Abstract
Although compositional data arise frequently in many different disciplines, relatively little is known about families of parametric densities that reside on a simplex, which is the natural sample space for such data. An exception to this is a recent paper by Gupta and Richards (1987) wherein the Liouville family is defined and discussed. They sought to characterize dependence properties within this new family using, among other ideas, multivariate total positivity. Further investigation reveals that their results are not very useful. In fact, it is shown that there does not exist a Liouville density, as defined by Gupta and Richards, that is also multivariate totally positive of order 2. Fat 2d flaws associated with an obvious alternative to their definition are discussed.
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References
ALTCHISON J., Inverse distributions and independent gamma-distributed products of random variables, Biometrika, 50 (1963), 505–509.
ALTCHISON, J., The statistical analysis of compositional data, J. Royal Statistical Society B, 44 (1982), 139–177.
ALTCHISON, J., A general class of distributions on the simplex, J. Royal Statistical Society B, 47 (1985), 136–146.
ALTCHISON, J., The Statistical Analysis of Compositional Data. New York: John Wiley (1986).
GUPTA, R.D., and RICHARDS, D. ST. P., Multivariate Liouville distributions, J. Multivariate Analysis, 23 (1987), 233–256.
KARLIN, S., Total Positivity, Stanford: Stanford University Press, Vol. 1 (1968).
KARLIN, S., and Rlnott, Y., Classes of orderings of measures and related correlation inequalities. I. Multivariate totally positive distributions, J. Multivariate Analysis, 10 (1980a), 467–498.
KARLIN, S., and RINOTT, Y., Classes of orderings of measures and related correlation inequalities. II. Multivariate reverse rule distributions, J. Multivariate Analysis, 10 (1980b), 499–516.
RAYENS, W. and SRINIVASAN, C., BOX-COX transformations in the analysis of compositional data, J. Chemometrics, 5 (1991a), 227–240.
RAYENS, W. and Srinivasan, C., Estimation in compositional data analysis, J. Chemometrics 5 (1991b), 361–374.
RAYENS, W. and SRINIVASAN, C., Dependence properties of Generalized Liouville Distributions on the simplex, University of Kentucky Technical Report 321 (1991c).
SLVAZLIAN, B.D., The generalized Dirichlet’s multiple integral, SI AM Review, 11 (1969), 285–288.
SLVAZLIAN, B.D., A class of multivariate distributions, Australian J. Statistics, 23, (1981a), 251–255.
SLVAZLIAN, B.D., On a multivariate extension of the gamma and beta distributions, SIAM J. Applied Mathematics, 41 (1981b), 205–209.
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© 1996 Springer-Verlag New York, Inc.
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Rayens, W.S. (1996). A Note Concerning Multivariate Total Positivity on the Simplex. In: Wheeler, M.F. (eds) Environmental Studies. The IMA Volumes in Mathematics and its Applications, vol 79. Springer, New York, NY. https://doi.org/10.1007/978-1-4613-8492-2_17
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DOI: https://doi.org/10.1007/978-1-4613-8492-2_17
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