Abstract
Several characterizations of the Generalized Inverse Gaussian (GIG) distribution on the positive real line have been proposed in the literature, especially over the past two decades. These characterization theorems are surveyed, and two new characterizations are established, one based on maximum likelihood estimation and the other is a Stein characterization.
Highlights
The Generalized Inverse Gaussian distribution on the positive real line has been proposed by Good [14] in his study of population frequencies, yet its first appearance can be traced back to Etienne Halphen in the forties [16], whence the GIG is sometimes called Halphen Type A distribution
A review of known characterizations of GIG distributions is presented in Section 2, while the short Section 3 is devoted to our two new characterizations
The proof is based on the fact that the GIG distribution is characterized by its moments, on a lemma by Kagan et al [20] giving a necessary and sufficient condition for the constancy of E(S|Λ), and on a careful manipulation of a differential equation satisfied by the function f (t) = E(1/X2) exp(itX)
Summary
The Generalized Inverse Gaussian (hereafter GIG) distribution on the positive real line has been proposed by Good [14] in his study of population frequencies, yet its first appearance can be traced back to Etienne Halphen in the forties [16], whence the GIG is sometimes called Halphen Type A distribution. The GIG distributions have been used in the modelization of diverse real phenomena such as, for instance, waiting time (Jørgensen [19]), neural activity (Iyengar and Liao [18]), or, most importantly, hydrologic extreme events (see Chebana et al [6] and references therein). Despite this popularity of GIG models, statistical aspects of the GIG distributions have received much less attention in the literature than their probabilistic counterparts. A review of known characterizations of GIG distributions is presented in Section 2, while the short Section 3 is devoted to our two new characterizations
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