A New Transformation and its Properties and Applications

Abstract

Motivated by a representation of starlike function, Jiang (1988, AIAM MaAn) first gave a new kind of univariate transformation and its convergence theorem. Jiang, Dickey, and Kuo (2004, StocProc) extend it to the multivariate transformation. This transformation is particularly useful for distributions that are difficult to deal with by Fourier transformation. The new multivariate transformation is shown to have many properties, e.g., uniqueness and convergence, which are similar to those of the Fourier transformation. In addition, it has a closed form for the filtered-variate Dirichlet distribution, which is important on the histogram smoothing problems (see, e.g., Dickey and Jiang, 1998, JASA). With this new multivariate transformation, we are also able to find the distribution, on the region bounded by an ellipse, of a random functional of a Ferguson-Dirichlet process over the boundary. This result generalized that given by Jiang (1991, StPrLet). Other new interesting properties and applications shall also be given and addressed.