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.     

Nonparametric test for the homogeneity of the overall variability

In this paper, we propose a nonparametric test for homogeneity of overall variabilities for two multi-dimensional populations. Comparisons between the proposed nonparametric procedure and the asymptotic parametric procedure and a permutation test based on standardized generalized variances are made when the underlying populations are multivariate normal. We also study the performance of these test procedures when the underlying populations are non-normal. We observe that the nonparametric procedure and the permutation test based on standardized generalized variances are not as powerful as the asymptotic parametric test under normality. However, they are reliable and powerful tests for comparing overall variability under other multivariate distributions such as the multivariate Cauchy, the multivariate Pareto and the multivariate exponential distributions, even with small sample sizes. A Monte Carlo simulation study is used to evaluate the performance of the proposed procedures. An example from an educational study is used to illustrate the proposed nonparametric test.     

DISPERSION CONTROL FOR MULTIVARIATE PROCESSES – SOME COMPARISONS

Tang & Barnett (1996) present some techniques for monitoring and controlling the dispersion of multivariate normal processes based on subgroup data. The current paper compares the proposed techniques and various competing procedures. Simulation results indicate that the proposed techniques are superior to existing procedures.     

A Phase I nonparametric Shewhart-type control chart based on the median

A nonparametric Shewhart-type control chart is proposed for monitoring the location of a continuous variable in a Phase I process control setting. The chart is based on the pooled median of the available Phase I samples and the charting statistics are the counts (number of observations) in each sample that are less than the pooled median. An exact expression for the false alarm probability (FAP) is given in terms of the multivariate hypergeometric distribution and this is used to provide tables for the control limits for a specified nominal FAP value (of 0.01, 0.05 and 0.10, respectively) and for some values of the sample size (n) and the number of Phase I samples (m). Some approximations are discussed in terms of the univariate hypergeometric and the normal distributions. A simulation study shows that the proposed chart performs as well as, and in some cases better than, an existing Shewhart-type chart based on the normal distribution. Numerical examples are given to demonstrate the implementation of the new chart.     

Testing for lack of fit in linear multiresponse models based on exact or near replicates

Three procedures for testing the adequacy of a proposed linear multiresponse regression model against unspecified general alternatives are considered. The model has an error structure with a matrix normal distribution which allows the vector of responses for a particular run to have an unknown covariance matrix while the responses for different runs are uncorrelated. Furthermore, each response variable may be modeled by a separate design matrix. Multivariate statistics corresponding to the classical univariate lack of fit and pure error sums of squares are defined and used to determine the multivariate lack of fit tests. A simulation study was performed to compare the power functions of the test procedures in the case of replication. Generalizations of the tests for the case in which there are no independent replicates on all responses are also presented.     

CUSUM control schemes for monitoring the covariance matrix of multivariate time series

Modified cumulative sum (CUSUM) control charts and CUSUM schemes for residuals are suggested to detect changes in the covariance matrix of multivariate time series. Several properties of these schemes are derived when the in-control process is a stationary Gaussian process. A Monte Carlo study reveals that the proposed approaches show similar or even better performance than the schemes based on the multivariate exponentially weighted moving average (MEWMA) recursion. We illustrate how the control procedures can be applied to monitor the covariance structure of developed stock market indices.     

The distribution of the ratio of independent central wishart determinants

Ratios of independent central Wishart determinants are useful statistics in multivariate analyses, particularly in the study of multivariate linear models. A method based on the inversion of characteristic functions is outlined for deriving new experessions for the probability distribution functions of the logarithms of these statistics. Accurate tables of the percentiles of these distributions have been obtained covering many bivariate and trivariate cases which have been computed by approximating these expression.     

Testing independence of several groups of variables

The null distribution of Wilks' likelihood-ratio criterion for testing independence of several groups of variables in a multivariate normal population is derived. Percentage points are tabulated for various values of the sample sizeN and partitions of p, the number of variables. This paper extends Mathai and Katiya's (1979) “sphericity” results and tables.     

Tables of percentage points of the k-variate normal distribution for large values of k

This paper gives tabulations of the upper α percentage points of the maximum absolute value of the k-variate normal distribution with common correlation ρ for values of k as high as 500. The tables are useful for performing multiple comparison procedures in experiments with large numbers of treatments.     

A Multiple Three-Decision Procedures for Comparing Several Treatments with a Control Under Heteroscedasticity

In this article, the design-oriented two-stage multiple three-decision procedure is proposed to classify a set of normal populations with respect to a control under heteroscedasticity. The statistical tables of percentage points and the power-related design constants, to implement our new two-stage procedure, are given. Sometimes when the sample for the second stage is not available, the one-stage data analysis procedure is proposed. Classifying a treatment better than control when it is actually worse (and vice versa) is known as type III error. Both the two-stage and one-stage procedures control the type III error rate at a specified level. The relationship between the two-stage and one-stage procedures is discussed. Finally, the application of the proposed procedures is illustrated with an example.     

Complex elliptically symmetric distributions

In this paper, the authors introduce a class of distributions known as complex elliptically symmetric distributions. The complex multivariate normal and complex multivariate t distributions are members of this class. Various properties of the complex elliptically symmetric distributions are studied. Finally, the robustness of certain test procedures are discussed when the assumption of complex multivariate normality is violated but the underlying distribution still belongs to the class of elliptically symmetric distributions.     

Wavelet-based estimation for multivariate stable laws

In this article, an estimation problem for multivariate stable laws using wavelets has been studied. The method of applying wavelets, which has already been done, to estimate parameters in univariate stable laws, has been extended to multivariate stable laws. The proposed estimating method is based on a nonlinear regression model on wavelet coefficients of characteristic functions. In particular, two parametric sub-classes of stable laws are considered: the class of multivariate stable laws with discrete spectral measure, and sub-Gaussian laws. Using a simulation study, the proposed method has been compared with well-known estimation procedures.     

A Multivariate Group Runs Control Chart for Process Dispersion

In this article, control charts for bivariate as well as for multivariate normal data are proposed to detect a shift in the process variability. Methods of obtaining design parameters and procedures of implementing the proposed charts are discussed. Performance of the proposed charts is compared with some existing control charts. It is verified that the proposed charts significantly reduce the out of control “average run length” (ARL) as compared to other charts considered in the study. Also, when the process variability decreases (process improvement), it is verified that the ARL of the proposed multivariate control chart increases as compared to other charts considered in the study.     

Regression for doubly inflated multivariate Poisson distributions

Dependent multivariate count data occur in several research studies. These data can be modelled by a multivariate Poisson or Negative binomial distribution constructed using copulas. However, when some of the counts are inflated, that is, the number of observations in some cells are much larger than other cells, then the copula-based multivariate Poisson (or Negative binomial) distribution may not fit well and it is not an appropriate statistical model for the data. There is a need to modify or adjust the multivariate distribution to account for the inflated frequencies. In this article, we consider the situation where the frequencies of two cells are higher compared to the other cells and develop a doubly inflated multivariate Poisson distribution function using multivariate Gaussian copula. We also discuss procedures for regression on covariates for the doubly inflated multivariate count data. For illustrating the proposed methodologies, we present real data containing bivariate count observations with inflations in two cells. Several models and linear predictors with log link functions are considered, and we discuss maximum likelihood estimation to estimate unknown parameters of the models.     

Robust multivariate mixture regression models with incomplete data

Multivariate mixture regression models can be used to investigate the relationships between two or more response variables and a set of predictor variables by taking into consideration unobserved population heterogeneity. It is common to take multivariate normal distributions as mixing components, but this mixing model is sensitive to heavy-tailed errors and outliers. Although normal mixture models can approximate any distribution in principle, the number of components needed to account for heavy-tailed distributions can be very large. Mixture regression models based on the multivariate t distributions can be considered as a robust alternative approach. Missing data are inevitable in many situations and parameter estimates could be biased if the missing values are not handled properly. In this paper, we propose a multivariate t mixture regression model with missing information to model heterogeneity in regression function in the presence of outliers and missing values. Along with the robust parameter estimation, our proposed method can be used for (i) visualization of the partial correlation between response variables across latent classes and heterogeneous regressions, and (ii) outlier detection and robust clustering even under the presence of missing values. We also propose a multivariate t mixture regression model using MM-estimation with missing information that is robust to high-leverage outliers. The proposed methodologies are illustrated through simulation studies and real data analysis.     

Bayesian selection of log-linear models

A general methodology is presented for finding suitable Poisson log-linear models with applications to multiway contingency tables. Mixtures of multivariate normal distributions are used to model prior opinion when a subset of the regression vector is believed to be nonzero. This prior distribution is studied for two- and three-way contingency tables, in which the regression coefficients are interpretable in terms of odds ratios in the table. Efficient and accurate schemes are proposed for calculating the posterior model probabilities. The methods are illustrated for a large number of two-way simulated tables and for two three-way tables. These methods appear to be useful in selecting the best log-linear model and in estimating parameters of interest that reflect uncertainty in the true model.     

On a class of partially sequential two-sample test procedures for multivariate continuous data

In a two-sample testing problem, sometimes one of the sample observations are difficult and/or costlier to collect compared to the other one. Also, it may be the situation that sample observations from one of the populations have been previously collected and for operational advantages we do not wish to collect any more observations from the second population that are necessary for reaching a decision. Partially sequential technique is found to be very useful in such situations. The technique gained its popularity in statistics literature due to its very nature of capitalizing the best aspects of both fixed and sequential procedures. The literature is enriched with various types of partially sequential techniques useable under different types of data set-up. Nonetheless, there is no mention of multivariate data framework in this context, although very common in practice. The present paper aims at developing a class of partially sequential nonparametric test procedures for two-sample multivariate continuous data. For this we suggest a suitable stopping rule adopting inverse sampling technique and propose a class of test statistics based on the samples drawn using the suggested sampling scheme. Various asymptotic properties of the proposed tests are explored. An extensive simulation study is also performed to study the asymptotic performance of the tests. Finally the benefit of the proposed test procedure is demonstrated with an application to a real-life data on liver disease.     

Multivariate analysis of variance for functional data

Functional data are being observed frequently in many scientific fields, and therefore most of the standard statistical methods are being adapted for functional data. The multivariate analysis of variance problem for functional data is considered. It seems to be of practical interest similarly as the one-way analysis of variance for such data. For the MANOVA problem for multivariate functional data, we propose permutation tests based on a basis function representation and tests based on random projections. Their performance is examined in comprehensive simulation studies, which provide an idea of the size control and power of the tests and identify differences between them. The simulation experiments are based on artificial data and real labeled multivariate time series data found in the literature. The results suggest that the studied testing procedures can detect small differences between vectors of curves even with small sample sizes. Illustrative real data examples of the use of the proposed testing procedures in practice are also presented.     

Integration of support vector machines and control charts for multivariate process monitoring

Statistical process control tools have been used routinely to improve process capabilities through reliable on-line monitoring and diagnostic processes. In the present paper, we propose a novel multivariate control chart that integrates a support vector machine (SVM) algorithm, a bootstrap method, and a control chart technique to improve multivariate process monitoring. The proposed chart uses as the monitoring statistic the predicted probability of class (PoC) values from an SVM algorithm. The control limits of SVM-PoC charts are obtained by a bootstrap approach. A simulation study was conducted to evaluate the performance of the proposed SVM–PoC chart and to compare it with other data mining-based control charts and Hotelling's T ² control charts under various scenarios. The results showed that the proposed SVM–PoC charts outperformed other multivariate control charts in nonnormal situations. Further, we developed an exponential weighed moving average version of the SVM–PoC charts for increasing sensitivity to small shifts.     

Some multivariate goodness-of-fit tests based on data depth

Based on data depth, three types of nonparametric goodness-of-fit tests for multivariate distribution are proposed in this paper. They are Pearson’s chi-square test, tests based on EDF and tests based on spacings, respectively. The Anderson–Darling (AD) test and the Greenwood test for bivariate normal distribution and uniform distribution are simulated. The results of simulation show that these two tests have low type I error rates and become more efficient with the increase in sample size. The AD-type test performs more powerfully than the Greenwood type test.