Default probability distributions are often defined in terms of their conditional default probability distribution, or their hazard rate. By their definition, they imply a unique probability density function. The applications of default probability distributions are varied, including the risk premium model used to price default bonds, reliability measurement models, insurance, etc. Fractional probability density functions (FPD), however, are not in general conventional probability density functions (Tapiero and Vallois, Physica A,. Stat. Mech. Appl. 462:1161–1177, 2016). As a result, a fractional FPD does not define a fractional hazard rate. However, a fractional hazard rate implies a unique and conventional FPD. For example, an exponential distribution fractional hazard rate implies a Weibull probability density function while, a fractional exponential probability distribution is not a conventional distribution and therefore does not define a fractional hazard rate. The purpose of this paper consists of defining fractional hazard rates implied fractional distributions and to highlight their usefulness to granular default risk distributions. Applications of such an approach are varied. For example, pricing default bonds, pricing complex insurance contracts, as well as complex network risks of various granularity, that have well defined and quantitative definitions of their hazard rates.

中文翻译：

---

隐含风险分数和违约风险分布

违约概率分布通常根据其有条件的违约概率分布或其危害率来定义。根据它们的定义，它们暗示着唯一的概率密度函数。违约概率分布的应用是多种多样的，包括用于对违约债券定价的风险溢价模型，可靠性度量模型，保险等。然而，分数概率密度函数（FPD）在常规常规概率密度函数中并不常见（Tapiero和Vallois ，Physica A，Stat。Mech。Appl。462：1161-1177，2016）。结果，FPD分数未定义危险分数。但是，部分危险率意味着独特而传统的FPD。例如，指数分布分数危险率表示威布尔概率密度函数，而 分数指数概率分布不是常规分布，因此未定义分数危险率。本文的目的包括定义隐含分数分布的分数危险率，并强调其对精细违约风险分布的有用性。这种方法的应用是多种多样的。例如，对违约债券进行定价，对复杂的保险合同进行定价以及各种粒度的复杂网络风险，这些风险已得到很好的定义和定量定义。本文的目的包括定义隐含分数分布的分数危险率，并强调其对精细违约风险分布的有用性。这种方法的应用是多种多样的。例如，对违约债券进行定价，对复杂的保险合同进行定价以及各种粒度的复杂网络风险，这些风险已得到很好的定义和定量定义。本文的目的包括定义隐含分数分布的分数危险率，并强调其对精细违约风险分布的有用性。这种方法的应用是多种多样的。例如，对违约债券进行定价，对复杂的保险合同进行定价以及各种粒度的复杂网络风险，这些风险已得到很好的定义和定量定义。