Power Comparison of Multivariate Wilcoxon-type Tests Based on the Jurečková–Kalina’s Ranks of Distances

A multivariate two-sample testing problem is one of the most important topics in nonparametric statistics. One of the multivariate two-sample testing problems based on the Jure?ková–Kalina ranks of distance is discussed in this article. Further, a multivariate Wilcoxon-type test is proposed for testing the equality of two continuous distribution functions. Simulations are used to investigate the power of this test for the two-sided alternative with various population distributions. The results show that the proposed test statistic is more suitable than various existing statistics for testing a shift in the locationt and location-scale parameters.     

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.     

Combined sample scores versus two-sample scores under censoring

For testing the equality of K-populations against ordered alternatives a quadratic form based on two-sample scores of (K-l) pairs of adjacent samples has been suggested as an alternative to the statistic proposed by Beslow(1970). The Pitman asymptotic efficiency of the proposed statistic relative to the Breslow's statistic is found to be one or very close to one in all cases considered. The Bahadur asymptotic efficiency has also been studied and found to be at least one for the situations taken up in this paper.     

Distribution-free test for symmetry based on Wilcoxon two-sample test

McWilliams introduced a test of symmetry based on a runs statistic for a continuous distribution about a known median. His test statistic R* performs better than other competitors considered in his study. In this paper, we present a conditional test for symmetry based on a Wilcoxon two-sample test. The proposed test turns out to be competitive with the test of McWilliams.     

An efficient nonparametric test for bivariate two-sample location problem

We propose a new test for the two-sample bivariate location problem. The proposed test statistic has a U-statistic representation with a degenerate kernel. The limiting distribution is found for the proposed test statistic. The power of the test is compared using Monte Carlo simulation to the tests of Blumen [I. Blumen, A new bivariate sign-test for location, Journal of the American Statistical Association 53 (1958) 448–456], Mardia [K.V. Mardia, A non-parametric test for the bivariate two-sample location problem, Journal of the Royal Statistical Society, Series B 29 (1967) 320–342], Peters and Randles [D. Peters, R.H. Randles, A bivariate signed-rank test for the two-sample location problem, Journal of the Royal Statistical Society, Series B 53 (1991) 493–504], LaRocque, Tardif and van Eeden [D. LaRocque, S. Tardif, C. van Eeden, An affine-invariant generalization of the Wilcoxon signed-rank test for the bivariate location problem, Australian and New Zealand Journal of Statistics 45 (2003) 153–165], and Baringhaus and Franz [L. Baringhaus, C. Franz, On a new multivariate two-sample test, Journal of Multivariate Analysis 88 (2004) 190–206]. Under the bivariate normal and bivariate t distributions the proposed test was more powerful than the competitors for almost every change in location. Under the other distributions the proposed test reached the desired power of one at a faster rate than the other tests in the simulation study. Application of the test is presented using bivariate data from a synthetic and a real-life data set.     

Rank statistics expressible as integrals under P–P-plots and receiver operating characteristic curves

Given a probability measure on the unit square, the measure of the region under an empirical P – P -plot defines a two-sample rank statistic. Instances include trimmed and censored versions of the Mann–Whitney–Wilcoxon statistic and a class of statistics with applications in the analysis of receiver operating characteristic (ROC) curves. A large sample distribution for such a statistic is obtained, which is valid under sampling from general populations. Explicit results are presented for comparing arbitrary quantile segments of two populations. The results are not restricted to continuous data and incorporate adjustments for tied values in the discrete case. A multivariate version of the large sample distribution extends the class of tractable statistics in ROC analysis and facilitates the use of methods based on partial areas when the data are discrete.     

A two-sample distribution-free scale test of the smirnov type

The two-sample, distribution-free statistics of Smirnov (1939) are used to define a new statistic. While the Smirnov statistics are used as a general goodness-of-fit test, a distribution-free scale test based on this new statistic is developed. It is shown that this new test has higher power than the two-sided Smirnov statistic in detecting differences in scale for some symmetric distributions with equal means/medians. The critical values of the proposed test statistic and its limiting distribution are given     

On a further generalization of the savage test

A statistic based on the frequencies within the k+1 intervals specified by k arbitrary quantiles is proposed for a LMP test against Lehmann alternatives generalizing the Savage test for the two-sample problem. The maximum efficiency relative to the Savage test for optimally chosen k quantiles is also provided for k=l(2)l5. The asymptotic normality of the statistic follows from the asymptotic multinomial distribution of the frequencies in the classes determined by the k quantiles.     

A class of k-sample distribution-free tests for location against ordered alternatives

This paper introduces a new class of distribution-free tests for testing the homogeneity of several location parameters against ordered alternatives. The proposed class of test statistics is based on a linear combination of two-sample U-statistics based on subsample extremes. The mean and variance of the test statistic are obtained under the null hypothesis as well as under the sequence of local alternatives. The optimal weights are also determined. It is shown via Pitman ARE comparisons that the proposed class of test statistics performs better than its competitor tests in case of heavy-tailed and long-tailed distributions     

A changepoint statistic with uniform type I error probabilities

ABSTRACT

Likelihood ratio tests for a change in mean in a sequence of independent, normal random variables are based on the maximum two-sample t-statistic, where the maximum is taken over all possible changepoints. The maximum t-statistic has the undesirable characteristic that Type I errors are not uniformly distributed across possible changepoints. False positives occur more frequently near the ends of the sequence and occur less frequently near the middle of the sequence. In this paper we describe an alternative statistic that is based upon a minimum p-value, where the minimum is taken over all possible changepoints. The p-value at any particular changepoint is based upon both the two-sample t-statistic at that changepoint and the probability that the maximum two-sample t-statistic is achieved at that changepoint. The new statistic has a more uniform distribution of Type I errors across potential changepoints and it compares favorably with respect to statistical power, false discovery rates, and the mean square error of changepoint estimates.     

A Distribution-free Multivariate Change-point Model for Statistical Process Control

This article develops a new distribution-free multivariate procedure for statistical process control based on minimal spanning tree (MST), which integrates a multivariate two-sample goodness-of-fit (GOF) test based on MST and change-point model. Simulation results show that our proposed procedure is quite robust to nonnormally distributed data, and moreover, it is efficient in detecting process shifts, especially moderate to large shifts, which is one of the main drawbacks of most distribution-free procedures in the literature. The proposed procedure is particularly useful in start-up situations. Comparison results and a real data example show that our proposed procedure has great potential for application.     

An algorithm for computing a two-sample test based on empirical probability measures

An algorithm is presented for computing a test statistic based on empirical probability measures that is appropriate for the two-sample problem.     

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.     

Optimal weighted two-sample t-test with partially paired data in a unified framework

In this paper, we provide a unified framework for two-sample t-test with partially paired data. We show that many existing two-sample t-tests with partially paired data can be viewed as special members in our unified framework. Some shortcomings of these t-tests are discussed. We also propose the asymptotically optimal weighted linear combination of the test statistics comparing all four paired and unpaired data sets. Simulation studies are used to illustrate the performance of our proposed asymptotically optimal weighted combinations of test statistics and compare with some existing methods. It is found that our proposed test statistic is generally more powerful. Three real data sets about CD4 count, DNA extraction concentrations, and the quality of sleep are also analyzed by using our newly introduced test statistic.     

Multivariate rank tests for the two-sample location problem

A multivariate affinc-invariant family of rank tests is proposed for the two sample location problem. The class of statistics introduced is built upon Randles' multivariate one-sample sign statistic based on interdirections and the multivariate one-sample signed-rank statistic of Peters and Randles. Asymptotic relative efficiencies are obtained which indicate that selected members of the class perform very well for a broad class of distributions. Further comparisons are made among several statistics using Monte Carlo results.     

Two-sample high-dimensional empirical likelihood

In this paper, we apply empirical likelihood for two-sample problems with growing high dimensionality. Our results are demonstrated for constructing confidence regions for the difference of the means of two p-dimensional samples and the difference in value between coefficients of two p-dimensional sample linear model. We show that empirical likelihood based estimator has the efficient property. That is, as p → ∞ for high-dimensional data, the limit distribution of the EL ratio statistic for the difference of the means of two samples and the difference in value between coefficients of two-sample linear model is asymptotic normal distribution. Furthermore, empirical likelihood (EL) gives efficient estimator for regression coefficients in linear models, and can be as efficient as a parametric approach. The performance of the proposed method is illustrated via numerical simulations.     

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.     

On empirical likelihood for semiparametric two-sample density ratio models

We consider estimation and test problems for some semiparametric two-sample density ratio models. The profile empirical likelihood (EL) poses an irregularity problem under the null hypothesis that the laws of the two samples are equal. We show that a dual form of the profile EL is well defined even under the null hypothesis. A statistical test, based on the dual form of the EL ratio statistic (ELRS), is then proposed. We give an interpretation for the dual form of the ELRS through φ$\phi$-divergences and duality techniques. The asymptotic properties of the test statistic are presented both under the null and the alternative hypotheses, and approximation of the power function of the test is deduced.     

Multivariate Control Chart Based on Multivariate Smirnov Test

Robust control charts are useful in statistical process control (SPC) when there is limited knowledge about the underlying process distribution, especially for multivariate observations. This article develops a new robust and self-starting multivariate procedure based on multivariate Smirnov test (MST), which integrates a multivariate two-sample goodness-of-fit (GOF) test based on multivariate empirical distribution function (MEDF) and the change-point model. As expected, simulation results show that our proposed control chart is robust to nonnormally distributed data, and moreover, it is efficient in detecting process shifts, especially large shifts, which is one of the main drawbacks of most robust control charts in the literature. As it avoids the need for a lengthy data-gathering step, the proposed chart is particularly useful in start-up or short-run situations. Comparison results and a real data example show that our proposed chart has great potential for application.     

Comparison of two weibull distributions under random censoring

The two-sample problem for comparing Weibull scale parameters is studied for randomly censored data. Three different test statistics are considered and their asymptotic properties are established under a sequence of local alternatives, It is shown that both the test statistic based on the mlefs (maximum likelihood estimators) and the likelihood ratio test are asymptotically optimum. The third statistic based only on the number of failures is not, Asymptotic relative efficiency of this statistic is obtained and its numerical values are computed for uniform and Weibull censoring, Effects of uniform random censoring on the censoring level of the experiment are illus¬trated, A direct proof for the joint asymptotic normality of the mlefs of the shape and the scale parameters is also given