A simple nonparametric test for bivariate symmetry about a line

Over the years many researchers have dealt with testing the hypotheses of symmetry in univariate and multivariate distributions in the parametric and nonparametric setup. In a multivariate setup, there are several formulations of symmetry, for example, symmetry about an axis, joint symmetry, marginal symmetry, radial symmetry, symmetry about a known point, spherical symmetry, and elliptical symmetry among others. In this paper, for the bivariate case, we formulate a concept of symmetry about a straight line passing through the origin in a plane and accordingly develop a simple nonparametric test for testing the hypothesis of symmetry about a straight line. The proposed test is based on a measure of deviance between observed counts of bivariate samples in suitably defined pairs of sets. The exact null distribution and non-null distribution, for specified classes of alternatives, of the test statistics are obtained. The null distribution is tabulated for sample size from n=5 up to n=30. The null mean, null variance and the asymptotic null distributions of the proposed test statistics are also obtained. The empirical power of the proposed test is evaluated by simulating samples from the suitable class of bivariate distributions. The empirical findings suggest that the test performs reasonably well against various classes of asymmetric bivariate distributions. Further, it is advocated that the basic idea developed in this work can be easily adopted to test the hypotheses of exchangeability of bivariate random variables and also bivariate symmetry about a given axis which have been considered by several authors in the past.     

A test for bivariate exponentiality against bifr alternative

A test is proposed for testing bivariate exponentiality against the Bivariate Increasing Failure Rate (BIFR) class of alternatives. The test statistic is a function of U-statistics and hence asymptotically normally distributed and consistent     

A nonparametric test for bivariate circular symmetry based on the empirical cdf

A nonparametric test for circular symmetry about 0 in a continuous bivariate distribution is proposed. The test is of the von Mises type, based on the empirical cdf of the sample, expressed in polar co-ordinates. However, the test is independent of the choice of the polar axis. The asymptotic form of the test statistic is obtained by considering the weak convergence of the empirical process to a limiting Gaussian process. The asymptotic distribution of the test statistic is found explicitly, both under the null hypothesis and under simple alternatives. The test is shown to be consistent against all alternatives.     

Tests of symmetry derived as components of pearson's phi-squared distance measure

Rank statistics which arise as estimates of the first and third components of a frequency decomposition of Pearson's phi-squared distance measure, introduced by Eubank, LaRiccia, and Rosenstein (1987), are examined for their usefulness as tests of symmetry about a known median against various asymmetric alternatives, Pitman asymptotic relative efficiencies are used to compare the efficacies of the new statistics with classical test procedures, and empirical powers of the new tests are compared via simulation for a variety of asymmetric distributions. Statistics which arise from components based on Legendre polynomials are of particular interest. The first component is the familiar Wilcoxon signed rank statistic. The third component, which is a new statistic for this problem, exhibits a high level of sensitivity to a variety of asymmetric alternatives both asymptotically and in small sample studies.     

Comparison of tests for bivariate symmetry versus location and/or scale alternatives

This paper studies the performance of tests which use a null hypothesis of bivariate symmetry and detect the broad class of location and/or scale alternatives . The conditionally distribution-free tests of Sen (1967) and Hollander (1971) and parametric tests related to those of Bell and Haller (1969) are compared in a Monte Carlo study which also includes a new conditionally distribution-free test.     

Extension of the log-rank test

For the comparison of two groups of survival times subject to censoring the log-rank test is widely used. The log-rank test is known to be asymptotically fully efficient for the proportional hazards alternatives. But if the ratio of the hazards changes, the log-rank test may not detect the difference between the two groups. In this article a new test procedure is proposed. Simulation results show that the proposed test procedure provides good power against alternatives, where the hazard ratio between the two groups changes across 1.     

On the Power of Bootstrapped Specification Tests

Abstract

Decisions based on econometric model estimates may not have the expected effect if the model is misspecified. Thus, specification tests should precede any analysis. Bierens' specification test is consistent and has optimality properties against some local alternatives. A shortcoming is that the test statistic is not distribution free, even asymptotically. This makes the test unfeasible. There have been many suggestions to circumvent this problem, including the use of upper bounds for the critical values. However, these suggestions lead to tests that lose power and optimality against local alternatives. In this paper we show that bootstrap methods allow us to recover power and optimality of Bierens' original test. Bootstrap also provides reliable p-values, which have a central role in Fisher's theory of hypothesis testing. The paper also includes a discussion of the properties of the bootstrap Nonlinear Least Squares Estimator under local alternatives.     

Testing Homogeneity in Gamma Mixture Models

This paper characterizes the asymptotic behaviour of the likelihood ratio test statistic (LRTS) for testing homogeneity (i.e. no mixture) against gamma mixture alternatives. Under the null hypothesis, the LRTS is shown to be asymptotically equivalent to the square of Davies's Gaussian process test statistic and diverges at a log n rate to infinity in probability. Based on the asymptotic analysis, we propose and demonstrate a computationally efficient method to simulate the null distributions of the LRTS for small to moderate sample sizes.     

Bivariate Sign Test for One-Sample Bivariate Location Model Using Ranked Set Sample

In this article, we introduce a bivariate sign test for the one-sample bivariate location model using a bivariate ranked set sample (BVRSS). We show that the proposed test is asymptotically more efficient than its counterpart sign test based on a bivariate simple random sample (BVSRS). The asymptotic null distribution and the non centrality parameter are derived. The asymptotic distribution of the vector of sample median as an estimator of the locations of the bivariate model is introduced. Theoretical and numerical comparisons of the asymptotic efficiency of the BVRSS sign test with respect to the BVSRS sign test are also given.     

A new test for bivariate distributions : exponential vs new-better-than-used alternative

This paper presents a new test for testing bivariate exponentiality against the bivariate new better than used (BNBU) class of non-exponential probability distributions. The test statistic is a function of U-statistics and hence asymptotically normally distributed and consistent.     

Testing for Circular Reflective Symmetry about a Known Median Axis

Circular data arise in many contexts, a particularly rich source being animal orientation experiments. Often, in the analysis of such data, a fundamental question of scientific interest is whether the underlying distribution is reflectively symmetric about some specific axis. In this paper, the situation in which the axis of interest is known to be a median axis is considered and a simple, asymptotically distribution- free test for circular reflective symmetry against skew alternatives is developed. The results from a simulation study lead to a testing strategy incorporating the new test and the circular analogue of the modified runs test of Modarres & Gastwirth (1996). The application of the testing strategy is illustrated using circular data arising from two animal orientation experiments.     

A simple approach for testing constant failure rate against different ageing classes for discrete data

In this paper, we develop simple non-parametric test based on U-statistics for testing constant failure rate against IFR, IFRA, DMRL, NBU and NBUE alternatives. The asymptotic properties of the test statistics are studied. In particular, the test statistics are shown to be asymptotically normal and consistent against the relevant alternatives. Some numerical results are presented to demonstrate the performance of the proposed tests.     

Testing for Granger-causality in quantiles

This paper proposes a consistent parametric test of Granger-causality in quantiles. Although the concept of Granger-causality is defined in terms of the conditional distribution, most articles have tested Granger-causality using conditional mean regression models in which the causal relations are linear. Rather than focusing on a single part of the conditional distribution, we develop a test that evaluates nonlinear causalities and possible causal relations in all conditional quantiles, which provides a sufficient condition for Granger-causality when all quantiles are considered. The proposed test statistic has correct asymptotic size, is consistent against fixed alternatives, and has power against Pitman deviations from the null hypothesis. As the proposed test statistic is asymptotically nonpivotal, we tabulate critical values via a subsampling approach. We present Monte Carlo evidence and an application considering the causal relation between the gold price, the USD/GBP exchange rate, and the oil price.     

A class of tests for exponentiality against decreasing mean residual life alternatives

A class of tests is proposed for testing Exponentiality against the Decreasing Mean Residual Life (DMRL) class of non-exponential probability distributions. These tests are consistent and asymptotically unbiased against all continuous DMRL alternatives. They are U - statistics and hence asymptotically normally distributed. The asymptotic relative efficiency (ARE) with respect to other tests for DMRL are quite high. Small sample powers are also comparable with small sample powers of the competitors.     

Robustness Properties of the Pitman–Morgan Test

Abstract

The classical Pitman–Morgan test is known to be optimal for testing equality of the variances of components of a bivariate normal vector. We first show that it is also optimal for a generalized model involving the matrix spherical distribution. Then we discuss and demonstrate, both analytically and empirically, that it is nonrobust, i.e., its type I error control is inexact both asymptotically and in moderate size bivariate random samples.     

On Threshold Choice in Hypothesis Testing for Dynamical Systems with Small Noise

We considered the problem of testing a simple hypothesis against composite one-sided alternative by the continuous time observations of diffusion process with small noise. Moreover, we propose a test which is asymptotically equivalent to the Neyman-Pearson test for local alternatives. The special choice of the threshold allows us to improve the rate of convergence of the first type error to the given value. The calculation of this threshold is based on the stochastic expansion of the test statistics and on the Edgeworth expansion of its distribution function.     

A rank test for stochastically ordered alternatives

A rank test based on the number of ‘near-matches’ among within-block rankings is proposed for stochastically ordered alternatives in a randomized block design with t treatments and b blocks. The asymptotic relative efficiency of this test with respect to the Page test is computed as number of blocks increases to infinity. A sequential analog of the above test procedure is also considered. A repeated significance test procedure is developed and average sample number is computed asymptotically under the null hypothesis as well as under a sequence of contiguous alternatives.     

Variance Components Testing in ANOVA‐Type Mixed Models

The purpose of this article is threefold. First, variance components testing for ANOVA ‐type mixed models is considered, in which response may not be divided into independent sub‐vectors, whereas most of existing methods are for models where response can be divided into independent sub‐vectors. Second, testing that a certain subset of variance components is zero. Third, as normality is often violated in practice, it is desirable to construct tests under very mild assumptions. To achieve these goals, an adaptive difference‐based test and an adaptive trace‐based test are constructed. The test statistics are asymptotically normal under the null hypothesis, are consistent against all global alternatives and can detect local alternatives distinct from the null at a rate as close to n ^{? 1 ∕ 2} as possible with n being the sample size. Moreover, when the dimensions of variance components in different sets are bounded, we develop a test with chi‐square as its limiting null distribution. The finite sample performance of the tests is examined via simulations, and a real data set is analysed for illustration.     

The two-sample problem with multivariate censored data: a new rank test family

A new rank test family is proposed to test the equality of two multivariate failure times distributions with censored observations. The tests are very simple: they are based on a transformation of the multivariate rank vectors to a univariate rank score and the resulting statistics belong to the familiar class of the weighted logrank test statistics. The new procedure is also applicable to multivariate observations in general, such as repeated measures, some of which may be missing. To investigate the performance of the proposed tests, a simulation study was conducted with bivariate exponential models for various censoring rates. The size and power of these tests against Lehmann alternatives were compared to the size and power of two other tests (Wei and Lachin, 1984 and Wei and Knuiman, 1987). In all simulations the new procedures provide a relatively good power and an accurate control over the size of the test. A real example from the National Cooperative Gallstone Study is given     

A test for exponentiatity against hnbue alternatives

Neyman's asymptotically most powerful test in the class of al1 similar tests for testing exponentiality against a parametric class of Makeham alternatives is shown to be consistent for the much wider class of harmonic new better than used in expectation distributions.