Vertical Density Representation And Center Similar Distribution

The concept of Vertical Density Representation (VDR) was first proposed by Troutt in 1991. With the development of the research work about VDR, the concept of Multivariate Vertical Density Representation (MVDR) and type 2 VDR were presented, while the Troutt's VDR proposed before was called as type 1 VDR. In this paper, we use the main results of type 2 VDR to analyze the spherical symmetric distribution. We obtain the two equivalent representations of the spherical symmetric distribution. They are both a product of a positive random variable and a random vector with uniform distribution, and the domain of uniform distribution are both related with sphere. With this conclusion, we deduce that when the random vector Xn comes from spherical symmetric distribution, the student statistic Tn defined by Xn has student distribution with n ?1 degree of freedom. Based on the thought of extending the domain on which the random vector Xn has uniform distribution to other sets, we construct the concept of Center Similar Distribution (CSD), and give some related theorems. Also we get two representations of CSD. Then using one of these representations, we work out the p.d.f of the student statistic Tn defined by some random vectors Xn come from CSD.