Modeling of ‎I‎nfinite Divisible Distributions Using Invariant and Equivariant Functions

‎Basu’s theorem is one of the most elegant results of classical statistics‎. ‎Succinctly put‎, ‎the theorem says‎: ‎if T is a complete sufficient statistic for a family of probability measures‎, ‎and V is an ancillary statistic‎, ‎then T and V are independent‎. ‎A very novel application of Basu’s theorem appears recently in proving the infinite divisibility of certain statistics‎. ‎In addition to Basu’s theorem‎, ‎this application requires a version of the Goldie-Steutel law‎. ‎By using Basu’s theorem that a large class of functions of random variables‎, ‎two of which are independent standard normal‎, ‎is infinitely divisible‎. ‎The next result provides a representation of functions of normal variables as the product of two random variables‎, ‎where one is infinitely divisible‎, ‎while the other is not‎, ‎and the two are independently distributed‎. 1049