A revisit of the distribution of linear combinations of Dirichlet components
Kun-Lin Kuo, Tom Jiang · Communication in Statistics- Theory and Methods · 2017
Provost and Cheong (2000 Provost, S. B., and Y.-H. Cheong. 2000. On the distribution of linear combinations of the components of a Dirichlet random vector. The Canadian Journal of Statistics 28 (2):417–25.[Crossref], [Web of Science ®] , [Google Scholar]) show the importance of the distribution of linear combinations of components of a Dirichlet random vector to quadratic forms and their ratios in statistics, which can be applied in a variety of contexts. The c-characteristic function has been shown to be very useful and more practical in some distributions that are hard to manage with the traditional characteristic functions. The importance of the distribution of linear combinations of components of a Dirichlet random vector to quadratic forms and their ratios in statistics, which can be applied in a variety of contexts, is well known. We first provide its inversion formula which is practical in determining the distribution function of a random variable when its c-characteristic function is known. We then use this inversion formula to find an expression of probability density function of linear combinations of components of any Dirichlet vector. This would generalize the currently well known results.