_ This paper presents a distribution free multivariate Kolmogorov-Smirnov goodness of fit test. The test uses an statistic which is built using Rosenblatt's transformation and an algorithm is developed to compute it in the bivariate case. An approximate test, that can be easily computed in any dimension, is also presented. The power of these multivariate tests is studied in a simulation study.

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A new portmanteau test for time series more powerful than the tests ofLjung and Box (1978) and Monti (1994} is proposed. The test is based on the pth root of the determinant of the pth autocorrelation matrix. It is shown that this statistic can be interpreted as the geometric mean of the squared multiple correlation coefficients with m lag values when m goes from 1 to p. It can also be interpreted as a geometric mean of the partial autocorrelation coefficients. The asymptotic distribution of the test statistic is obtained. This distribution is a linear combination of chi-squared distributions and it is shown that it can be approximated by a gamma distribution. The power of the test is compared with that of the Ljung and Box and Monti tests and it is shown that the proposed test can be up to 50% more powerful depending upon the model and sample size.The test is applied to the detection of nonlinearity by using the same matrix but with coefficients that are now the autocorrelations of the squared residuals. The new test is more powerful than the test of McLeod and Li (1983) (1978) and Monti (1994) is proposed. The test is based on the pth root of the determinant of the pth autocorrelation matrix. It is shown that this statistic can be interpreted as the geometric mean of the squared multiple correlation coefficients with m lag values when m goes from 1 to p. It can also be interpreted as a geometric mean of the partial autocorrelation coefficients. The asymptotic distribution of the test statistic is obtained. This distribution is a linear combination of chi-squared distributions and it is shown that it can be approximated by a gamma distribution. The power of the test is compared with that of the Ljung and Box and Monti tests and it is shown that the proposed test can be up to 50% more powerful depending upon the model and sample size.The test is applied to the detection of nonlinearity by using the same matrix but with coefficients that are now the autocorrelations of the squared residuals. The new test is more powerful than the test of McLeod and Li (1983) for nonlinearity. An example is presented in which this test detects nonlinearity in the residuals of the sunpot series.

In this article, we present a simple multivariate outlier-detection procedure and a robust estimator for the covariance matrix, based on the use of information obtained from projections onto the directions that maximize and minimize the kurtosis coef cient of the projected data. The properties of this estimator (computationa l cost, bias) are analyzed and compared with those of other robust estimators described in the literature through simulation studies. The performance of the outlier-detection procedure is analyzed by applying it to a set of well-known examples.KEY WORDS: Kurtosis; Linear projection; Multivariate statistics.The detection of outliers in multivariate data is recognized to be an important and dif cult problem in the physical, chemical, and engineering sciences. Whenever multiple measurements are obtained, there is always the possibility that changes in the measurement process will generate clusters of outliers. Most standard multivariate analysis techniques rely on the assumption of normality and require the use of estimates for both the location and scale parameters of the distribution. The presence of outliers may distort arbitrarily the values of these estimators and render meaningless the results of the application of these techniques. According to Rocke and Woodruff (1996), the problem of the joint estimation of location and shape is one of the most dif cult in robust statistics. Wilks (1963) proposed identifying sets of outliers of size j in normal multivariate data by checking the minimum values of the ratios -A 4I 5 -=-A-, where -A 4I 5 -is the internal scatter of a modi ed sample in which the set of observations I of size j has been deleted and -A-is the internal scatter of the complete sample. The internal scatter is proportional to the determinant of the covariance matrix and the ratios are computed for all possible sets of size j. Wilks computed the distribution of the statistic for j equal to 1 and 2. It is well known that this procedure is a likelihood ratio test and that for j D 1 the method is equivalent to selecting the observation with the largest Mahalanobis distance from the center of the data.Because a direct extension of this idea to sets of outliers larger than 2 or 3 is not practical, Gnanadesikan and Kettenring (1972) proposed to reduce the multivariate detection problem to a set of univariate problems by looking at projections of the data onto some direction. They chose the direction of maximum variability of the data and, therefore, they proposed to obtain the principal components of the data and search for outliers in these directions. Although this method provides the correct solution when the outliers are located close to the directions of the principal components, it may fail to identify outliers in the general case.An alternative approach is to use robust location and scale estimators. Maronna (1976) studied af nely equivariant M estimators for covariance matrices, and Campbell (1980) proposed using the Mahalanobis distance computed using M estimators for the mean...

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This paper considers outliers in multivariate time series analysis. It generalizes four types of disturbances commonly used in the univariate time series analysis to the multivariate case, and investigates dynamic effects of a multivariate outlier on individual components if marginal models are used. An innovational outlier of a vector series can introduce a patch of outliers for the marginal component models. The paper also proposes an iterative procedure to detect and handle multiple outliers. By comparing and contrasting results of univariate and multivariate outlier detections, one can gain insights into the characteristics of an outlier. An outlier in a component series mayor may not have significant impacts on the other components. We use real examples to demonstrate the proposed analysis.

This artiele describes a procedure to identify elusters in multivariate data using information obtained from the univariate proj~ctions of the sample data onto certain directions. The directions are chosen as those that minimize and maXlmlze the kurtOSlS coefficlent of the projected data. It is shown that, under certain conditions, these directions provide the largest separatlOn for the dlfferent clusters. The projected univariate data are used to group the observations according to the values of the gaps or spacmgs between consecutIveordered observations. These groupings are then combined over all projection directions. The behavlOr of the method lS tested on ~everal examples, and compared to k-means, MCLUST, and the procedure proposed by Jones and Sibson in 1987. The proposed algonthm lS iterative, affine equivariant, flexible, robust to outliers, fast to implement, and seems to work well m practIce.KEY WORDS: Classification; Kurtosis; Multivariate analysis; Robustness; Spacings. . INTRODUCTIONLet us suppose we have a sample of multivariate observations generated from several different populations. One of the most important problems of cluster analysis is the partitioning of the points of this sample into nonoverlapping clusters. The most commonly used algorithms as sume that the number of clusters, G, is known and the partition of the data is carried out by maximizing some optimality criterion. These algorithms start with an initial classification of the points into clusters and then reassign each point in tum to increase the criterion. The process is repeated until a local optimum of the criterion is reached. The most often used criteria can be derived from the application of likelihood ratio tests to mixtures of multivariate normal populations with different means. It is well known that (i) when all the covariance matrices are assumed to be egual to the identity matrix, the criterio n obtained corresponds to minimizing tr(W), where W is the within-groups covariance matrix, this is the criterion used in the standard k-means procedure; (ii) when the covariance matrices are assumed to be egual, without other restrictions, the criterion obtained is minimizing ¡W¡ (Friedman and Rubin 1967); (iii) when the covariance matrices are allowed to be different, the criterion obtained is minimizing Lf= 1 n j log I W j / n J where W j is the sample cross-product matrix for the jth cluster (see Seber 1984, and Gordon 1994, for other criteria). These algorithms may present two main limitations: (i) we have to choose the criterion a priori, without knowing the covariance structure of the data and different criteria can lead to very different answers; and (ii) they usually reguire large amounts of computer time, which makes them difficult to apply to large data sets.Banfield and Raftery (1993), and Dasgupta and Raftery (1998) have proposed a model-based approach to clustering that has several advantages over previous procedures. They as sume a mixture model and use the EM algorithm to estimate the parameters. The initial esti...