Abstract
LetE be the class ofpdf'sf on (0,∞) such that, for eachu>0,f(uv) f (u/v) is completely monotone as a function ofw=v+v −1. This class includes many familiarpdf's and is closed with respect to multiplication and division of independentrv's. Further,E ⊂T, whereT is the class of generalized Gamma convolutions (GGC) introduced by O. Thorin. Moreover,E coincides with the class ofpdf's of the form $$C \cdot x^{\beta - 1} \cdot \mathop \Pi \limits_{i = 1}^N (1 + c_i x)^{y_i }$$ (all parameters positive) or limits thereof. The Laplace transform ϕ of aGGC is characterized by complete monotonicity of ϕ(uv) ϕ(u/v) as a function ofw. This characterization has many consequences and applications. It follows that also the classT has simple multiplicative properties.
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
