In the era of big data, testing the equality of multiple high-dimensional covariance matrices is a fundamental yet challenging task in modern multivariate analysis. This paper constructs a testing framework that addresses two practically important issues simultaneously: sparse differences between covariance matrices and small disturbances between covariance matrices. Based on this framework, we propose two new multi-sample test statistics, each targeting one of these distinct structural characteristics. To further improve the adaptivity of our approach, we combine the p-values of the two tests using Fisher’s method, thereby constructing a Fisher adaptive test statistic. Theoretically, we establish the asymptotic distributions of the two proposed component test statistics and rigorously prove their asymptotic independence in the multi-sample setting, providing a solid theoretical foundation for the Fisher adaptive test statistic. Simulation studies and real data analyses consistently demonstrate that the proposed test statistics outperform existing methods across various performance metrics.
Exchangeability is an important data structure and, if appropriately discovered, may help in discernible dimension reduction. This aspect carries special significance for high-dimensional data. For certain data sets, such as when same units are repeatedly observed, not only across all time points, but further under all experimental conditions, exchangeability is strongly expected, and a test to determine its presence is desired. For example, the same patients might be observed across several time points, under both drug and placebo. In another setting, expressions from normal and tumor cells from the same individuals may be evaluated. The notion of exchangeability in such cases implies that the distinction of different conditions is of no significance, or that the sub-vectors under different conditions are exchangeable. The aim of this article is to present tests for multivariate exchangeability when the distributions may not necessarily be normal and the dimensions of the vectors, even of sub-vectors, may exceed the sample size. The test statistics are composed of consistent and computationally highly efficient estimators, and their properties are studied under a few mild assumptions. Extension to two-layer pattern of exchangeability, i.e., when the sub-vectors are further partitioned into sub-sub-vectors, based on an additional variable, is also considered. Simulation results are used to show the accuracy of the proposed tests. Applications on real data examples are demonstrated.
A smooth simultaneous confidence band (SCB) is constructed for the cumulative distribution function of nonstationary spatial series based on a plug-in kernel distribution estimator. The kernel distribution estimator is computed from residuals after removing the bivariate spline trend over a smaller number of observation locations, while the bivariate trend estimator of spatial series is computed from the remaining larger set of observation locations. Under mild assumptions, it is shown that the kernel distribution estimator is a uniformly oracle-efficient estimator of the marginal error distribution, and the SCB has the same asymptotic properties as the classic Kolmogorov–Smirnov SCB based on the infeasible empirical cumulative distribution function of unobserved errors. Simulation experiments corroborate the theoretical findings. Specifically, the proposed SCB method is applied to real-world datasets, including carbon dioxide concentration data and Black Sea data, to evaluate the marginal error distribution.
Patients affected by Parkinson’s disease (PD) usually experience a range of symptoms, including movement disorder, sleep disturbance, and brain structural changes, and may start treatment at certain time points throughout clinical course. Current research focusing on a single modality (e.g., movement disorder) fails to display the full picture of the disease progression and its connection with timing of symptomatic therapy. In this paper, we integrate longitudinal multi-type measurements from multiple modalities (e.g., binary/ordinal clinical measures, continuous neuroimaging biomarkers) and uncover their relationship with the initiation of PD treatment using a latent Gaussian process joint model. The dependence structure between observed multi-type biomarkers is characterized by several underlying unobserved Gaussian processes through a generalized factor model. Instead of assuming a certain parametric form a priori, the Gaussian processes, with their philosophy to let data speak for themselves, are expected to capture the possibly nonlinear and complex progression profiles of PD-related measurements. The obtained Gaussian processes are also incorporated into a varying coefficient Cox model so as to jointly monitor the longitudinal PD-related measurements and their time-varying effects on time to initiation of PD treatment. The application of our proposed method to the Parkinson’s Progression Markers Initiative dataset yields a comprehensive understanding of PD progression by synthesizing information from clinical, biological, and neuroimaging perspectives.
