Comparison of unweighted and weighted rank based tests for an ordered alternative in randomized complete block designs

In randomized complete block designs, a monotonic relationship among treatment groups may already be established from prior information, e.g., a study with different dose levels of a drug. The test statistic developed by Page and another from Jonckheere and Terpstra are two unweighted rank based tests used to detect ordered alternatives when the assumptions in the traditional two-way analysis of variance are not satisfied. We consider a new weighted rank based test by utilizing a weight for each subject based on the sample variance in computing the new test statistic. The new weighted rank based test is compared with the two commonly used unweighted tests with regard to power under various conditions. The weighted test is generally more powerful than the two unweighted tests when the number of treatment groups is small to moderate.     

A Flexible Nonparametric Test for Ordered and Umbrella Alternatives

A flexible and robust test for the ordered and umbrella alternatives is proposed in this paper. A two-step procedure is presented to make the proposed test be easily applicable. Type I error and power of the given approach are thoroughly investigated by extensive Monte Carlo studies.     

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.     

Precedence-type test based on Kaplan–Meier estimator of cumulative distribution function

In this paper, we introduce a precedence-type test based on Kaplan–Meier estimator of cumulative distribution function (CDF) for testing the hypothesis that two distribution functions are equal against a stochastically ordered hypothesis. This test is an alternative to the precedence life-test proposed first by Nelson (1963). After deriving the null distribution of the test statistic, we present its exact power function under the Lehmann alternative, and compare the exact power as well as simulated power (under location-shift) of the proposed test with other precedence-type tests. Next, we extend this test to the case of progressively Type-II censored data. Critical values for some combination of sample sizes and progressive censoring schemes are presented. We then examine the power properties of this test procedure and compare them to those of the weighted precedence and weighted maximal precedence tests under a location-shift alternative by means of Monte Carlo simulations. Finally, we present two examples to illustrate all the test procedures discussed here, and then make some concluding remarks.     

An omnibus two-sample test for ranked-set sampling data

We develop an omnibus two-sample test for ranked-set sampling (RSS) data. The test statistic is the conditional probability of seeing the observed sequence of ranks in the combined sample, given the observed sequences within the separate samples. We compare the test to existing tests under perfect rankings, finding that it can outperform existing tests in terms of power, particularly when the set size is large. The test does not maintain its level under imperfect rankings. However, one can create a permutation version of the test that is comparable in power to the basic test under perfect rankings and also maintains its level under imperfect rankings. Both tests extend naturally to judgment post-stratification, unbalanced RSS, and even RSS with multiple set sizes. Interestingly, the tests have no simple random sampling analog.     

Data Driven Rank Test for Two-Sample Problem

Traditional linear rank tests are known to possess low power for large spectrum of alternatives. In this paper we introduce a new rank test possessing a considerably larger range of sensitivity than linear rank tests. The new test statistic is a sum of squares of some linear rank statistics while the number of summands is chosen via a data-based selection rule. Simulations show that the new test possesses high and stable power in situations when linear rank tests completely break down, while simultaneously it has almost the same power under alternatives which can be detected by standard linear rank tests. Our approach is illustrated by some practical examples. Theoretical support is given by deriving asymptotic null distribution of the test statistic and proving consistency of the new test under essentially any alternative.     

Testing the information matrix equality with robust estimators

The information matrix (IM) equality can be used to test for misspecification of a parametric model. We study the behavior of the IM test when the maximum-likelihood (ML) estimators used in the construction of this test are replaced with robust estimators. The latter do not suffer from the masking effect in the presence of outliers and can improve the power of the IM test. At the normal location-scale model, the IM test using the ML estimators is known as the Jarque–Bera test, and uses skewness and kurtosis to detect deviations from normality. When robust estimators are employed to test the IM equality, a robust version of the Jarque–Bera test emerges. We investigate in detail the local asymptotic power of the IM test, for various estimators and under a variety of local alternatives. For the normal regression model, it is shown by simulations under fixed alternatives that in many cases the use of robust estimators substantially increases the power of the IM test.     

Power comparisons between the F-test,the studentised range test,and an optimal test of the equality of several normal means

The power assessment of tests of the equality of k normal means such as the k treatment means in a one-way fixed effects analysis of variance model is addressed. Power assessment is considered in terms of a constraint on the range of the treatment means. The power properties of the standard F-test and Studentised range test are compared with those of an optimal (minimax) test procedure, which is known to maximise power levels under this constraint. It is shown that the standard test procedures compare well with the optimal test procedure, and in particular, the Studentised range test is shown to be practically as good as optimal in this setting.     

Non‐Parametric Change‐Point Tests for Long‐Range Dependent Data

Abstract. We propose a non‐parametric change‐point test for long‐range dependent data, which is based on the Wilcoxon two‐sample test. We derive the asymptotic distribution of the test statistic under the null hypothesis that no change occurred. In a simulation study, we compare the power of our test with the power of a test which is based on differences of means. The results of the simulation study show that in the case of Gaussian data, our test has only slightly smaller power minus.3pt than the ‘difference‐of‐means’ test. For heavy‐tailed data, our test outperforms the ‘difference‐of‐means’ test.     

On testing the bioequivalence of several treatments using the measure of distance

A studentized range test is proposed to test the hypothesis of bioequivalence of normal means in terms of a standardized distance among means. A least favourable configuration (LFC) of means to guarantee the maximum level at a null hypothesis and an LFC of means to guarantee the minimum power at an alternative hypothesis are obtained. This level and power of the test are fully independent of the unknown means and variances. For a given level, the critical value of the test under a null hypothesis can be determined. Furthermore, if the power under an alternative is also required at a given level, then both the critical value and the required sample size for an experiment can be simultaneously determined. In situations where the common population variance is unknown and the bioequivalence is the actual distance between means without standardization, a two-stage sampling procedure can be employed to find these solutions.     

A generalization of mosteller's test

A generalization of Mosteller's test for slippage of the location parameter is proposed. The distribution of the test statistic under the null hypothesis is obtained and the power of the test is compared with that of Mosteller's test     

Two-sample spatial rank test using projection

This paper explores in high-dimensional settings how to test the equality of two location vectors. We introduce a rank-based projection test under elliptical symmetry. Optimal projection direction is derived according to asymptotically and locally best power criteria. Data-splitting strategy is used to estimate optimal projection and construct test statistics. The limiting null distribution and power function of the proposed statistics are thoroughly investigated under some mild assumptions. The test is shown to keep type I error rates pretty well and outperforms several existing methods in a broad range of settings, especially in the presence of large correlation structures. Simulation studies are conducted to confirm the asymptotic results and a real data example is applied to demonstrate the advantage of the proposed procedure.     

Testing additivity in two-way classifications with no replications:the locally best invariant test

Row x column interaction is frequently assumed to be negligible in two-way classifications having one observation per cell. Absence of interaction allows the researcher to estimate experimental error and to proceed with making inferences about row and column effects. If additivity is suspect, it is conventional to test it against a structured alternative. If the structured alternative missspecifies the existing nonadditivity, then the power of the test is low, even if the magnitude of the existing nonadditivity is large. The locally best invariant (LBI) test of additivity is less subject to model misspecification because a particular structural alternative need not be hypothesized. This paper illustrates the LBI test of additivity and compares its power to that of the Johnson-Graybill likelihood ratio (LR) test. The LBI test performs as well as the LR test under a Johnson-Graybill alternative and performs better than the LR test under more general alternatives.     

WILCOXON-TYPE RANK-SUM PRECEDENCE TESTS

This paper introduces Wilcoxon‐type rank‐sum precedence tests for testing the hypothesis that two life‐time distribution functions are equal. They extend the precedence life‐test first proposed by Nelson in 1963. The paper proposes three Wilcoxon‐type rank‐sum precedence test statistics—the minimal, maximal and expected rank‐sum statistics—and derives their null distributions. Critical values are presented for some combinations of sample sizes, and the exact power function is derived under the Lehmann alternative. The paper examines the power properties of the Wilcoxon‐type rank‐sum precedence tests under a location‐shift alternative through Monte Carlo simulations, and it compares the power of the precedence test, the maximal precedence test and Wilcoxon rank‐sum test (based on complete samples). Two examples are presented for illustration.     

Methods for Testing Population Signal-to-Noise Ratio

This article considered several test statistics for testing the population signal-to-noise ratio based on parametric, nonparametric, and modified methods. To compare the performance of the proposed test statistics, a simulation study has been conducted under both symmetric and skewed distributions. The performance of the test statistics is compared based on the empirical size and power of the test. It is evident for large sample that some of our proposed test statistics are performing better in the sense of high power and have been recommended for the researchers.     

A simulation study on the influence of ties on uniform scores test for circular data

Uniform scores test is a rank-based method that tests the homogeneity of k-populations in circular data problems. The influence of ties on the uniform scores test has been emphasized by several authors in several articles and books. Moreover, it is suggested that the uniform scores test should be used with caution if ties are present in the data. This paper investigates the influence of ties on the uniform scores test by computing the power of the test using average, randomization, permutation, minimum, and maximum methods to break ties. Monte Carlo simulation is performed to compute the power of the test under several scenarios such as having 5% or 10% of ties and tie group structures in the data. The simulation study shows no significant difference among the methods under the existence of ties but the test loses its power when there are many ties or complicated group structures. Thus, randomization or average methods are equally powerful to break ties when applying uniform scores test. Also, it can be concluded that k-sample uniform scores test can be used safely without sacrificing the power if there are only less than 5% of ties or at most two groups of a few ties.     

Non-Parametric Testing of Conditional Variance Functions in Time Series

This paper proposes a non‐parametric test for examining hypotheses about variance functions under stationarity and ergodicity conditions. Special cases of nonlinear time series models are studied, and it is found that under mild conditions the test is consistent. Its power is examined in a simulation study.     

A new nonparametric bivariate test for two sample location problem

A strictly nonparametric bivariate test for two sample location problem is proposed. The proposed test is easy to apply and does not require the stringent condition of affine-symmetry or elliptical symmetry which is required by some of the major tests available for the same problem. The power function of the proposed test is calculated. The asymptotic distribution of the proposed test statistic is found to be normal. The power of proposed test is compared with some of the well-known tests under various distributions using Monte Carlo simulation technique. The power study shows that the proposed test statistic performs better than most of the test statistics for almost all the distributions considered here. As soon as the underlying population structure deviates from normality, the ability of the proposed test statistic to detect the smallest shift in location increases as compared to its competitors. The application of the test is shown by using a data set.     

Likelihood ratio tests based on subglobal optimization: A power comparison in exponential mixture models

The paper compares several versions of the likelihood ratio test for exponential homogeneity against mixtures of two exponentials. They are based on different implementations of the likelihood maximization algorithm. We show that global maximization of the likelihood is not appropriate to obtain a good power of the LR test. A simple starting strategy for the EM algorithm, which under the null hypothesis often fails to find the global maximum, results in a rather powerful test. On the other hand, a multiple starting strategy that comes close to global maximization under both the null and the alternative hypotheses leads to inferior power.     

A new test for two-sample location problem based on empirical distribution function

We propose a new test for testing the equality of location parameter of two populations based on empirical distribution function (ECDF). The test statistics is obtained as a power divergence between two ECDFs. The test is shown to be distribution free, and its null distribution is obtained. We conducted empirical power comparison of the proposed test with several other available tests in the literature. We found that the proposed test performs better than its competitors considered here under several population structures. We also used two real datasets to illustrate the procedure.