Abstraet _During the past fifteen years, the ordinary least squares estimator and the corresponding pivotal statistic have been widely used for testing the unit root hypothesis in autoregressive processes. Recently, several new criteriia, based on the maximum likelihood estimators and weighted symmetric estimators, have been proposed. In this article, we describe several different test criteria. Results from a Monte Carlo study that compares the power of the different criteria indicates that the new tests are more powerful against the stationary alternative. Of the procedures studied, the weighted symmetric estimator and the unconditional maximum likelihood estimator provide the most powerful tests against the stationary alternative.As an illustration, we analyze the quarterly change in busine;ss investories.

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AMS subject classifications: 62H99 62E05 15A23 15A52 PACS: 62H10 62E15 15A09 15A52 Keywords: Statistical shape theory Configuration density Elliptically contoured distributions Zonal and invariant polynomials Matrix variate Kotz Pearson Bessel and Jensen-logistic distributions a b s t r a c tThe noncentral configuration density, derived under an elliptical model, generalizes and corrects the Gaussian configuration and some Pearson results. Partition theory is then used to obtain explicit configuration densities associated with matrix variate symmetric Kotz type distributions (including the normal distribution), matrix variate Pearson type VII distributions (including t and Cauchy distributions), the matrix variate symmetric Bessel distribution (including the Laplace distribution) and the matrix variate symmetric Jensenlogistic distribution.

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For a singular random matrix X ; we find the Jacobians associated to the following decompositions: QR; Polar, Singular Value ðSVDÞ; L 0 U; L 0 DM and modified QR (QDR). Similarly, for the cross-product matrix S ¼ X 0 X we find the Jacobians of the Spectral, Cholesky's, L 0 DL and symmetric nonnegative definite square root decompositions. r 2004 Elsevier Inc. All rights reserved.

Assuming that Y has a singular matrix variate elliptically contoured distribution with respect to the Hausdorff measure, the distributions of several matrices associated to QR, modified QR, SV and polar decompositions of matrix Y are determined, for central and non-central, non-singular and singular cases, as well as their relationship to the Wishart and pseudo-Wishart generalized singular and non-singular distributions. Some of these results are also applied to two particular subfamilies of elliptical distributions, the singular matrix variate normal distribution and the singular matrix variate symmetric Pearson type VII distribution.