On the infinite divisibility of inverse Beta distributions
نویسندگان
چکیده
We show that all negative powers β a,b of the Beta distribution are infinitely divisible. The case b ≤ 1 follows by complete monotonicity, the case b > 1, s ≥ 1 by hyperbolically complete monotonicity and the case b > 1, s < 1 by a Lévy perpetuity argument involving the hypergeometric series. We also observe that β a,b is self-decomposable if and only if 2a + b + s + bs ≥ 1, and that in this case it is not necessarily a generalized Gamma convolution. On the other hand, we prove that all negative powers of the Gamma distribution are generalized Gamma convolutions, answering to a recent question of L. Bondesson.
منابع مشابه
Free infinite divisibility for beta distributions and related ones ∗
We prove that many of beta, beta prime, gamma, inverse gamma, Student tand ultraspherical distributions are freely infinitely divisible, but some of them are not. The latter negative result follows from a local property of probability density functions. Moreover, we show that the Gaussian, many of ultraspherical and Student t-distributions have free divisibility indicator 1.
متن کاملModeling 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 ...
متن کاملOn Free and Classical Type G Distributions
There is a one-to-one correspondence between classical one-dimensional infinitely divisible distributions and free infinitely divisible distributions. In this work we study the free infinitely divisible distributions corresponding to the one-dimensional type distributions. A new characterization of classical type distributions is given first and the class of type classical infinitely divi...
متن کاملInfinitely Divisible Distributions for Rectangular Free Convolution: Classification and Matricial Interpretation
In a previous paper ([B-G1]), we defined the rectangular free convolution ⊞ λ . Here, we investigate the related notion of infinite divisibility, which happens to be closely related the classical infinite divisibility: there exists a bijection between the set of classical symmetric infinitely divisible distributions and the set of ⊞ λ -infinitely divisible distributions, which preserves limit t...
متن کاملOn the Non-classical Infinite Divisibility of Power Semicircle Distributions
The family of power semicircle distributions defined as normalized real powers of the semicircle density is considered. The marginals of uniform distributions on spheres in high-dimensional Euclidean spaces are included in this family and a boundary case is the classical Gaussian distribution. A review of some results including a genesis and the so-called Poincaré’s theorem is presented. The mo...
متن کامل