Jiandong Ren's Discussion on “Size-Biased Risk Measures of Compound Sums,” by Michel Denuit, January 2020

Abstract

2. SIZE-BIASED TRANSFORM FOR DISTRIBUTIONS IN ða,b, 0Þ CLASS The concept of ða, b, 0Þ class distributions is well known to actuaries, mainly because of the popularity of Panjer’s recursive formulas for calculating the distribution of the corresponding compound sums. For detailed introductions and applications, refer to Klugman, Panjer, and Willmot (2019) and Sundt and Vernic (2009). In this section, we present a result for the sizebiased transform of distributions in the class. For completeness, we begin with two definitions. Definition 1. Let PNðkÞ denote the probability function of a discrete random variable N; it is a member of the ða, b, 0Þ class of distributions if there exist constants a and b such that