A general class of non parametric tests for comparing scale parameters

In this paper, a general class of non parametric tests is proposed for the two-sample scale problem. Testing of the scale parameter is very useful in real-life situations commonly faced in engineering, trade, cultivation, industries, medicine, etc. In all these fields, one will prefer the method that gives more consistent results. Thus, it is worthwhile to test the equality of scale parameters. The distribution of the proposed test is established. To assess the performance of the proposed test, the asymptotic efficacies are studied for some underlying distributions and the results are interpreted with useful information. To see the working of the proposed test, an illustrative example for the real-life data set is provided. The simulation study is also carried out to find the asymptotic power of the proposed test. An extension of the general class of tests to the multiple-sample problem is also discussed.     

Nonparametric testing for simple regression under progressive censoring with stagering entry and random withdrawal

In the context of (multi-center) clinical trials and life testins problems, a general model incorporating both the staggering entry and random withdrawal and pertaining to a simple regression problem (including the two-sample location problem as a special case) is conceived, and, within this framework, a scheme allowing progressive censoring (continuous monitoring of experimentation from the beginning) is developed along with the proposal for and study of some nonparametric testing procedures, The proposed tests rest on the construction of certain two-dimensional time-parameter stochastic processes from a triangular array of progressively censored linear rank statistics and their weak convergence to appropriate Gaussian functions. Asymptotic properties of these procedures are studied. A computer program pertaining to the numerical computations and practical administrations of these testing procedures is also provided at the end.     

Modified Baumgartner statistics for the two-sample and multisample problems: a numerical comparison

Various non-parametric rank tests based on the Baumgartner statistic have been proposed for testing the location, scale and location–scale parameters. The modified Baumgartner statistics are not suitable for the scale shifts for a two-sample problem. Two modified Baumgartner statistics are proposed by changing the weight function. The suggested statistics are extended to the multisample problem. Some exact critical values of the suggested test statistics are evaluated. Simulations are used to investigate the power of the modified Baumgartner statistics.     

A note on the nonparametric test based on theL 1-version of the Cramér-von Mises statistic

We consider the test based on theL ₁-version of the Cramér-von Mises statistic for the nonparametric two-sample problem. Some quantiles of the exact distribution under H₀ of the test statistic are computed for small sample sizes. We compare the test in terms of power against general alternatives to other two-sample tests, namely the Wilcoxon rank sum test, the Smirnov test and the Cramér-von Mises test in the case of unbalanced small sample sizes. The computation of the power is rather complicated when the sample sizes are unequal. Using Monte Carlo power estimates it turns out that the Smirnov test is more sensitive to non stochastically ordered alternatives than the new test. And under location-contamination alternatives the power estimates of the new test and of the competing tests are equal.     

The generalized Cucconi test statistic for the two-sample problem

When testing hypotheses in two-sample problem, the Lepage test statistic is often used to jointly test the location and scale parameters, and this test statistic has been discussed by many authors over the years. Since two-sample nonparametric testing plays an important role in biometry, the Cucconi test statistic is generalized to the location, scale, and location–scale parameters in two-sample problem. The limiting distribution of the suggested test statistic is derived under the hypotheses. Deriving the exact critical value of the test statistic is difficult when the sample sizes are increased. A gamma approximation is used to evaluate the upper tail probability for the proposed test statistic given finite sample sizes. The asymptotic efficiencies of the proposed test statistic are determined for various distributions. The consistency of the original Cucconi test statistic is shown on the specific cases. Finally, the original Cucconi statistic is discussed in the theory of ties.     

Robustness and power of modified Lepage,Kolmogorov-Smirnov and Crame´r-von Mises two-sample tests

For the two-sample problem with location and/or scale alternatives, as well as different shapes, several statistical tests are presented, such as of Kolmogorov-Smirnov and Cramér-von Mises type for the general alternative, and such as of Lepage type for location and scale alternatives. We compare these tests with the t-test and other location tests, such as the Welch test, and also the Levene test for scale. It turns out that there is, of course, no clear winner among the tests but, for symmetric distributions with the same shape, tests of Lepage type are the best ones whereas, for different shapes, Cramér-von Mises type tests are preferred. For extremely right-skewed distributions, a modification of the Kolmogorov-Smirnov test should be applied.     

On the Usefulness or Lack Thereof of Optimality Criteria for Structural Change Tests

Elliott and Müller (2006) considered the problem of testing for general types of parameter variations, including infrequent breaks. They developed a framework that yields optimal tests, in the sense that they nearly attain some local Gaussian power envelop. The main ingredient in their setup is that the variance of the process generating the changes in the parameters must go to zero at a fast rate. They recommended the so-called qL?L test, a partial sums type test based on the residuals obtained from the restricted model. We show that for breaks that are very small, its power is indeed higher than other tests, including the popular sup-Wald (SW) test. However, the differences are very minor. When the magnitude of change is moderate to large, the power of the test is very low in the context of a regression with lagged dependent variables or when a correction is applied to account for serial correlation in the errors. In many cases, the power goes to zero as the magnitude of change increases. The power of the SW test does not show this non-monotonicity and its power is far superior to the qL?L test when the break is not very small. We claim that the optimality of the qL?L test does not come from the properties of the test statistics but the criterion adopted, which is not useful to analyze structural change tests. Instead, we use fixed-break size asymptotic approximations to assess the relative efficiency or power of the two tests. When doing so, it is shown that the SW test indeed dominates the qL?L test and, in many cases, the latter has zero relative asymptotic efficiency.     

Resampling schemes with low resampling intensity and their applications in testing hypotheses

The paper explores statistical features of different resampling schemes under low resampling intensity. The original sample is considered in a very general framework of triangular arrays, without independence or equally distributed assumptions, although improvements under such conditions are also provided. We show that low resampling schemes have very interesting and flexible properties, providing new insights into the performance of widely used resampling methods, including subsampling, two-sample unbalanced permutation statistics or wild bootstrap. It is shown that, under regularity assumptions, resampling tests with critical values derived by the appertaining low resampling procedures are asymptotically valid and there is no loss of power compared with the power function of an ideal (but unfeasible) parametric family of tests. Moreover we show that in several contexts, including regression models, they may act as a filter for the normal part of a limit distribution, turning down the influence of outliers.     

Permutation tests for homogeneity based on some characterizations

In this work, non parametric tests are proposed for testing the homogeneity of two or more populations. The tests are based on recently obtained characterizations. The test procedure is based on the permutation bootstrap technique. For the two-sample case the new tests are compared with permutation tests based on the empirical characteristic function and some other tests. The comparison is fulfilled via a Monte Carlo simulation.     

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.     

An Adaptive Test for the Two-Sample Location Problem Based on U-Statistics

For the two-sample location problem with continuous data we consider a general class of tests, all members of it are based on U-statistics. The asymptotic efficacies are investigated in detail. We construct an adaptive test where all statistics involved are suitably chosen U-statistics. It is shown that the proposed adaptive test has good asymptotic and finite sample power properties.     

A Framework for Monte Carlo based Multiple Testing

We are concerned with a situation in which we would like to test multiple hypotheses with tests whose p‐values cannot be computed explicitly but can be approximated using Monte Carlo simulation. This scenario occurs widely in practice. We are interested in obtaining the same rejections and non‐rejections as the ones obtained if the p‐values for all hypotheses had been available. The present article introduces a framework for this scenario by providing a generic algorithm for a general multiple testing procedure. We establish conditions that guarantee that the rejections and non‐rejections obtained through Monte Carlo simulations are identical to the ones obtained with the p‐values. Our framework is applicable to a general class of step‐up and step‐down procedures, which includes many established multiple testing corrections such as the ones of Bonferroni, Holm, Sidak, Hochberg or Benjamini–Hochberg. Moreover, we show how to use our framework to improve algorithms available in the literature in such a way as to yield theoretical guarantees on their results. These modifications can easily be implemented in practice and lead to a particular way of reporting multiple testing results as three sets together with an error bound on their correctness, demonstrated exemplarily using a real biological dataset.     

Rank tests in the two-sample scale problem with unequal and unknown locations

The two-sample scale problem is studied in the case of unequal and unknown location parameters. The method proposed is based on the idea of Moses (1963) and it is distribution-free. The two samples are separated into random subgroups of the same sizek. It is proposed to choosek=4 and to apply the Wilconxon test or the Savage test to the ranges or sample variances of the subgroups. The asymptotic power functions of the tests are compared. For small and moderate sample sizes simulations are carried out. Relations to some other procedures, especially to the method of Compagnone and Denker (1996) are briefly discussed.     

Nonparametric Simultaneous Tests for Location and Scale Testing: A Comparison of Several Methods

The two-sample location-scale problem arises in many situations like climate dynamics, bioinformatics, medicine, and finance. To address this problem, the nonparametric approach is considered because in practice, the normal assumption is often not fulfilled or the observations are too few to rely on the central limit theorem, and moreover outliers, heavy tails and skewness may be possible. In these situations, a nonparametric test is generally more robust and powerful than a parametric test. Various nonparametric tests have been proposed for the two-sample location-scale problem. In particular, we consider tests due to Lepage, Cucconi, Podgor-Gastwirth, Neuhäuser, Zhang, and Murakami. So far all these tests have not been compared. Moreover, for the Neuhäuser test and the Murakami test, the power has not been studied in detail. It is the aim of the article to review and compare these tests for the jointly detection of location and scale changes by means of a very detailed simulation study. It is shown that both the Podgor–Gastwirth test and the computationally simpler Cucconi test are preferable. Two actual examples within the medical context are discussed.     

Adaptive choice of scale tests in flexible two-stage designs with applications in experimental ecology and clinical trials

In this paper, the two-sample scale problem is addressed within the rank framework which does not require to specify the underlying continuous distribution. However, since the power of a rank test depends on the underlying distribution, it would be very useful for the researcher to have some information on it in order to use the possibly most suitable test. A two-stage adaptive design is used with adaptive tests where the data from the first stage are used to compute a selector statistic to select the test statistic for stage 2. More precisely, an adaptive scale test due to Hall and Padmanabhan and its components are considered in one-stage and several adaptive and non-adaptive two-stage procedures. A simulation study shows that the two-stage test with the adaptive choice in the second stage and with Liptak combination, when it is not more powerful than the corresponding one-stage test, shows, however, a quite similar power behavior. The test procedures are illustrated using two ecological applications and a clinical trial.     

Smoothed alternatives of the two-sample median and Wilcoxon's rank sum tests

We discuss smoothed rank statistics for testing the location shift parameter of the two-sample problem. They are based on discrete test statistics – the median and Wilcoxon's rank sum tests. For the one-sample problem, Maesono et al. [Smoothed nonparametric tests and their properties. arXiv preprint. 2016; ArXiv:1610.02145] reported that some nonparametric discrete tests have a problem with their p-values because of their discreteness. The p-values of Wilcoxon's test are frequently smaller than those of the median test in the tail area. This leads to an arbitrary choice of the median and Wilcoxon's rank sum tests. To overcome this problem, we propose smoothed versions of those tests. The smoothed tests inherit the good properties of the original tests and are asymptotically equivalent to them. We study the significance probabilities and local asymptotic powers of the proposed tests.     

A Modified Hall-Padmanabhan Test for the Homogeneity of Scales

We consider a simple adaptive test for the homogeneity of scales due to Hall and Padmanabhan (1997 Hall , P. , Padmanabhan , A. R. ( 1997 ). Adaptive inference for the two-sample scale problem . Technometrics 23 : 351 – 361 . [Google Scholar]) which is based on the ratio of adaptively trimmed sample variances. We develop a modification of this test that has the form of a combined bootstrap test and that is obtained within the nonparametric combination of dependent tests framework (Pesarin, 2001 Pesarin , F. ( 2001 ). Multivariate Permutation Tests with Applications in Biostatistics . Chichester : Wiley . [Google Scholar]). We considered other approaches for the combination of tests concluding that the Liptak method is the most suitable one for the problem at hand. We compare the modified test with the original one in terms of robustness of significance level and power in a simulation study which considers distributions that range from symmetric to skewed and from light to heavy tailed ones. We show that the modified test is more powerful than the original one under heavier than normal tailed and very skewed distributions. A practical application to detect difference in scale of energy intake of lactating and non pregnant and not lactating women is presented.     

On a class of partially sequential two-sample test procedures for multivariate continuous data

In a two-sample testing problem, sometimes one of the sample observations are difficult and/or costlier to collect compared to the other one. Also, it may be the situation that sample observations from one of the populations have been previously collected and for operational advantages we do not wish to collect any more observations from the second population that are necessary for reaching a decision. Partially sequential technique is found to be very useful in such situations. The technique gained its popularity in statistics literature due to its very nature of capitalizing the best aspects of both fixed and sequential procedures. The literature is enriched with various types of partially sequential techniques useable under different types of data set-up. Nonetheless, there is no mention of multivariate data framework in this context, although very common in practice. The present paper aims at developing a class of partially sequential nonparametric test procedures for two-sample multivariate continuous data. For this we suggest a suitable stopping rule adopting inverse sampling technique and propose a class of test statistics based on the samples drawn using the suggested sampling scheme. Various asymptotic properties of the proposed tests are explored. An extensive simulation study is also performed to study the asymptotic performance of the tests. Finally the benefit of the proposed test procedure is demonstrated with an application to a real-life data on liver disease.     

Nonparametric repeated significance tests for one-way anova with adoptation to multiple comparisons

Based on two-sample rank order statistics, a repeated significance testing procedure for a multi-sample location problem is considered. The asymptotic distribution theory of the proposed tests is given under the null hypothesis as well as under local alternatives. A Bahadur efficiency result of the repeated significance test relative to the terminal test based solely on the target sample size is presented. In the adaptation of the proposed tests to multiple comparisons, an asymptotically equivalent test statistic in terms of the rank estimators of the location parameters is derived from which the Scheffé method of multiple comparisons can be obtained in a convinient way.     

Closed testing procedures for all pairwise comparisons in a randomized block design

We consider multiple comparison test procedures among treatment effects in a randomized block design. We propose closed testing procedures based on maximum values of some two-sample t test statistics and based on F test statistics. It is shown that the proposed procedures are more powerful than single-step procedures and the REGW (Ryan/Einot–Gabriel/Welsch)-type tests. Next, we consider the randomized block design under simple ordered restrictions of treatment effects. We propose closed testing procedures based on maximum values of two-sample one-sided t test statistics and based on Batholomew’s statistics for all pairwise comparisons of treatment effects. Although single-step multiple comparison procedures are utilized in general, the power of these procedures is low for a large number of groups. The closed testing procedures stated in the present article are more powerful than the single-step procedures. Simulation studies are performed under the null hypothesis and some alternative hypotheses. In this studies, the proposed procedures show a good performance.