Abstract
A family of generalised Planck (GP) laws is defined and its structural properties explored. Sometimes subject to parameter restrictions, a GP law is a randomly scaled gamma law; it arises as the equilibrium law of a perturbed version of the Feller mean reverting diffusion; the density functions can be decreasing, unimodal or bimodal; it is infinitely divisible. It is argued that the GP law is not a generalised gamma convolution. Characterisations are obtained in terms of invariance under random contraction of a weighted version of a related law. The GP law is a particular instance of equilibrium laws obtained from a recursion suggested by a genetic mutation-selection balance model. Some related infinitely divisible laws are exhibited.
Highlights
In 1900 Planck derived a formula describing the energy spectrum of black body radiation
Our objective in this paper is to investigate some structural properties of the generalised Planck (GP) laws and point out limits of this investigation
We examine modal behaviour of the GP laws
Summary
In 1900 Planck derived a formula describing the energy spectrum of black body radiation. K is a normalisation factor which is evaluated at (6) below Their motivation seems to be that the case c = 0 yields the gamma family, and that the generalised family gives a more malleable family of pdf ’s. This property opens the possibility of characterising L(Y ) through a relation of the form Y =d V LrY where 0 ≤ V ≤ 1 is a random variable and the factors in the product are independent. There is some duplication of notation between sections, but no confusion will result from this
Sign-in/Register to view highlights & summary 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 callMore From: Journal of Statistical Distributions and Applications
Disclaimer: 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.
