examples. We show that under the null hypothesis the test statistics of these two families converge weakly to Gumbel distributions, and propose tests that control the type I error in the high-dimensional setting where $d>n$. We further show that the two tests are rate-optimal in terms of power against sparse alternatives, and outperform competitors in simulations, especially when $d$ is large.Comment: to appear in Biometrik

MATS: Inference for potentially Singular and Heteroscedastic MANOVA

In many experiments in the life sciences, several endpoints are recorded per subject. The analysis of such multivariate data is usually based on MANOVA models assuming multivariate normality and covariance homogeneity. These assumptions, however, are often not met in practice. Furthermore, test statistics should be invariant under scale transformations of the data, since the endpoints may be measured on different scales. In the context of high-dimensional data, Srivastava and Kubokawa (2013) proposed such a test statistic for a specific one-way model, which, however, relies on the assumption of a common non-singular covariance matrix. We modify and extend this test statistic to factorial MANOVA designs, incorporating general heteroscedastic models. In particular, our only distributional assumption is the existence of the group-wise covariance matrices, which may even be singular. We base inference on quantiles of resampling distributions, and derive confidence regions and ellipsoids based on these quantiles. In a simulation study, we extensively analyze the behavior of these procedures. Finally, the methods are applied to a data set containing information on the 2016 presidential elections in the USA with unequal and singular empirical covariance matrices

A One-Sample Test for Normality with Kernel Methods

We propose a new one-sample test for normality in a Reproducing Kernel Hilbert Space (RKHS). Namely, we test the null-hypothesis of belonging to a given family of Gaussian distributions. Hence our procedure may be applied either to test data for normality or to test parameters (mean and covariance) if data are assumed Gaussian. Our test is based on the same principle as the MMD (Maximum Mean Discrepancy) which is usually used for two-sample tests such as homogeneity or independence testing. Our method makes use of a special kind of parametric bootstrap (typical of goodness-of-fit tests) which is computationally more efficient than standard parametric bootstrap. Moreover, an upper bound for the Type-II error highlights the dependence on influential quantities. Experiments illustrate the practical improvement allowed by our test in high-dimensional settings where common normality tests are known to fail. We also consider an application to covariance rank selection through a sequential procedure

Aligned Rank Tests As Robust Alternatives For Testing Interactions In Multiple Group Repeated Measures Designs With Heterogeneous Covariances

Data simulation was used to investigate whether tests performed on aligned ranks (Beasley, 2002) could be used as robust alternatives to parametric methods for testing a split-plot interaction with non-normal data and heterogeneous covariance matrices. Results indicated the aligned rank method do not have any distinct advantage over parametric methods in this situation

Multivariate small sample tests for two-way designs with applications to industrial statistics

In this paper, we present a novel nonparametric approach for multivariate analysis of two-way crossed factorial design based on NonParametric Combination applied to Synchronized Permutation tests. This nonparametric hypothesis testing procedure not only allows to overcome the shortcomings of MANOVA test like violation of assumptions such as multivariate normality or covariance homogeneity, but, in an extensive simulation study, reveals to be a powerful instrument both in case of small sample size and many response variables. We contextualize its application in the field of industrial experiments and we assume a linear additive model for the data set analysis. Indeed, the linear additive model interpretation well adapts to the industrial production environment because of the way control of production machineries is implemented. The case of small sample size reflects the frequent needs of practitioners in the industrial environment where there are constraints or limited resources for the experimental design. Furthermore, an increase in rejection rate can be observed under alternative hypothesis when the number of response variables increases with fixed number of observed units. This could lead to a strategical benefit considering that in many real problems it could be easier to collect more information on a single experimental unit than adding a new unit to the experimental design. An application to industrial thermoforming processes is useful to illustrate and highlight the benefits of the adoption of the herein presented nonparametric approach

A kinematic analysis of hand configurations in static and dynamic fingerspelling

The focus of this study was the investigation of target handshapes in American Sign Language fingerspelling in order to determine whether there was a difference between static canonical structures and structures produced in the context of a movement sequence. This was achieved by measuring the joint angles of a signing hand with an 18-sensor CyberGlove® by Virtual Technologies, Inc. A discriminant analysis was used to identify targets that occurred at points of minimum angular joint velocity. A multivariate analysis of variance with planned compansons was then applied to these dynamic data along with the static data to test the hypothesis. The results showed that there was a significant difference between handshapes produced statically and those produced dynamically, which suggested that a simple, cipher model of static handshapes produced within the context of a movement sequence is not sufficient to account for the production and perception of fingerspelling. These findings may be applied to future research in sign language recognition, so that consideration of the variability of target handshapes, as influenced by the spatiotemporal environment, might be incorporated into future models