A monte carlo study of the LN statistic for the multivariate nonparametric median and rank sum tests for two populations

Small sample tables are not available for the multisample multivariate rank sum test (MMRST) or the multisample multivariate median test (MMMT) L_N statistic. Consequently, the statistic usually is compared to its asymptotic Chi-square value. To investigate the appropriateness of this procedure a Monte Carlo study is used to measure both significance level and relative power for a variety of multivariate dispersion structures.     

Simultaneous tests for finite families of hypotheses

A procedure is proposed whereby R test statistics F=(F₁F₂…F_r)together with "randomly generated critical points" (C₁C₂…C_r) may be used to construct a simultaneous test for

a family containing R hypotheses. This procedure provides simultaneous tests having an exact prescribed type I error rate; the procedure does not require the distribution of F to be known. The simultaneous test is illustrated for making all pairwise comparisons in a one-way ANOVA model.     

A class of nonparametric tests for location and scale parameters

A class of nonparametric two-sample tests for testing identity of distributions versus alternatives containing both location and scale parameters is proposed and some properties are derived. A recursion formula for the exact distribution under the hypothesis is presented and, the asymptotic distribution is given under both the hypothesis and a contiguous sequence of alternatives. Some asymptotic optimality properties are deduced for particular tests of the class and finally, the asymptotic efficiency is found.     

Saddlepoint approximations to exact tests and improved likelihood ratio tests for the gamma distribution

For the most common one-sample and two-sample tests in the gamma distribution we derive the log likelihood ratio tests and the improved versions obtained by a Bartlett adjustment. For most of these tests an exact test exists and we give the saddlepoint approximation to the latter. The tests are compared with previously published tests and a small simulation study is included.     

On the sample breakdown robustness of some nonparametric tests

In this paper, relying on the sample breakdown points, we investigate the sample breakdown properties of some nonparametric tests. It is shown that the sample breakdown points of the sign test asymptotically dominate those of the Wilcoxon test for one–sided hypotheses, However, the different conclusion is derived in the case of testing some shrinking neighborhood hypotheses. The breakdown behaviors of the Kolmogorov test and X²–test are also explored. These studies unify or refine some existing breakdown analyses of tests.     

Power of depth-based nonparametric tests for multivariate locations

Li and Liu [New nonparametric tests of multivariate locations and scales. Statist Sci. 2004;19(4):686–696] introduced two tests for a difference in locations of two multivariate distributions based on the concept of data depth. Using the simplicial depth [Liu RY. On a notion of data depth based on random simplices. Ann Stat. 1990;18(1):405–414], they studied the performance of these tests for symmetric distributions, namely, the normal and the Cauchy, in a simulation study. However, to the best of our knowledge, the performance of these tests for skewed distributions has not been studied in the current literature. This paper is a contribution in that direction and examines the performance of these depth-based tests in an extensive simulation study involving ten distributions belonging to five well-known families of multivariate skewed distributions. The study includes a comparison of the performance of these tests for four popular affine-invariant depth functions. Conclusions and recommendations are offered.     

Some count-based nonparametric tests for circular symmetry of a bivariate distribution

In this paper, we revisit the problem of testing of the hypothesis of circular symmetry of a bivariate distribution. We propose some nonparametric tests based on sector counts. These include tests based on chi-square goodness-of-fit test, the classical likelihood ratio, mean deviation, and the range. The proposed tests are easy to implement and the exact null distributions for small sample sizes of the test statistics are obtained. Two examples with small and large data sets are given to illustrate the application of the tests proposed. For small and moderate sample sizes, the performances of the proposed tests are evaluated using empirical powers (empirical sizes are also reported). Also, we evaluate the performance of these count-based tests with adaptations of several well-known tests such as the Kolmogorov–Smirnov-type tests, tests based on kernel density estimator, and the Wilcoxon-type tests. It is observed that among the count-based tests the likelihood ratio test performs better.     

On a class of nonparametric tests for the treatment vs control problem

The classical problem of testing treatment versus control is revisited by considering a class of test statistics based on a kernel that depends on a constant ‘a’. The proposed class includes the celebrated Wilcoxon-Mann-Whitnet statistics as a special case when ‘a’=1. It is shown that, with optimal choice of ‘a’ depending on the underlying distribution, the optimal member performs better (in terms of Pitman efficiency) than the Wilcoxon-Mann-Whitney and the Median tests for a wide range of underlying distributions. An extended Hodges-Lehmann type point estimator of the shift prameter corresponding to the proposed ‘optimal’ test statistic is also derived.     

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     

On the corrections to information matrix tests

This paper addresses the issue of designing finite-sample corrections to information matrix tests. We review a Cornish-Fisher correction that has been propowed elsewhere and propose an alternative, Bartlett-type correction. Simulation results for skewness, excess kurtosis, normality and heteroskedasticity tests are given.     

A consistent test for heteroscedasticity in semi-parametric regression with nonparametric variance function based on the kernel method

It is important to detect the variance heterogeneity in regression models. Heteroscedasticity tests have been well studied in parametric and nonparametric regression models. This paper presents a consistent test for heteroscedasticity for nonlinear semi-parametric regression models with nonparametric variance function based on the kernel method. The properties of the test are investigated through Monte Carlo simulations. The test methods are illustrated with a real example.     

Small sample power of multivariate nonparametric tests for monotone trend

Bhattacharyya and Kioiz (1966) propose two multivariate nonparametric tests for monotone trend, one involving coordinate-wise Mann statistics and the other, coordinate-wise Spearman statistics. Dietz and Killeen (1981) propose a different test statistic based on coordinate-wise Mann statistics. The Pitman asymptotic relative efficiency of all three tests with respect to a normal theory competitor equals the cube root of the efficiency of a multivariate signed rank test with respect to Hotelling's T2. In this article, the small sample power of the nonparametric tests, the normal theory test, and a Bonferroni approach involving coordinate-wise univariate Mann or Spearman tests is examined in a simulation study. The Mann statistic of Dietz and Killeen and the Spearman statistic of Bhattacharyya and Klotz are found to perform well under both null and alternative hypotheses     

Likelihood ratio tests for fuzzy hypotheses testing

In hypotheses testing, such as other statistical problems, we may confront imprecise concepts. One case is a situation in which hypotheses are imprecise. In this paper, we recall and redefine some concepts about fuzzy hypotheses testing, and then we introduce the likelihood ratio test for fuzzy hypotheses testing. Finally, we give some applied examples.     

On the role of minque in testing of hypotheses under hiked linear models

Testing of hypotheses under balanced ANOVA models is fairly simple and generally based on the usual ANOVA sums of squares. Difficulties may arise in special cases when these sums of squares do not form a complete sufficient statistic. There is a huge literature on this subject which was recently surveyed in Seifert's contribution to the book of Mumak (1904). But there are only a few results about unbalanced models. In such models the consideration of likelihood ratios leads to more complex sums of squares known from MINQUE theory.

Uniform optimality of testsusually reduces to local optimality. Here we prespnt a small review of methods proposed for testing of hypotheses in unbalanced models. where MINQUEI playb a major role. We discuss the use of iterated MINQUE for the construction of asymptotically optimal tests described in Humak (1984) and approximate tests based on locally uncorrelated linear combinations of MINQUE estimators by Seifert (1985), We show that the latter tests coincide with robust locally optimal invariant tests proposeci by Kariya and Sinha and Das and Sinha, if the number of variance components is two. Explicit expressions for corresponding tests are given for the unbalanced two-way cross classification random model, which covers some other models as special cases. A simulation study under lines the relevance of MINQUE for testing of hypotheses problems.     

Some pitfalls of tests of separate families of hypotheses: normality vs. lognormality

It is often desirable to test non-nested hypotheses. Cox (1961, 1962) proposed forming a log-likelihood ratio from their maxima and then comparing this value to its expected value under the null hypothesis. Pitfalls exists when we apply Cox's test to the special case of testing normality versus lognormality. Pesaran (1981) and Kotz (1973) pointed out the slow convergence rate of the Cox's test. In this paper, this fact has been reemphasized; moreover, we propose an alternative likelihood ratio test which remedies problems arising from negative estimates of the asymptotic variance of Cox's test statistic and is uniformly more powerful than most commonly used tests.     

A nonparametric test for Type I censored data: r samples

This paper is concerned with testing that r random samples are all from the same population when the data are left (or right) censored. A statistic is developed which has elements of a goodness-of-fit statistic and of a modified Kruskal-Wallis statistic. The efficiency of this statistic relative to some other commonly used statistics is calculated. Some Monte Carlo comparisons are given.     

On the asymptotic distribution of the Dickey Fuller-GLS test statistic

In a very influential paper, Elliott et al. [Efficient tests for an autoregressive unit root. Econometrica. 1996;64:813–836] show that no uniformly most powerful test for the unit root testing problem exits, derive the relevant power envelope and characterize a family of point-optimal tests. As a by-product, they also propose a ‘generalized least squares (GLS) detrended’ version of the conventional Dickey–Fuller test, denoted DF-GLS, that has since then become very popular among practitioners, much more so than the point-optimal tests. In view of this, it is quite strange to find that, while conjectured in Elliott et al. [Efficient tests for an autoregressive unit root. Econometrica. 1996;64:813–836], so far there seems to be no formal proof of the asymptotic distribution of the DF-GLS test statistic. By providing three separate proofs, the current paper not only substantiates the required result, but also provides insight regarding the pros and cons of different methods of proof.