A test for the complete independence of high-dimensional random vectors

ABSTRACT

This paper discusses the problem of testing the complete independence of random variables when the dimension of observations can be much larger than the sample size. It is reported that two typical tests based on, respectively, the biggest off-diagonal entry and the largest eigenvalue of the sample correlation matrix lose their control of type I error in such high-dimensional scenarios, and exhibit distinct behaviours in type II error under different types of alternative hypothesis. Given these facts, we propose a permutation test procedure by synthesizing these two extreme statistics. Simulation results show that for finite dimension and sample size the proposed test outperforms the existing methods in various cases.     

Multiple comparisons of mean vectors with large dimension under general conditions

ABSTRACT

Multiple comparisons for two or more mean vectors are considered when the dimension of the vectors may exceed the sample size, the design may be unbalanced, populations need not be normal, and the true covariance matrices may be unequal. Pairwise comparisons, including comparisons with a control, and their linear combinations are considered. Under fairly general conditions, the asymptotic multivariate distribution of the vector of test statistics is derived whose quantiles can be used in multiple testing. Simulations are used to show the accuracy of the tests. Real data applications are also demonstrated.     

On the independence between two generalized second degree polynomial statistics and their covariance

The condition of independence between two generalized second degree polynomial statistics is worked out in this paper under the assumption that the covariance matrix of the underlying normal distribution may be singular. The covariance between two such forms is also worked out without the assumption of normality. The results are indicated in compact matrix form.     

The non-null distribution for the problem of tests of independence

The problem of testing independence in the multinormal case is considered in this paper. The non-null distribution of the likelihood ratio criterion is obtained for the case of two subvectors by using a simple straightforward technique. The null case as well as the known cases are also verified.     

Testing for pairwise serial independence via the empirical distribution function

Built on Skaug and Tjøstheim's approach, this paper proposes a new test for serial independence by comparing the pairwise empirical distribution functions of a time series with the products of its marginals for various lags, where the number of lags increases with the sample size and different lags are assigned different weights. Typically, the more recent information receives a larger weight. The test has some appealing attributes. It is consistent against all pairwise dependences and is powerful against alternatives whose dependence decays to zero as the lag increases. Although the test statistic is a weighted sum of degenerate Cramér–von Mises statistics, it has a null asymptotic N (0, 1) distribution. The test statistic and its limit distribution are invariant to any order preserving transformation. The test applies to time series whose distributions can be discrete or continuous, with possibly infinite moments. Finally, the test statistic only involves ranking the observations and is computationally simple. It has the advantage of avoiding smoothed nonparametric estimation. A simulation experiment is conducted to study the finite sample performance of the proposed test in comparison with some related tests.     

Empirical approximations for Hoeffding’s test of bivariate independence using two Weibull extensions

The sampling distributions are generally unavailable in exact form and are approximated either in terms of the asymptotic distributions, or their correction using expansions such as Edgeworth, Laguerre or Cornish–Fisher; or by using transformations analogous to that of Wilson and Hilferty. However, when theoretical routes are intractable, in this electronic age, the sampling distributions can be reasonably approximated using empirical methods. The point is illustrated using the null distribution of Hoeffding’s test of bivariate independence which is important because of its consistency against all dependence alternatives. For constructing the approximations we employ two Weibull extensions, the generalized Weibull and the exponentiated Weibull families, which contain a rich variety of density shapes and tail lengths, and have their distribution functions and quantile functions available in closed form, making them convenient for obtaining the necessary percentiles and p-values. Both approximations are seen to be excellent in terms of accuracy, but that based on the generalized Weibull is more portable.     

Some blum-kiefer-rosenblatt type tests for the joint independence of variables

Testing the joint independence of variables has long been an interesting issue in statistical inferences. Blum, Kiefer and Rosenblatt (1961) suggested a test based on a sample distribution function. To overcome the sparseness of data points in high-dimensional space and deal with general cases, we in this paper suggest several extended versions of B-K-R tests via projection pursuit. Bootstrap method is applied to determine the critical values and for computational reason, an approximation derived by Number-theoretic method, for the bootstrap statistics is suggested. Several simulation experiments are performed and a real-life example is investigated.     

A statistical analysis of the screening measure of multiple recursive random number generators of orders one and two

This paper presents the results of a comprehensive empirical analysis of the screening measure of multiple recursive generators (MRGs) of orders one and two. Two kinds of screening measures are distinguished: spectral test and lattice test. With regard to these screening measures, two exhaustive searches of the twenty best MRGs of orders one and two are conducted. Some empirical comparisons reveal that the screening procedure with maximum spectral value criterion is preferred in terms of efficiency and thus, is a good way of obtaining ideal MRGs of higher orders. Several extensively tested second-order MRGs are also presented and are therefore recommended.     

A one-sample location test based on weighted averaging of two test statistics when the dimension and the sample size are large

We discuss a one-sample location test that can be used when the dimension and the sample size are large. It is well-known that the power of Hotelling’s test decreases when the dimension is close to the sample size. To address this loss of power, some non exact approaches were proposed, e.g., Dempster (1958 Dempster, A.P. (1958). A high dimensional two sample significance test. Ann. Math. Stat. 29:995–1010.[Crossref] , [Google Scholar], 1960 Dempster, A.P. (1960). A significance test for the separation of two highly multivariate small samples. Biometrics 16:41–50.[Crossref], [Web of Science ®] , [Google Scholar]), Bai and Saranadasa (1996 Bai, Z.D., Saranadasa, H. (1996). Effect of high dimension: by an example of a two sample problem. Stat. Sin. 6:311–329.[Web of Science ®] , [Google Scholar]), and Srivastava and Du (2008 Srivastava, M.S., Du, M. (2008). A test for the mean vector with fewer observations than the dimension. J. Multivariate Anal. 99:386–402.[Crossref], [Web of Science ®] , [Google Scholar]). In this article, we focus on Hotelling’s test and Dempster’s test. The comparative merits and demerits of these two tests vary according to the local parameters. In particular, we consider the situation where it is difficult to determine which test should be used, that is, where the two tests are asymptotically equivalent in terms of local power. We propose a new statistic based on the weighted averaging of Hotelling’s T²-statistic and Dempster’s statistic that can be applied in such a situation. Our weight is determined on the basis of the maximum local asymptotic power on a restricted parameter space that induces local asymptotic equivalence between Hotelling’s test and Dempster’s test. Numerical results show that our test is more stable than Hotelling’s T²-statistic and Dempster’s statistic in most parameter settings.     

Testing the equality of two normal percentiles

Two statistics are suggested for testing the equality of two normal percentiles where population means and variances are unknown. The first is based on the generalized likelihood ratio test (LRT), the second on Cochran's statistic used in the Behrens-Fisher problem. Size and power comparisons are made by using simulation and asympototic theory.     

Testing equality of location parameters of two exponential distributions from censored samples

A modification to Tiku's (1981) test, which may be seriously biased, is proposed. The modified test is only marginally biased if at all and is substantially more powerful. A ratio test based on Tiku’s (1967) modified likelihood function is also proposed, and shown to have power comparable to the power of the ratio test based on the likelihood function. The proposed ratio test is, however, much easier from a computational viewpoint.     

Testing for lack of dependence in the functional linear model

The authors consider the linear model Y_n = ψX_n + ?_n relating a functional response with explanatory variables. They propose a simple test of the nullity of ψ based on the principal component decomposition. The limiting distribution of their test statistic is chi‐squared, but this distribution is also an excellent approximation in finite samples. The authors illustrate their method using data from terrestrial magnetic observatories.     

Testing the equality of two independent regression models

In some situations, for example in agriculture, biology, hydrology, and psychology, researchers wish to determine whether the relationship between response variable and predictor variables differs in two populations. In other words, we are interested in comparing two regression models for two independent datasets. In this work, we will use the parametric and nonparametric methods to establish hypothesis testing for the equality of two independent regression models. Then the simulation study is provided to investigate the performance of the proposed method.     

Testing hypotheses about the common mean of two normal populations

Two simple tests which allow for unequal sample sizes are considered for testing hypothesis for the common mean of two normal populations. The first test is an exact test of size a based on two available t-statistics based on single samples made exact through random allocation of α among the two available t-tests. The test statistic of the second test is a weighted average of two available t-statistics with random weights. It is shown that the first test is more efficient than the available two t-tests with respect to Bahadur asymptotic relative efficiency. It is also shown that the null distribution of the test statistic in the second test, which is similar to the one based on the normalized Graybill-Deal test statistic, converges to a standard normal distribution. Finally, we compare the small sample properties of these tests, those given in Zhou and Mat hew (1993), and some tests given in Cohen and Sackrowitz (1984) in a simulation study. In this study, we find that the second test performs better than the tests given in Zhou and Mathew (1993) and is comparable to the ones given in Cohen and Sackrowitz (1984) with respect to power..     

A characterization of the independence - distribution - preserving covariance structure for the multivariate maximum squared - radii statistic

Necessary and sufficient conditions on the observation covariance structure and on the set of linear transformations are given for which the distribution of the multivariate maximum squared - radii statistic for detecting a single multivariate outlier is invariant from the distribution assuming the usual independence covariance structure. Thus, we extend the work of Baksalary and Puntanen (1990), who have given necessary and sufficient conditions for an independence-distribution-preserving covariance structure for Grubbs' statistic for detecting a univariate outlier. We also extend the work of Marco, Young, and Turner (1987) and Pavur and Young (1991), who have given sufficient conditions for an independence-distribution-preserving dependency structure for the multivariate squared - radii statistic.     

Comparing the mean vectors of two independent multivariate log-normal distributions

The multivariate log-normal distribution is a good candidate to describe data that are not only positive and skewed, but also contain many characteristic values. In this study, we apply the generalized variable method to compare the mean vectors of two independent multivariate log-normal populations that display heteroscedasticity. Two generalized pivotal quantities are derived for constructing the generalized confidence region and for testing the difference between two mean vectors. Simulation results indicate that the proposed procedures exhibit satisfactory performance regardless of the sample sizes and heteroscedasticity. The type I error rates obtained are consistent with expectations and the coverage probabilities are close to the nominal level when compared with the other method which is currently available. These features make the proposed method a worthy alternative for inferential analysis of problems involving multivariate log-normal means. The results are illustrated using three examples.     

Testing for the parametric component of partially linear EV models under random censorship

This paper investigates the hypothesis test of the parametric component in partially linear errors-in-variables (EV) model with random censorship. We construct two test statistics based on the difference of the corrected residual sum of squares and empirical likelihood ratio under the null and alternative hypotheses. It is shown that the limiting distributions of the proposed test statistics are both weighted sum of independent standard chi-squared distribution with one degree of freedom under the null hypothesis. Based on the adjusted test statistics, we further develop two new types of test procedures. Finite sample performance of the proposed test procedures is evaluated by extensive simulation studies.     

Step-Down diagnostic analysis for monitoring the covariance matrix of bivariate normal processes

ABSTRACT

A comparison among VMIX, VMAX and the adapted step-down Sullivan et al. (SE) tests for covariance matrix under bivariate normal assumption is presented. The type I error and power estimates were obtained by using Monte Carlo simulation under different scenarios with respect to covariance and correlation structures. In general, VMIX was more powerful than VMAX being SE more powerful than both, with few exceptions. SE test is more general since it can be used for normal and non-normal data, with no restriction with respect to the pattern of the covariance matrix shifts, and for larger dimension than the bivariate case.     

Testing the equality of several log-normal means based on a computational approach

In this study, we propose a new test based on a computational approach to test the equality of several log-normal means. We compare this test with some existing methods in terms of the type-I error rate and power using Monte Carlo simulations for varying values of number of groups and sample sizes. The simulation results indicate that the proposed test could be suggested as a good alternative for testing the equality of several log-normal means.