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Statistical depth provides a center–outward ordering of multivariate observations and is widely used in nonparametric inference. We study depth-based tests for multivariate two-sample problems and examine the behaviour of different depth notions using the DD plot (data-depth plot) across a variety of distributional space. The DD plot illustrates that depth functions differ in their sensitivity to distributional differences, emphasizing the importance of depth selection in two-sample testing. We propose a new two-sample test statistic, log DDR, constructed from ratios of numerical depth values rather than depth-induced ranks. Simulation studies under multiple scenarios and for three representative depth functions indicate that log DDR achieves improved power relative to several competing depth-based nonparametric tests. The results further demonstrate that the performance of log DDR and existing methods depends strongly on the chosen depth function, consistent with insights from the DD plot. These findings support a two-stage testing approach in which the DD plot is used to guide the choice of depth notion before applying log DDR for homogeneity testing.

We propose a robust methodology to evaluate the performance and computational efficiency of non-parametric two-sample tests, specifically designed for high-dimensional generative models in scientific applications such as in particle physics. The study focuses on tests built from univariate integral probability measures: the sliced Wasserstein distance and the mean of the Kolmogorov–Smirnov (KS) statistics, already discussed in the literature, and the novel sliced KS statistic. These metrics can be evaluated in parallel, allowing for fast and reliable estimates of their distribution under the null hypothesis. We also compare these metrics with the recently proposed unbiased Fréchet Gaussian distance and the unbiased quadratic Maximum Mean Discrepancy, computed with a quartic polynomial kernel. We evaluate the proposed tests on various distributions, focusing on their sensitivity to deformations parameterized by a single parameter ε . Our experiments include correlated Gaussians and mixtures of Gaussians in 5, 20, and 100 dimensions, and a particle physics dataset of gluon jets from the JetNet dataset, considering both jet- and particle-level features. Our results demonstrate that one-dimensional-based tests provide a level of sensitivity comparable to other multivariate metrics, but with significantly lower computational cost, making them ideal for evaluating generative models in high-dimensional settings. This methodology offers an efficient, standardized tool for model comparison and can serve as a benchmark for more advanced tests, including machine-learning-based approaches.

This article states a condition monitoring strategy for wind turbines using a statistical data-driven modeling approach by means of supervisory control and data acquisition (SCADA) data. Initially, a baseline data-based model is obtained from the healthy wind turbine by means of multiway principal component analysis (MPCA). Then, when the wind turbine is monitorized, new data is acquired and projected into the baseline MPCA model space. The acquired SCADA data are treated as a random process given the random nature of the turbulent wind. The objective is to decide if the multivariate distribution that is obtained from the wind turbine to be analyzed (healthy or not) is related to the baseline one. To achieve this goal, a test for the equality of population means is performed. Finally, the results of the test can determine that the hypothesis is rejected (and the wind turbine is faulty) or that there is no evidence to suggest that the two means are different, so the wind turbine can be considered as healthy. The methodology is evaluated on a wind turbine fault detection benchmark that uses a 5 MW high-fidelity wind turbine model and a set of eight realistic fault scenarios. It is noteworthy that the results, for the presented methodology, show that for a wide range of significance, α ∈ [ 1 % , 13 % ] , the percentage of correct decisions is kept at 100%; thus it is a promising tool for real-time wind turbine condition monitoring.

A high-pressure CO2 process applied to ready-to-eat food products guarantees an increase of both their microbial safety and shelf-life. However, the treatment often produces unwanted changes in the visual appearance of products depending on the adopted process conditions. Accordingly, the alteration of the visual appearance influences consumers’ perception and acceptability. This study aims at identifying the optimal treatment conditions in terms of visual appearance by using an artificial vision system. The developed methodology was applied to fresh-cut carrots (Daucus carota) as the test product. The results showed that carrots packaged in 100% CO2 and subsequently treated at 6 MPa and 40 °C for 15 min maintained an appearance similar to the fresh product for up to 7 days of storage at 4 °C. Mild appearance changes were identified at 7 and 14 days of storage in the processed products. Microbiological analysis performed on the optimal treatment condition showed the microbiological stability of the samples up to 14 days of storage at 4 °C. The artificial vision system, successfully applied to the CO2 pasteurization process, can easily be applied to any food process involving changes in the appearance of any food product.

In the paper we present a new test for comparison of the means of multivariate samples with unknown distributions. The test is based on the comparison of the distributions of the distances between the samples’ elements and their means using univariate two-sample Kolmogorov–Smirnov test. The activity of the suggested method is illustrated by numerical analysis of the real-world and simulated data.

Complex multivariate testing problems are frequently encountered in many scientific disciplines, such as engineering, medicine and the social sciences. As a result, modern statistics needs permutation testing for complex data with low sample size and many variables, especially in observational studies.

Refractive state can be regarded as a dynam-ic quantity. Multiple measurements of refractive state can be determined easily and rapidly on a number of different occasions using an autore-fractor. In an experimental trial undertaken by Gillan, a 30-year-old female was subjected to 30 autorefractor measurements each taken at vari-ous intervals before and after the instillation of Mydriacyl 1% (tropicamide) into her right eye. The purpose of this paper is to apply multivar-iate analysis of variance (MANOVA) to Gillan’s sample data in order to assess whether instillation of Mydriacyl into the eye affects variability of distance refractive state as well as mean distance refractive state as measured by an autorefractor. In  five  of  the  seven  cases  where  pairwise hypotheses  tests  were  performed,  it  is  con-cluded that at a 99% level of confidence there is no difference in variability of distance refrac-tive state before and after cycloplegia. In two of the three cases where MANOVA was applied, there is a significant difference at a 95% and at a 99% level of confidence in both variability of distance refractive state and mean distance refractive  state  with  and  without  cycloplegia.

This paper derives a duality result for a general class of hypothesis testing problems in multivariate analysis utilizing the relationship between convex cones and their polar cones together with the properties of minimum norm problems between points and cones in Euclidian space. Special cases of this result yield generalizations of a well-known duality relation in multivariate equality constraints testing. For example, any multivariate inequality constraints test on the parameters of a multivariate normal random vector has an equivalent multivariate one-sided test in terms of the vector of dual variables associated with the constraints. Also, any combination multivariate inequality and equality constraints test has an equivalent combination multivariate one-sided and two-sided test in terms of the vector of dual variables associated with both sets of constraints.

In multivariate analysis under normality assumptions, many likelihood ratio criteria (λ(n)) are distributed as$k\\prod^m_{i=1} Z^a_{li}(1 - Z_{li})^{b_i}\\prod^{m'}_{j=1} Z^{c_j}_{2j}$for some constants, k, m, m', ai, bi, and cjwhen their associated null hypotheses are true, where Zijare independently distributed beta variates. Let T(n)= -n-1ln λ(n). This paper shows that a sequence {T(n)} of this kind is asymptotically optimal in the sense of exact slopes. Explicit forms of the exact slopes are obtained.

The multivariate t probability of a special set Ar(c), defined in ξ1, is of interest in problems of confidence bounding and of testing multivariate hypotheses. This probability is derived analytically in terms of F distribution probabilities. Tables for use with up to five-variate t distributions are provided.