Power laws and power laws with exponential cut-off are two distinct families of distributions on the positive real half-line. In the present paper, we propose a unified treatment of both families by building a family of distributions that interpolates between them, which we call Interpolating Family (IF) of distributions. Our original construction, which relies on techniques from statistical physics, provides a connection for hitherto unrelated distributions like the Pareto and Weibull distributions, and sheds new light on them. The IF also contains several distributions that are neither of power law nor of power law with exponential cut-off type. We calculate quantile-based properties, moments and modes for the IF. This allows us to review known properties of famous distributions on $\mathbb{R}^+$ and to provide in a single sweep these characteristics for various less known (and new) special cases of our Interpolating Family.
翻译:具有指数截断作用的权力法和权力法是两个在正正正正正正正中线上分配的不同家庭。在本文件中,我们建议对两个家庭实行统一处理,办法是建立一个在两个家庭之间相互交织的分布式家庭,我们称之为分配式家庭(IF),我们称之为分配式家庭(IF),我们最初的建筑依靠统计物理学的技术,为Pareto和Weibull的分布式等迄今无关的分布式提供连接,并为它们提供了新的线索。《IF》还包含一些既非权力法也非指数截断型权力法的分配式的分布式家庭。我们计算了基于量基特性、时间和模式的IF,从而使我们能够审查在$\mathb{R ⁇ $上著名的分配式家庭的已知特性,并一次性地列出这些特性,供我们内插家庭各种不为人所知的(和新的)特殊案例使用。