Distribution-free Two-sample Tests Research Articles

A control chart based on data depth for monitoring the variability in a multivariate process

Majority of the existing statistical procedures for monitoring the covariance matrix of the multivariate processes are based on normality or other distributional assumption. In order to isolate the change in the process variation, these procedures are less powerful due to distributional assumptions. In this article, we study the control charts for monitoring covariance matrix critically. We provide a distribution-free two-sample test for multivariate scale problem called as modified rank test based on data depth. A control chart is proposed based on the data depth based modified rank test. A modified rank test based on data depth is the extension of the modified rank test for univariate two-sample location and scale problem. Data depth based center outward ranking is used for defining test statistic. The performance study of the proposed control charts reveals that the proposed control chart is a better alternative to traditional control charts for variability when a distribution of the process is not known but some historical in-control data is available. The control chart is illustrated with the real-life CSTR process data.

Two-Sample Testing for Tail Copulas with an Application to Equity Indices

A novel, general two-sample hypothesis testing procedure is established for testing the equality of tail copulas associated with bivariate data. More precisely, using a martingale transformation of a natural two-sample tail copula process, a test process is constructed, which is shown to converge in distribution to a standard Wiener process. Hence, from this test process a myriad of asymptotically distribution-free two-sample tests can be obtained. The good finite-sample behavior of our procedure is demonstrated through Monte Carlo simulations. Using the new testing procedure, no evidence of a difference in the respective tail copulas is found for pairs of negative daily log-returns of equity indices during and after the global financial crisis.

Two new distribution-free two-sample tests for versatile alternative

This paper introduces two alternative distribution-free tests for the combined classical-location-scale and Lehmann alternatives, known as the versatile alternative. Recently, two such statistics were proposed, one based on the Euclidean distance and the other based on the Mahalanobis distance, combining the three test statistics, the Wilcoxon statistic for the location parameter, the Ansari-Bradley statistic for the scale parameter and a Savage-type statistic for the Lehmann alternative. However, noting that there is some practical advantage of using the Mood statistic in place of the Ansari-Bradley statistic, two new tests are designed in the present paper. We derive the limiting distributions of the proposed statistics and investigate their power performances in different situations. Simulation studies based on Monte-Carlo also show that the proposed tests are good competitors of the existing tests and have some advantage in certain situations. We include illustrations based on real-life data and finally offer some concluding remarks.

Two-Sample Testing for Tail Copulas with an Application to Equity Indices

A novel, general two-sample hypothesis testing procedure is established for testing the equality of tail copulas associated with bivariate data. More precisely, using an ingenious transformation of a natural two-sample tail copula process, a test process is constructed, which is shown to converge in distribution to a standard Wiener process. Hence, from this test process a myriad of asymptotically distribution-free two-sample tests can be obtained. The good finite-sample behavior of our procedure is demonstrated through Monte Carlo simulations. Using the new testing procedure, no evidence of a difference in the respective tail copulas is found for pairs of negative daily log-returns of equity indices during and after the global financial crisis.

On Matched Filtering for Statistical Change Point Detection

Non-parametric and distribution-free two-sample tests have been the foundation of many change point detection algorithms. However, randomness in the test statistic as a function of time makes them susceptible to false positives and localization ambiguity. We address these issues by deriving and applying filters matched to the expected temporal signatures of a change for various sliding window, two-sample tests under IID assumptions on the data. These filters are derived asymptotically with respect to the window size for the Wasserstein quantile test, the Wasserstein-1 distance test, Maximum Mean Discrepancy squared (MMD <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> ), and the Kolmogorov-Smirnov (KS) test. The matched filters are shown to have two important properties. First, they are distribution-free, and thus can be applied without prior knowledge of the underlying data distributions. Second, they are peak-preserving, which allows the filtered signal produced by our methods to maintain expected statistical significance. Through experiments on synthetic data as well as activity recognition benchmarks, we demonstrate the utility of this approach for mitigating false positives and improving the test precision. Our method allows for the localization of change points without the use of ad-hoc post-processing to remove redundant detections common to current methods. We further highlight the performance of statistical tests based on the Quantile-Quantile (Q-Q) function and show how the invariance property of the Q-Q function to order-preserving transformations allows these tests to detect change points of different scales with a single threshold within the same dataset.

Levene type tests for the ratio of two scales

Tests for the equality of variances are of interest in many areas such as quality control, agricultural production systems, experimental education, pharmacology, biology, as well as a preliminary to the analysis of variance, dose–response modelling or discriminant analysis. The literature is vast. Traditional non-parametric tests are due to Mood, Miller and Ansari–Bradley. A test which usually stands out in terms of power and robustness against non-normality is the W50 Brown and Forsythe [Robust tests for the equality of variances, J. Am. Stat. Assoc. 69 (1974), pp. 364–367] modification of the Levene test [Robust tests for equality of variances, in Contributions to Probability and Statistics, I. Olkin, ed., Stanford University Press, Stanford, 1960, pp. 278–292]. This paper deals with the two-sample scale problem and in particular with Levene type tests. We consider 10 Levene type tests: the W50, the M50 and L50 tests [G. Pan, On a Levene type test for equality of two variances, J. Stat. Comput. Simul. 63 (1999), pp. 59–71], the R-test [R.G. O'Brien, A general ANOVA method for robust tests of additive models for variances, J. Am. Stat. Assoc. 74 (1979), pp. 877–880], as well as the bootstrap and permutation versions of the W50, L50 and R tests. We consider also the F-test, the modified Fligner and Killeen [Distribution-free two-sample tests for scale, J. Am. Stat. Assoc. 71 (1976), pp. 210–213] test, an adaptive test due to Hall and Padmanabhan [Adaptive inference for the two-sample scale problem, Technometrics 23 (1997), pp. 351–361] and the two tests due to Shoemaker [Tests for differences in dispersion based on quantiles, Am. Stat. 49(2) (1995), pp. 179–182; Interquantile tests for dispersion in skewed distributions, Commun. Stat. Simul. Comput. 28 (1999), pp. 189–205]. The aim is to identify the effective methods for detecting scale differences. Our study is different with respect to the other ones since it is focused on resampling versions of the Levene type tests, and many tests considered here have not ever been proposed and/or compared. The computationally simplest test found robust is W50. Higher power, while preserving robustness, is achieved by considering the resampling version of Levene type tests like the permutation R-test (recommended for normal- and light-tailed distributions) and the bootstrap L50 test (recommended for heavy-tailed and skewed distributions). Among non-Levene type tests, the best one is the adaptive test due to Hall and Padmanabhan.

A two-sample empirical likelihood ratio test based on samples entropy

Powerful entropy-based tests for normality, uniformity and exponentiality have been well addressed in the statistical literature. The density-based empirical likelihood approach improves the performance of these tests for goodness-of-fit, forming them into approximate likelihood ratios. This method is extended to develop two-sample empirical likelihood approximations to optimal parametric likelihood ratios, resulting in an efficient test based on samples entropy. The proposed and examined distribution-free two-sample test is shown to be very competitive with well-known nonparametric tests. For example, the new test has high and stable power detecting a nonconstant shift in the two-sample problem, when Wilcoxon's test may break down completely. This is partly due to the inherent structure developed within Neyman-Pearson type lemmas. The outputs of an extensive Monte Carlo analysis and real data example support our theoretical results. The Monte Carlo simulation study indicates that the proposed test compares favorably with the standard procedures, for a wide range of null and alternative distributions.

Adaptive bootstrap tests and its competitors in the c-sample scale problem

This paper deals with a study of different types of tests for the two-sided c-sample scale problem. We consider the classical parametric test of Bartlett [M.S. Bartlett, Properties of sufficiency and statistical tests, Proc. R. Stat. Soc. Ser. A. 160 (1937), pp. 268–282] several nonparametric tests, especially the test of Fligner and Killeen [M.A. Fligner and T.J. Killeen, Distribution-free two-sample tests for scale, J. Amer. Statist. Assoc. 71 (1976), pp. 210–213], the test of Levene [H. Levene, Robust tests for equality of variances, in Contribution to Probability and Statistics, I. Olkin, ed., Stanford University Press, Palo Alto, 1960, pp. 278–292] and a robust version of it introduced by Brown and Forsythe [M.B. Brown and A.B. Forsythe, Robust tests for the equality of variances, J. Amer. Statist. Assoc. 69 (1974), pp. 364–367] as well as two adaptive tests proposed by Büning [H. Büning, Adaptive tests for the c-sample location problem – the case of two-sided alternatives, Comm. Statist.Theory Methods. 25 (1996), pp. 1569–1582] and Büning [H. Büning, An adaptive test for the two sample scale problem, Nr. 2003/10, Diskussionsbeiträge des Fachbereich Wirtschaftswissenschaft der Freien Universität Berlin, Volkswirtschaftliche Reihe, 2003]. which are based on the principle of Hogg [R.V. Hogg, Adaptive robust procedures. A partial review and some suggestions for future applications and theory, J. Amer. Statist. Assoc. 69 (1974), pp. 909–927]. For all the tests we use Bootstrap sampling strategies, too. We compare via Monte Carlo Methods all the tests by investigating level α and power β of the tests for distributions with different strength of tailweight and skewness and for various sample sizes. It turns out that the test of Fligner and Killeen in combination with the bootstrap is the best one among all tests considered.

A statistical model of cluster stability

In the current paper we present a method for assessing cluster stability. This method, combined with a clustering algorithm, yields an estimate of the data partition, namely, the number of clusters. We adopt the cluster stability standpoint where clusters are imagined as islands of “high” density in a sea of “low” density. Explicitly, a cluster is associated with its high density core. Our approach offers to evaluate the goodness of a cluster by the similarity amongst the entire cluster and its core. We propose to measure this resemblance by two-sample tests or by probability distances between appropriate probability distributions. The distances are calculated on clustered samples drawn from the source population according to two different distributions. The first law is the underlying set distribution. The second law is constructed so that it represents the clusters’ cores. Here, a variant of the k-nearest neighbor density estimation is applied, so that items belonging to cores have a much higher chance to be selected. As the sample distribution is unknown a distribution-free two-sample test is required to examine the mentioned correspondence. For constructing such a test, we use distance functions built on negative definite kernels. In practice, outliers in the samples and limitations of the clustering algorithm heavily contribute to the noise level. As a result of this shortcoming the distance values have to be determined for many pairs of samples and therefore an empirical distance's distribution is obtained. The distribution is dependent on the examined number of clusters. To prevent this property for biasing the results we normalize the distances. It is conjectured that the true number of clusters yields the most concentrated normalized distribution. To measure the concentration we use the sample mean and the sample 25th percentile. The paper exhibits the good performance of the proposed method on synthetic and real-world data.

Distribution-Free Two-Sample Tests Based on Rank Spacings

Abstract For the two-sample problem, rank spacings of one sample are the positive integer distances between combined-sample ordered ranks from that sample. Elementary mathematical development yields distribution-free orthogonal components analogous to L statistics used by Kaigh for assessing one-sample uniformity. Rank spacings components are linear combinations of ordered ranks with Hahn polynomial vector weight functions. The first four rank spacings components provide nonparametric measures of location, scale, skewness, and kurtosis. Asymptotically normal rank spacings components are related to linear rank statistic components obtained by Pettitt. Aggregates of squared rank spacings components yield a component decomposition of the Dixon statistic and provide omnibus chi-squared statistics analogous to those of Pettitt and Boos.

Distribution-Free Two-Sample Tests Based on Rank Spacings

Abstract For the two-sample problem, rank spacings of one sample are the positive integer distances between combined-sample ordered ranks from that sample. Elementary mathematical development yields distribution-free orthogonal components analogous to L statistics used by Kaigh for assessing one-sample uniformity. Rank spacings components are linear combinations of ordered ranks with Hahn polynomial vector weight functions. The first four rank spacings components provide nonparametric measures of location, scale, skewness, and kurtosis. Asymptotically normal rank spacings components are related to linear rank statistic components obtained by Pettitt. Aggregates of squared rank spacings components yield a component decomposition of the Dixon statistic and provide omnibus chi-squared statistics analogous to those of Pettitt and Boos.

長期的トレンド、周期変化、ステップ的変化のラページ検定による検出

Power of the Lepage test, a distribution-free two-sample test, is studied by means of statistical simulation. The results show that the Lepage test detects such changes as long term trends, cyclic variations and step-like changes in different ways for each type of phenomenon. The Lepage test can be a very good tool for statistical researches in climate change.

A Class of Distribution-Free Two-Sample Tests Based on Placements

A Class of Distribution-Free Two-Sample Tests Based on Placements

A Class of Distribution-Free Two-Sample Tests Based on Placements

Abstract We consider a class of two-sample distribution-free tests that are appropriate for situations where one of the sample sizes is large relative to the other. These procedures are based on the placements of the observations in the smaller sample among the ordered observations in the larger sample, and this class of tests generalizes the Mann-Whitney (1947) procedure in much the same way that the class of linear rank tests generalizes the equivalent Wilcoxon (1945) rank sum form. Optimality criteria for choosing a test from this class are discussed and limiting distributions for the associated class of test statistics are determined for the case where only one of the sample sizes goes to infinity.

Corrected and Extended Tables for Tukey's Quick Test

Tukey has presented a quick distribution-free two-sample test of location together with tables of exact critical values (Tukey 1959). Corrected and extended tables are given, which now include exact 5%/1%/.l% two-sided critical values for all cases in which the sample sizes differ by no more than 20.

Distribution-Free Two-Sample Tests for Scale

Abstract Several linear rank statistics have been proposed in the literature for the two-sample scale problem. We propose a new class of statistics which are distribution free when the populations are identical, but are not linear rank statistics. Our analogs of the Ansari-Bradley, Mood, and Klotz tests are of particular interest. Each has the same Pitman efficiency as its corresponding linear rank statistic, and yet our small-sample power is significantly higher. In addition, our tests are consistent for scale differences with unequal location in the case when the populations are symmetric and the sample sizes are equal.

Distribution-Free Two-Sample Tests for Scale

Abstract Several linear rank statistics have been proposed in the literature for the two-sample scale problem. We propose a new class of statistics which are distribution free when the populations are identical, but are not linear rank statistics. Our analogs of the Ansari-Bradley, Mood, and Klotz tests are of particular interest. Each has the same Pitman efficiency as its corresponding linear rank statistic, and yet our small-sample power is significantly higher. In addition, our tests are consistent for scale differences with unequal location in the case when the populations are symmetric and the sample sizes are equal.

A Distribution-Free Two-Sample Test on a Circle

Abstract : Samples of n sub 1 and n sub 2 are taken from two populations of unit vectors or directions in two dimensions. A distribution-free test is here provided of the null hypothesis that the two populations have the same form. It could be useful for detecting a difference of mean directions in unimodal populations. (Author)