ORDER STATISTICS OF MUTTIVARIATE SYMMETRIC DISTRIBUTIONS RELATED TO AN EXPONENTIAL DISTRIBUTION AND THEIR APPLICATIONS

Let \boldsymbolu be uniformly distributed in the region \left\\boldsymbolu=\left(u_1, \cdots, u_n\right): \sum_i=1^n u_i=1, u_n \geqslant 0, i=1, \cdots, n\right\ . We have introduced in 3 a family of multivariate symmetric distributions related to exponential distribution \mathscrF_n=\\boldsymbolz: \boldsymbolz \stackreld= r \boldsymbolu, r \geqslant 0 \text and is independent of \boldsymbolu\ . Let z_(1) \leqslant \cdots \leqslant z_(n) be the order statistics of \boldsymbolz \boldsymbol-\left(z_1, \cdots, z_n\right) \in \mathscrF_n.In this paper, the joint distribution of z_(1), \cdots, z_(n), the marginal distribution of z_(0) and marginal distribution of z_(i) and z_(j), i=1, \cdots, n-1, j=i+1, \cdots, n, are obtained.The distributions of range and midrange of \left\z_(i), i=1, \cdots, n\right\ are also given.The moments of \left\z_(i)\right\ are found and some applications are discussed.