Tests for Trend: A Simulation Study

In this study, we use the wavelet analysis to construct a test statistic to test for the existence of a trend in the series. We also propose a new approach for testing the presence of trend based on the periodogram of the data. Since we are also interested in the presence of a long-memory process among the data, we study the properties of our test statistics under different degrees of dependency. We compare the results when using the band periodogram test and the wavelet test with results obtained by applying the ordinary least squares (OLS) method under the same conditions.     

Testing for additivity and joint effects in multivariate nonparametric regression using Fourier and wavelet methods

We consider the problem of testing for additivity and joint effects in multivariate nonparametric regression when the data are modelled as observations of an unknown response function observed on a d-dimensional (d 2) lattice and contaminated with additive Gaussian noise. We propose tests for additivity and joint effects, appropriate for both homogeneous and inhomogeneous response functions, using the particular structure of the data expanded in tensor product Fourier or wavelet bases studied recently by Amato and Antoniadis (2001) and Amato, Antoniadis and De Feis (2002). The corresponding tests are constructed by applying the adaptive Neyman truncation and wavelet thresholding procedures of Fan (1996), for testing a high-dimensional Gaussian mean, to the resulting empirical Fourier and wavelet coefficients. As a consequence, asymptotic normality of the proposed test statistics under the null hypothesis and lower bounds of the corresponding powers under a specific alternative are derived. We use several simulated examples to illustrate the performance of the proposed tests, and we make comparisons with other tests available in the literature.     

Closed testing procedures for all pairwise comparisons in a randomized block design

We consider multiple comparison test procedures among treatment effects in a randomized block design. We propose closed testing procedures based on maximum values of some two-sample t test statistics and based on F test statistics. It is shown that the proposed procedures are more powerful than single-step procedures and the REGW (Ryan/Einot–Gabriel/Welsch)-type tests. Next, we consider the randomized block design under simple ordered restrictions of treatment effects. We propose closed testing procedures based on maximum values of two-sample one-sided t test statistics and based on Batholomew’s statistics for all pairwise comparisons of treatment effects. Although single-step multiple comparison procedures are utilized in general, the power of these procedures is low for a large number of groups. The closed testing procedures stated in the present article are more powerful than the single-step procedures. Simulation studies are performed under the null hypothesis and some alternative hypotheses. In this studies, the proposed procedures show a good performance.     

Goodness‐of‐fit Test for Directional Data

In this paper, we study the problem of testing the hypothesis on whether the density f of a random variable on a sphere belongs to a given parametric class of densities. We propose two test statistics based on the L² and L¹ distances between a non‐parametric density estimator adapted to circular data and a smoothed version of the specified density. The asymptotic distribution of the L² test statistic is provided under the null hypothesis and contiguous alternatives. We also consider a bootstrap method to approximate the distribution of both test statistics. Through a simulation study, we explore the moderate sample performance of the proposed tests under the null hypothesis and under different alternatives. Finally, the procedure is illustrated by analysing a real data set based on wind direction measurements.     

Comment on testing for spurious and cointegrated regressions: a wavelet approach

In a recent paper, Leong and Huang [6] proposed a wavelet-correlation-based approach to test for cointegration between two time series. However, correlation and cointegration are two different concepts even when wavelet analysis is used. It is known that statistics based on non-stationary integrated variables have non-standard asymptotic distributions. However, wavelet analysis offsets the integrating order of non-stationary series so that traditional asymptotics on stationary variables suffices to ascertain the statistical properties of wavelet-based statistics. Based on this, this note shows that wavelet correlations cannot be used as a test of cointegration.     

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     

Parametric bootstrap and approximate tests for two Poisson variates

The parametric bootstrap tests and the asymptotic or approximate tests for detecting difference of two Poisson means are compared. The test statistics used are the Wald statistics with and without log-transformation, the Cox F statistic and the likelihood ratio statistic. It is found that the type I error rate of an asymptotic/approximate test may deviate too much from the nominal significance level α under some situations. It is recommended that we should use the parametric bootstrap tests, under which the four test statistics are similarly powerful and their type I error rates are all close to α. We apply the tests to breast cancer data and injurious motor vehicle crash data.     

On testing for causality in variance between two multivariate time series

Verifying the existence of a relationship between two multivariate time series represents an important consideration. In this article, the procedure developed by Cheung and Ng [A causality-in-variance test and its application to financial market prices, J. Econom. 72 (1996), pp. 33–48] designed to test causality in variance for univariate time series is generalized in several directions. A first approach proposes test statistics based on residual cross-covariance matrices of squared (standardized) residuals and cross products of (standardized) residuals. In a second approach, transformed residuals are defined for each residual vector time series, and test statistics are constructed based on the cross-correlations of these transformed residuals. Test statistics at individual lags and portmanteau-type test statistics are developed. Conditions are given under which the new test statistics converge in distribution towards chi-square distributions. The proposed methodology can be used to determine the directions of causality in variance, and appropriate test statistics are presented. Monte Carlo simulation results show that the new test statistics offer satisfactory empirical properties. An application with two bivariate financial time series illustrates the methods.     

TESTING HYPOTHESES ABOUT THE COMMON MEAN OF TWO NORMAL POPULATIONS

Three tests are considered concerning the common mean of two normal populations: (1) an F test based on a sample from one population, (2) a test based on the addition of the F statistics from independent samples from two popultions (proposed), and (3) a test based on the maximum of the F statistics from two independent samples from two populations. A condition under which test (2) is locally more powerful than test (1) is given. As the test statistic in test (2) does not follow a standard distribution, a formula for approximating the observed significance level is provided. A simulation study is used to compare the power of these tests.     

Testing Serial Correlation in Partially Linear Single-Index Errors-in-Variables Models

In this article, we propose two test statistics for testing the underlying serial correlation in a partially linear single-index model Y = η(Z ^τα) + X ^τβ + ? when X is measured with additive error. The proposed test statistics are shown to have asymptotic normal or chi-squared distributions under the null hypothesis of no serial correlation. Monte Carlo experiments are also conducted to illustrate the finite sample performance of the proposed test statistics. The simulation results confirm that these statistics perform satisfactorily in both estimated sizes and powers.     

The performance of tests when observations have different variances

It is common to test if there is an effect due to a treatment. The commonly used tests have the assumption that the observations differ in location, and that their variances are the same over the groups. Different variances can arise if the observations being analyzed are means of different numbers of observations on individuals or slopes of growth curves with missing data. This study is concerned with cases in which the unequal variances are known, or known to a constant of proportionality. It examines the performance of the ttest, the Mann–Whitney–Wilcoxon Rank Sum test, the Median test, and the Van der Waerden test under these conditions. The t-test based on the weighted means is the likelihood ratio test under normality and has the usual optimality properties. The other tests are compared to it. One may align and scale the observations by subtracting the mean and dividing by the standard deviation of each point. This leads to other, analogous test statistics based on these adjusted observations. These statistics are also compared. Finally, the regression scores tests are compared to the other procedures.     

Generalized discriminant analysis based on distances

This paper describes a method of generalized discriminant analysis based on a dissimilarity matrix to test for differences in a priori groups of multivariate observations. Use of classical multidimensional scaling produces a low‐dimensional representation of the data for which Euclidean distances approximate the original dissimilarities. The resulting scores are then analysed using discriminant analysis, giving tests based on the canonical correlations. The asymptotic distributions of these statistics under permutations of the observations are shown to be invariant to changes in the distributions of the original variables, unlike the distributions of the multi‐response permutation test statistics which have been considered by other workers for testing differences among groups. This canonical method is applied to multivariate fish assemblage data, with Monte Carlo simulations to make power comparisons and to compare theoretical results and empirical distributions. The paper proposes classification based on distances. Error rates are estimated using cross‐validation.     

Location‐invariant Multi‐sample U‐tests for Covariance Matrices with Large Dimension

For two or more multivariate distributions with common covariance matrix, test statistics for certain special structures of the common covariance matrix are presented when the dimension of the multivariate vectors may exceed the number of such vectors. The test statistics are constructed as functions of location‐invariant estimators defined as U‐statistics, and the corresponding asymptotic theory is used to derive the limiting distributions of the proposed tests. The properties of the test statistics are established under mild and practical assumptions, and the same are numerically demonstrated using simulation results with small or moderate sample sizes and large dimensions.     

Second‐order analysis of marked inhomogeneous spatiotemporal point processes: Applications to earthquake data

To analyze interactions in marked spatiotemporal point processes (MSTPPs), we introduce marked second‐order reduced moment measures and K‐functions for inhomogeneous second‐order intensity‐reweighted stationary MSTPPs. These summary statistics, which allow us to quantify dependence between different mark‐based classifications of points, depend on the specific mark space and mark reference measure chosen. Unbiased and consistent minus‐sampling estimators are derived for all statistics considered, and a test for random labeling is indicated. In addition, we treat Voronoi intensity estimators for MSTPPs. These new statistics are finally employed to analyze an Andaman Sea earthquake data set.     

The exact hypothesis test for the shape parameter of a new two-parameter distribution with the bathtub shape or increasing failure rate function under progressive censoring with random removals

This study considers the exact hypothesis test for the shape parameter of a new two-parameter distribution with the shape of a bathtub or increasing failure rate function under type II progressive censoring with random removals, where the number of units removed at each failure time follows a binomial or a uniform distribution. Several test statistics are proposed and one numerical example is provided to illustrate the proposed hypothesis test for the shape parameter. Finally, a simulation study is performed to compare the power performances of all proposed test statistics. We concluded that the test statistic w ₁ is more attractive than other methods as it has better performance than other test statistics for most cases based on the criteria of maximum power.     

Semi-supervised Bayesian adaptive multiresolution shrinkage for wavelet-based denoising

We can use wavelet shrinkage to estimate a possibly multivariate regression function g under the general regression setup, y = g + ε. We propose an enhanced wavelet-based denoising methodology based on Bayesian adaptive multiresolution shrinkage, an effective Bayesian shrinkage rule in addition to the semi-supervised learning mechanism. The Bayesian shrinkage rule is advanced by utilizing the semi-supervised learning method in which the neighboring structure of a wavelet coefficient is adopted and an appropriate decision function is derived. According to decision function, wavelet coefficients follow one of two prespecified Bayesian rules obtained using varying related parameters. The decision of a wavelet coefficient depends not only on its magnitude, but also on the neighboring structure on which the coefficient is located. We discuss the theoretical properties of the suggested method and provide recommended parameter settings. We show that the proposed method is often superior to several existing wavelet denoising methods through extensive experimentation.     

Testing for spherical symmetry via the empirical characteristic function

Kolmogorov–Smirnov-type and Cramér–von Mises-type goodness-of-fit tests are proposed for the null hypothesis that the distribution of a random vector X is spherically symmetric. The test statistics utilize the fact that X has a spherical symmetric distribution if, and only if, the characteristic function of X is constant over surfaces of spheres centred at the origin. Both tests come in convenient forms that are straightforwardly applicable with the computer. The asymptotic null distribution of the test statistics as well as the consistency of the tests is investigated under general conditions. Since both the finite sample and the asymptotic null distribution depend on the unknown distribution of the Euclidean norm of X, a conditional Monte Carlo procedure is used to actually carry out the tests. Results on the behaviour of the test in finite-samples are included along with a real-data example.     

Computationally efficient familywise error rate control in genome-wide association studies using score tests for generalized linear models

In genetic association studies, detecting phenotype–genotype association is a primary goal. We assume that the relationship between the data—phenotype, genetic markers and environmental covariates—can be modeled by a generalized linear model. The number of markers is allowed to be far greater than the number of individuals of the study. A multivariate score statistic is used to test each marker for association with a phenotype. We assume that the test statistics asymptotically follow a multivariate normal distribution under the complete null hypothesis of no phenotype–genotype association. We present the familywise error rate order k approximation method to find a local significance level (alternatively, an adjusted p-value) for each test such that the familywise error rate is controlled. The special case k=1 gives the Šidák method. As a by-product, an effective number of independent tests can be defined. Furthermore, if environmental covariates and genetic markers are uncorrelated, or no environmental covariates are present, we show that covariances between score statistics depend on genetic markers alone. This not only leads to more efficient calculations but also to a local significance level that is determined only by the collection of markers used, independent of the phenotypes and environmental covariates of the experiment at hand.     

Likelihood ratio tests for testing for multiple contaminants in the shocks and labelled slippage models

In the literature related to the study of lifelengths of experimental units, little attention has been paid to the models where shocks to the units generate outliers. In the present article, we consider a situation where n experimental units under investigation receive shocks at several time points. The parameter values of the lifelength distribution may change due to each shock, resulting in the generation of outliers. We derive the likelihood ratio test statistic to investigate if the shocks have significantly altered the parameter values. We also derive a likelihood ratio test under the labelled slippage alternative with multiple contaminations. Monte Carlo studies have been carried out to investigate the power of the proposed test statistics.     

Using principal components to test normality of high-dimensional data

Many multivariate statistical procedures are based on the assumption of normality and different approaches have been proposed for testing this assumption. The vast majority of these tests, however, are exclusively designed for cases when the sample size n is larger than the dimension of the variable p, and the null distributions of their test statistics are usually derived under the asymptotic case when p is fixed and n increases. In this article, a test that utilizes principal components to test for nonnormality is proposed for cases when p/n → c. The power and size of the test are examined through Monte Carlo simulations, and it is argued that the test remains well behaved and consistent against most nonnormal distributions under this type of asymptotics.