A permutation test for multivariate data with grouped components

In this paper, we consider a nonparametric test procedure for multivariate data with grouped components under the two sample problem setting. For the construction of the test statistic, we use linear rank statistics which were derived by applying the likelihood ratio principle for each component. For the null distribution of the test statistic, we apply the permutation principle for small or moderate sample sizes and derive the limiting distribution for the large sample case. Also we illustrate our test procedure with an example and compare with other procedures through simulation study. Finally, we discuss some additional interesting features as concluding remarks.     

On the robustness of the extreme deviate test for a single multivariate outlier against heavy-tailed distributions

In this paper we assess the sensitivity of the multivariate extreme deviate test for a single multivariate outlier to non-normality in the form of heavy tails. We find that the empirical significance levels can be markedly affected by even modest departures from multivariate normality. The effects are particularly severe when the sample size is large relative to the dimension. Finally, by way of example we demonstrate that certain graphical techniques may prove useful in identifying the source of rejection for the multivariate extreme deviate test.     

Moment inequalities for truncated multivariate distributions

In this note some properties of the absolute moments of a doubly truncated arbitrary multivariate distribution are studied and several moment inequalities are derived.     

On power and sample size of the ANOVA-type rank test

When using nonparametric methods to analyze factorial designs with repeated measures, the ANOVA-type rank test has gained popularity due to its robustness and appropriate type I error control. This article proposes power and sample size calculation formulas under two scenarios where the nonparametric regression coefficients are known or they are unknown but a pilot study is available. When a pilot study is available, the formulas do not need any assumption on the underlying population distributions. Simulation results confirm the accuracy of the proposed methods. An STZ rat excisional wound study is used to demonstrate the application of the methods.     

An empirical goodness-of-fit test for multivariate distributions

An empirical test is presented as a tool for assessing whether a specified multivariate probability model is suitable to describe the underlying distribution of a set of observations. This test is based on the premise that, given any probability distribution, the Mahalanobis distances corresponding to data generated from that distribution will likewise follow a distinct distribution that can be estimated well by means of a large sample. We demonstrate the effectiveness of the test for detecting departures from several multivariate distributions. We then apply the test to a real multivariate data set to confirm that it is consistent with a multivariate beta model.     

A note on the multivariate multisample rank sum and median tests

The muitivariate nonparametric tests analogous to the univar-iate rank sum test and median test are contained in Puri and Sen (1970). These tests provided a practical alternative for the analysis of multivariate data when the assumptions of parametric methods are not satisfied.

In this paper maximum values for L_Nthe asymptotic chi-Square test statistic for both the Multivariate Multisample Rank Sum Test (MMRST) and the Multivariate Multisample Median Test (MMMT) are developed.     

Maximum-likelihood estimates and likelihood-ratio criteria for multivariate elliptically contoured distributions

For a class of multivariate elliptically contoured distributions the maximum-likelihood estimators of the mean vector and covariance matrix are found under certain conditions. Likelihood-ratio criteria are obtained for a class of null hypotheses. These have the same form as in the normal case.     

Parametric estimation under a class of multivariate distributions

In this paper the parameters of some members of a class of multivariate distributions, which was constructed by AL-Hussaini and Ateya (2003), are estimated by using the maximum likelihood and Bayes methods.     

A strategy for constructing multivariate distributions

Given a random vector (X₁,…, X_n) for which the univariate and bivariate marginal distributions belong to some specified families of distributions, we present a procedure for constructing families of multivariate distributions with the specified univariate and bivariate margins. Some general properties of the resulting families of multivariate distributions are reviewed. This procedure is illustrated by generalizing the bivariate Plackett (1965) and Clayton (1978) distributions to three dimensions. In addition to providing rich families of models for data analysis, this method of construction provides a convenient way of simulating observations from multivariate distributions with specific types of univariate and bivariate marginal distributions. A general algorithm for simulating random observations from these families of multivariate distributions is presented     

Bayesian multivariate GARCH models with dynamic correlations and asymmetric error distributions

The main goal in this paper is to develop and apply stochastic simulation techniques for GARCH models with multivariate skewed distributions using the Bayesian approach. Both parameter estimation and model comparison are not trivial tasks and several approximate and computationally intensive methods (Markov chain Monte Carlo) will be used to this end. We consider a flexible class of multivariate distributions which can model both skewness and heavy tails. Also, we do not fix tail behaviour when dealing with fat tail distributions but leave it subject to inference.     

A covariance-based test for shared frailty in multivariate lifetime data

We decompose the score statistic for testing for shared finite variance frailty in multivariate lifetime data into marginal and covariance-based terms. The null properties of the covariance-based statistic are derived in the context of parametric lifetime models. Its non-null properties are estimated using simulation and compared with those of the score test and two likelihood ratio tests when the underlying lifetime distribution is Weibull. Some examples are used to illustrate the covariance-based test. A case is made for using the covariance-based statistic as a simple diagnostic procedure for shared frailty in a parametric exploratory analysis of multivariate lifetime data and a link to the bivariate Clayton–Oakes copula model is shown.     

Lower dimensional marginal density functions of absolutely continuous truncated multivariate distributions

The lower dimensional marginal density functions of a truncated multivariate density function is derived in general, and shown that it is a function of untruncated marginal density function, appropriately defined conditional distribution function and size of the multivariate truncation region. As a special case, lower dimensional marginal density function of a truncated multivariate normal distribution is given.     

Conditional specifications of multivariate pareto and student distributions

A random vector has a multivariate Pareto distribution if one of its univariate conditional distribution is Pareto and some of its marginals are identically distributed.A general method developed in the course of the proof of this result is applied also to characterize the multivariate Student (Cauchy) measure by one univariate Student conditional distribution.     

A survey on multivariate chi-square distributions and their applications in testing multiple hypotheses

We are concerned with three different types of multivariate chi-square distributions. Their members play important roles as limiting distributions of vectors of test statistics in several applications of multiple hypotheses testing. We explain these applications and consider the computation of multiplicity-adjusted p-values under the respective global hypothesis. By means of numerical examples, we demonstrate how much gain in level exhaustion or, equivalently, power can be achieved with corresponding multivariate multiple tests compared with approaches which are only based on univariate marginal distributions and do not take the dependence structure among the test statistics into account. As a further contribution of independent value, we provide an overview of essentially all analytic formulas for computing multivariate chi-square probabilities of the considered types which are available up to present. These formulas were scattered in the previous literature and are presented here in a unified manner.     

A randomized Baumgartner statistic for multivariate two-sample testing hypothesis

In this paper, multivariate two-sample testing problems were examined based on the Jure?ková–Kalina's ranks of distances. The multivariate two-sample rank test based on the modified Baumgartner statistic for the two-sided alternative was proposed. The proposed statistic was a randomized statistic. Simulations were used to investigate the power of the suggested statistic for various population distributions.     

Sample size calculation for the one‐sample log‐rank test

In this paper, an exact variance of the one‐sample log‐rank test statistic is derived under the alternative hypothesis, and a sample size formula is proposed based on the derived exact variance. Simulation results showed that the proposed sample size formula provides adequate power to design a study to compare the survival of a single sample with that of a standard population. Copyright © 2014 John Wiley & Sons, Ltd.     

Model selection criteria for reduced rank multivariate time series: a simulation study

The main focus of our paper is to compare the performance of different model selection criteria used for multivariate reduced rank time series. We consider one of the most commonly used reduced rank model, that is, the reduced rank vector autoregression (RRVAR (p, r)) introduced by Velu et al. [Reduced rank models for multiple time series. Biometrika. 1986;7(31):105–118]. In our study, the most popular model selection criteria are included. The criteria are divided into two groups, that is, simultaneous selection and two-step selection criteria, accordingly. Methods from the former group select both an autoregressive order p and a rank r simultaneously, while in the case of two-step criteria, first an optimal order p is chosen (using model selection criteria intended for the unrestricted VAR model) and then an optimal rank r of coefficient matrices is selected (e.g. by means of sequential testing). Considered model selection criteria include well-known information criteria (such as Akaike information criterion, Schwarz criterion, Hannan–Quinn criterion, etc.) as well as widely used sequential tests (e.g. the Bartlett test) and the bootstrap method. An extensive simulation study is carried out in order to investigate the efficiency of all model selection criteria included in our study. The analysis takes into account 34 methods, including 6 simultaneous methods and 28 two-step approaches, accordingly. In order to carefully analyse how different factors affect performance of model selection criteria, we consider over 150 simulation settings. In particular, we investigate the influence of the following factors: time series dimension, different covariance structure, different level of correlation among components and different level of noise (variance). Moreover, we analyse the prediction accuracy concerned with the application of the RRVAR model and compare it with results obtained for the unrestricted vector autoregression. In this paper, we also present a real data application of model selection criteria for the RRVAR model using the Polish macroeconomic time series data observed in the period 1997–2007.     

Linear signed rank tests for hultivariate location

Previously proposed linear signed rank tests for multivariate location are not invariant under linear transformations of the observations, The asymptotic relative efficiencies of the tests 2 with respect to Hotelling's T²test depend on the direction of shift and the covariance matrix of the alternative distributions. For distributions with highly correlated components, the efficiencies of some of these tests can be arbitrarily low; they approach zero for certain multivariate normal alternatives, This article proposes a transformation of the data to be performed prior to standard linear signed rank tests, The resulting procedures have attractive power and efficiency properties compared to the original tests, In particular, for elliptically symmetric contiguous alternafives, the efficiencies of the new tests equal those of corresponding univariate linear signed rank tests with respect to the t test.     

Tables for the Friedman rank test

This note presents tables for Friedman's test for two-way analysis of variance by ranks. These tables are more accurate than those that are presented in the literature. After intensive simulations, we have found for particular critical values some discrepancies with tables published earlier. The tables are also more extensive than those previously available.