Corrections to “Nonparametric Two-Sample Testing by Betting”

Nonparametric vs Parametric Tests of Location in Biomedical Research

Given samples from two distributions, a non-parametric two-sample test aims at determining whether the two distributions are equal or not, based on a test statistic. Classically, this statistic is computed on the whole data set, or is computed on a subset of the data set by a function trained on its complement. We consider methods in a third tier, so as to deal with large (possibly infinite) data sets, and to automatically determine the most relevant scales to work at, making two contributions. First, we develop a generic sequential non-parametric testing framework, in which the sample size need not be fixed in advance. This makes our test a truly sequential non-parametric multivariate two-sample test. Under information theoretic conditions qualifying the difference between the tested distributions, consistency of the two-sample test is established. Second, we instantiate our framework using nearest neighbor regressors, and show how the power of the resulting two-sample test can be improved using Bayesian mixtures and switch distributions. This combination of techniques yields automatic scale selection, and experiments performed on challenging data sets show that our sequential tests exhibit comparable performances to those of state-of-the-art non-sequential tests.

A new nonparametric test is proposed for the multivariate two-sample problem. Similar to Rosenbaum’s cross-match test, each observation is considered to be a vertex of a complete undirected weighted graph; interpoint distances are edge weights. A minimum-weight, r-regular subgraph is constructed, and the mean cross-count test statistic is equal to the number of edges in the subgraph containing one observation from the first group and one from the second, divided by r. Unequal distributions will tend to result in fewer edges that connect vertices between different groups. The mean cross-count test is sensitive to a wide range of distribution differences and has impressive power characteristics. We derive the first and second moments of the mean cross-count test, and note that simulation studies suggest this test statistic is asymptotically normal regardless of underlying data distributions. A small simulation study compares the power of the mean cross-count test to Hotelling’s T2 test and to the cross-match test. This new test is a more powerful generalization of Rosenbaum’s test (the cross-match test is the case r = 1) and constitutes a noteworthy addition to the class of multivariate, nonparametric two-sample tests.

A data-adaptive methodology for finding an optimal weighted generalized Mann–Whitney–Wilcoxon statistic

A novel method for color image retrieval based on statistical non-parametric tests such as two-sample Wald Test for equality of variance and Man-Whitney U test is proposed in this paper. The proposed method tests the deviation, i.e. distance in terms of variance between the query and target images; if the images pass the test, then it is proceeded to test the spectrum of energy, i.e. distance between the mean values of the two images; otherwise, the test is dropped. If the query and target images pass the tests then it is inferred that the two images belong to the same class, i.e. both the images are same; otherwise, it is assumed that the images belong to different classes, i.e. both images are different. The obtained test statistic values are indexed in ascending order and the image corresponds to the least value is identified as same or similar images. Here, either the query image or the target image is treated as sample; the other is treated as population. Also, some other features such as Coefficient of Variation, Skewness, Kurtosis, Variance, and Spectrum of Energy are compared between the query and target images color-wise. The proposed method is robust for scaling and rotation, since it adjusts itself and treats either the query image or the target image is the sample of other. The results obtained are comparable with the existing methods. Keywords—Variance, mean, query image, target image, non-parametric tests.

The two‐group Pitman Test, proposed by E. J. G. Pitman in 1937, is one of the earliest nonparametric or distribution‐free tests. The test has applications both to random samples drawn independently from two (large) populations, and to the randomization of available cases between two treatments. Although the mechanics of the two applications are identical, the hypotheses tested and the interpretation of those tests differ markedly between the two applications.

The objective of this research was to compare the performance of seven statistics for mean difference testing between two populations when data did not follow assumptions whereas the simulation conditions were determined as 5 distributions, variance, and sample size in both cases are equal and unequal. The results showed that when the population had log-normal distribution, gamma distribution andpoisson distribution and equal variance, the Welch Based on Rank test (WBR test) were most effective. When the population had log-normal distribution, gamma distribution,exponential distribution,poisson distribution and uniform distribution and unequal variance, the Welch t test was distinctively found to have a higher performance than others testing statistics. Keywords: Two-sample location test, Parametric test, Non-parametric test, Non-parametric bootstrap test, t test, Welch t test, Welch Based on Rank test, Brunner-Munzel test, Yuen-Welch test, Exact Wilcoxon signed-rank test

Journal Article A class of location-scale nonparametric tests Get access B. S. DURAN, B. S. DURAN Department of Mathematics, Texas Tech UniversityLubbock Search for other works by this author on: Oxford Academic Google Scholar W. S. TSAI, W. S. TSAI Department of Mathematics, Texas Tech UniversityLubbock Search for other works by this author on: Oxford Academic Google Scholar T. O. LEWIS T. O. LEWIS Department of Mathematics, Texas Tech UniversityLubbock Search for other works by this author on: Oxford Academic Google Scholar Biometrika, Volume 63, Issue 1, 1976, Pages 173–176, https://doi.org/10.1093/biomet/63.1.173 Published: 01 April 1976 Article history Received: 01 February 1975 Revision received: 01 October 1975 Published: 01 April 1976

Testing for differences between two groups is among the most frequently carried out statistical methods in empirical research. The traditional frequentist approach is to make use of null hypothesis significance tests which use p values to reject a null hypothesis. Recently, a lot of research has emerged which proposes Bayesian versions of the most common parametric and nonparametric frequentist two-sample tests. These proposals include Student’s two-sample t-test and its nonparametric counterpart, the Mann–Whitney U test. In this paper, the underlying assumptions, models and their implications for practical research of recently proposed Bayesian two-sample tests are explored and contrasted with the frequentist solutions. An extensive simulation study is provided, the results of which demonstrate that the proposed Bayesian tests achieve better type I error control at slightly increased type II error rates. These results are important, because balancing the type I and II errors is a crucial goal in a variety of research, and shifting towards the Bayesian two-sample tests while simultaneously increasing the sample size yields smaller type I error rates. What is more, the results highlight that the differences in type II error rates between frequentist and Bayesian two-sample tests depend on the magnitude of the underlying effect.

The main idea of this paper is to approximate the exact p-value of a class of non-parametric, two-sample location-scale tests. In this paper, the most famous non-parametric two-sample location-scale tests are formulated in a class of linear rank tests. The permutation distribution of this class is derived from a random allocation design. This allows us to approximate the exact p-value of the non-parametric two-sample location-scale tests of the considered class using the saddlepoint approximation method. The proposed method shows high accuracy in approximating the exact p-value compared to the normal approximation method. Moreover, the proposed method only requires a few calculations and time, as in the case of the simulated method. The procedures of the proposed method are clarified through four sets of real data that represent applications for a number of different fields. In addition, a simulation study compares the proposed method with the traditional methods to approximate the exact p-value of the specified class of the non-parametric two-sample location-scale tests.

In statistics, two-sample tests are used to determine whether two samples have been drawn from the same population. An example of such a test is the widely used Kolmogorov–Smirnov two-sample test. There are other distribution-free tests that might be applied in similar occasions. In this article, we describe a two-sample omnibus test introduced by Epps and Singleton, which usually has a greater power than the Kolmogorov–Smirnov test although it is distribution free. The superiority of the Epps–Singleton characteristic function test is illustrated in two examples. We compare the two tests and supplement this contribution with a Stata implementation of the omnibus test.

SYNOPTIC ABSTRACTThe Wilcoxon two sample nonparametric test is modified to analyze the data with missing observations. The approach is to weigh each complete observation with its own estimated weight score, which is the inverse of probability of completing the observation. The proposed modified method is applied to a recently conducted clinical trial data. A small simulation study shows that the modified method is unbiased under the null hypothesis, while the Wilcoxon two sample test is biased when applied only to complete observations.

Nonparametric two-sample or homogeneity testing is a decision theoretic problem that involves identifying differences between two random variables without making parametric assumptions about their underlying distributions. The literature is old and rich, with a wide variety of statistics having being designed and analyzed, both for the unidimensional and the multivariate setting. In this short survey, we focus on test statistics that involve the Wasserstein distance. Using an entropic smoothing of the Wasserstein distance, we connect these to very different tests including multivariate methods involving energy statistics and kernel based maximum mean discrepancy and univariate methods like the Kolmogorov–Smirnov test, probability or quantile (PP/QQ) plots and receiver operating characteristic or ordinal dominance (ROC/ODC) curves. Some observations are implicit in the literature, while others seem to have not been noticed thus far. Given nonparametric two-sample testing’s classical and continued importance, we aim to provide useful connections for theorists and practitioners familiar with one subset of methods but not others.

Power analysis and type I and type II error rates of Bayesian nonparametric two-sample tests for location-shifts based on the Bayes factor under Cauchy priors

In kernel non-parametric two-sample test, we aim to determine whether two sets of precise observations (i.e., samples) are from the same distribution based on a selected kernel. However, in real world, precise observations may be unavailable. For example, readings on an analogue measurement equipment are not precise numbers but intervals since there is only a finite number of decimals available. Hence, we consider a new and more realistic problem setting-two-sample test on imprecise observations. We show that the test power of existing kernel two- sample tests will drop significantly if they do not take care of the vagueness of the imprecise observations, and to this end, we propose a fuzzy-based maximum mean discrepancy (F-MMD), a powerful two-sample test on imprecise observations. F-MMD is based on a novel fuzzy-based kernel function that can measure the discrepancy between two imprecise observations. This novel kernel function takes care of the vagueness of the imprecise observations and its parameters are optimized to maximize the approximate test power of F-MMD. Experiments demonstrate that F-MMD significantly outperforms competitive two-sample test methods when facing imprecise observations.