A modification of the kruskal-wallis statistic for the generalized behrens-fisher problem

The problem of testing the equality of the medians of several populations is considered. Standard distribution-free procedures for this problem require that the populations have the same shape in order to maintain their nominal significance level, ever asymptotically, under the null hypothesis of equal medians , A modification of the Kruskal-Wallis test statistic is proposed which is exactly distribution-free under the usual nonparanetric asswnption that the continuous populations are identical with any shape. It is asymptotically distribution-free when the Continuous populations are asswned to be syrmnetric with equal medians.     

DISTRIBUTION-FREE CONFIDENCE PROCEDURE FOR UMBRELLA ORDERINGS

Responses in a one-factor experiment with A; ordered treatments follow an umbrella, ordering if they consist of two piecewise monotone segments, i.e. increasing and then decreasing, or the converse. This paper proposes a non-parametric distribution-free confidence procedure for umbrella orderings, the aim being to identify the treatments that correspond to the optimal effects. It uses a method that joins the seemingly unrelated theories of U-statistics and isotonic regression. A random confidence subset of the ordered treatments is constructed, such that it contains all the unknown peaks (optimal treatments) of an umbrella ordering with any prespecified confidence level. The paper demonstrates that the proposed confidence procedure is nonparametric distribution-free and, further, that the proposed procedure naturally implies a test for umbrella alternatives. Since the proposed confidence procedure is always more informative than tests for umbrella alternatives, it should be used in their place in practice. An example illustrates the proposed procedure.     

Distribution-free tests for umbrella alternatives

In this paper we propose distribution-free teots ebi ujabreiia aAiuatives based on the decomposition of umbrella alternatives into a union of simple order alternatives, The proposed tests are shown to have more balanced power performances, Tables of critical points for the proposed tests are provided.     

Comparison of Two Distribution-Free Procedures for Multiple Comparisons With a Control

Chakraborti and Desu (1988) presented a distribution-free procedure for testing that k (≥1) distributions are equal to a control distribution. They compared their procedure, a generalization of the test proposed by Mathisen (1943), to the procedure proposed by Slivka (1970). They asserted that their procedure has shorter expected duration than Slivka's procedure in life-testing experiments where observations become available in an ordered manner. Here it is proven that, in fact, Slivka's procedure has shorter duration in such circumstances. Normal approximations are presented which indicate that their procedure requires a smaller sample size to guarantee a specified power for Lehmann alternatives and proportional hazard alternatives when all observations are to be observed.     

k-sample rank tests using pair-specific scoring functions

In this paper we consider generalizations of the distribution-free one-way ANOVA test procedures studied by Puri (1965), such generalizations allowing for possibly different scoring functions for different pairs of samples. The asymptotic distributions of the test statistics are derived and formulas for the asymptotic relative efficiencies of these tests with respect to a parametric competitor are obtained. Applications of these procedures to settings of ordered and umbrella alternatives are presented and potential advantages of this pair-specific scoring scheme are discussed.     

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     

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.     

Bayes two-decision procedures for two-sample problems and rank-order data

A two-sample problem for rank-order data is formulated as a two-decision problem. Using the general Bayes solution, Bayes procedures are derived for several configurations of the set of states of nature including some for which the problem is distribution-free. It is shown that for certain prior distributions these procedures reduce to classical LMP rank tests. Some devices for selection of prior distributions are suggested. It is shown that the Bayes risk of these procedures tends to zero as sample sizes increase.     

Tests for a simple tree order restriction with application to dose–response studies

Summary.  We propose 'Dunnett-type' test procedures to test for simple tree order restrictions on the means of p independent normal populations. The new tests are based on the estimation procedures that were introduced by Hwang and Peddada and later by Dunbar, Conaway and Peddada. The procedures proposed are also extended to test for 'two-sided' simple tree order restrictions. For non-normal data, nonparametric versions based on ranked data are also suggested. Using computer simulations, we compare the proposed test procedures with some existing test procedures in terms of size and power. Our simulation study suggests that the procedures compete well with the existing procedures for both one-sided and two-sided simple tree alternatives. In some instances, especially in the case of two-sided alternatives or for non-normally distributed data, the gains in power due to the procedures proposed can be substantial.     

Extensions of empirical likelihood and chi-squared-based tests for ordered alternatives

Several methods for comparing k populations have been proposed in the literature. These methods assess the same null hypothesis of equal distributions but differ in the alternative hypothesis they consider. We focus on two important alternative hypotheses: monotone and umbrella ordering. Two new families of test statistics are proposed, including two known tests, as well as two new powerful tests under monotone ordering. Furthermore, these families are adapted for testing umbrella ordering. We compare some members of the families with respect to power and Type I errors under different simulation scenarios. Finally, the methods are illustrated in several applications to real data.     

An Automatic Test for the Umbrella Alternatives

The paper proposes a new test for detecting the umbrella pattern under a general non‐parametric scheme. The alternative asserts that the umbrella ordering holds while the hypothesis is its complement. The main focus is put on controlling the power function of the test outside the alternative. As a result, the asymptotic error of the first kind of the constructed solution is smaller than or equal to the fixed significance level α on the whole set where the umbrella ordering does not hold. Also, under finite sample sizes, this error is controlled to a satisfactory extent. A simulation study shows, among other things, that the new test improves upon the solution widely recommended in the literature of the subject. A routine, written in R, is attached as the Supporting Information file.     

The limit test statistic distribution of the maximum value test for right-censored data

ABSTRACT

In this paper, the maximum value test is proposed and considered for two-sample problem solving with lifetime data. This test is a distribution-free test under non-censoring and is a not distribution-free test under censoring. The formula of the limit distribution of the proposed maximal value test is represented in the general case. The distribution of the test statistic has been studied experimentally. Also, we propose the estimate of a p-value calculation of the maximum value test instead of the Monte-Carlo simulation. This test is useful and applicable in case of choosing among the logrank test, the Cox–Mantel test, the Q test and Generalized Wilcoxon tests, for instance, the Gehan's Generalized Wilcoxon test and the Peto and Peto's Generalized Wilcoxon test.     

Rank-based empirical likelihood inference on medians of k populations

We propose a nonparametric method, called rank-based empirical likelihood (REL), for making inferences on medians and cumulative distribution functions (CDFs) of k populations. The standard distribution-free approach to testing the equality of k medians requires that the k population distributions have the same shape. Our REL-ratio (RELR) test for this problem requires fewer assumptions and can effectively use the symmetry information when the distributions are symmetric. Furthermore, our RELR statistic does not require estimation of variance, and achieves asymptotic pivotalness implicitly. When the k populations have equal medians we show that the REL method produces valid inferences for the common median and CDFs of k populations. Simulation results show that the REL approach works remarkably well in finite samples. A real data example is used to illustrate the proposed REL method.     

Use of the squared ranks test to test for the equality of the coefficients of variation

A distribution-free test for the equality of the coefficients of variation from k populations is obtained by using the squared ranks test for variances, as presented by Conover and Iman (1978) and Conover (1980), on the original observations divided by their respective expected values. Substitution of the sample mean in place of the expected value results in the test being only asymptotically distribution-free. Results of a simulation study evaluating the size of the test for various coefficient of variation values and probability distributions are presented.     

A modified Mack–Wolfe test for the umbrella alternative problem

ABSTRACT

The Mack–Wolfe test is the most frequently used non parametric procedure for the umbrella alternative problem. In this paper, modifications of the Mack–Wolfe test are proposed for both known peak and unknown peak umbrellas. The exact mean and variance of the proposed tests in the null hypothesis are also derived. We compare these tests with some of the existing tests in terms of the type I error rate and power. In addition, a real data example is presented.     

An extensive power evaluation of a novel two-sample density-based empirical likelihood ratio test for paired data with an application to a treatment study of attention-deficit/hyperactivity disorder and severe mood dysregulation

In many case-control studies, it is common to utilize paired data when treatments are being evaluated. In this article, we propose and examine an efficient distribution-free test to compare two independent samples, where each is based on paired observations. We extend and modify the density-based empirical likelihood ratio test presented by Gurevich and Vexler [7] to formulate an appropriate parametric likelihood ratio test statistic corresponding to the hypothesis of our interest and then to approximate the test statistic nonparametrically. We conduct an extensive Monte Carlo study to evaluate the proposed test. The results of the performed simulation study demonstrate the robustness of the proposed test with respect to values of test parameters. Furthermore, an extensive power analysis via Monte Carlo simulations confirms that the proposed method outperforms the classical and general procedures in most cases related to a wide class of alternatives. An application to a real paired data study illustrates that the proposed test can be efficiently implemented in practice.     

Distribution-Free Runs Test for Conditional Symmetry

A distribution-free runs test for conditional symmetry is proposed. The null distribution of the test statistics is derived. Intensive simulation is conducted to examine the power of the proposed test for different sample sizes and different alternatives. Data on the bilirubin level of babies in neonatal intensive care is used to illustrate the method.     

Stochastic equivalence of ranking methods

The asymptotic behavior of linear rank statistics for comparing the locations of two populations, where the observations are ranked jointly with other populations, is considered. Under certain conditions, the asymptotic behavior of these statistics does not depend on which other populations are included in the ranking. In particular, the difference of a pair of these statistics, with the same score function, but based on two different rankings, converges to zero in probability under Pitman alternatives and Chernoff-Savage conditions on the scores and underlying distributions.     

Nonparametric tests of hypotheses for umbrella alternatives

The author proposes a general method for constructing nonparametric tests of hypotheses for umbrella alternatives. Such alternatives are relevant when the treatment effect changes in direction after reaching a peak. The author's class of tests is based on the ranks of the observations. His general approach consists of defining two sets of rankings: the first is induced by the alternative and the other by the data itself. His test statistic measures the distance between the two sets. The author determines the asymptotic distribution for some special cases of distances under both the null and the alternative hypothesis when the location of the peak is known or unknown. He shows the good power of his tests through a limited simulation study     

Merging data from multiple sources: pretest and shrinkage perspectives

In this article, we have developed asymptotic theory for the simultaneous estimation of the k means of arbitrary populations under the common mean hypothesis and further assuming that corresponding population variances are unknown and unequal. The unrestricted estimator, the Graybill-Deal-type restricted estimator, the preliminary test, and the Stein-type shrinkage estimators are suggested. A large sample test statistic is also proposed as a pretest for testing the common mean hypothesis. Under the sequence of local alternatives and squared error loss, we have compared the asymptotic properties of the estimators by means of asymptotic distributional quadratic bias and risk. Comprehensive Monte-Carlo simulation experiments were conducted to study the relative risk performance of the estimators with reference to the unrestricted estimator in finite samples. Two real-data examples are also furnished to illustrate the application of the suggested estimation strategies.