The Tail Stein's Identity with Actuarial Applications

Abstract

In this article, we examine a generalized version of an made famous by Stein (1981) who constructed the so-called Stein's Lemma in the special case of a normal distribution. Other works later followed to extend the lemma to the larger class of elliptical distributions, e.g. Landsman (2006) and Landsman and Neslehova (2008). The lemma has had many applications in statistics, finance, insurance and actuarial science. In an attempt to broaden the application of this generalized identity, we consider the version in the case where we investigate only the portion of the distribution of a random variable. Understanding the tails of a distribution is widely important in actuarial science and insurance. Our paper therefore introduces the concept of the tail Stein's identity to the case of any random variable defined on an appropriate probability space with a Lebesque density function satisfying certain regularity conditions. We also examined this tail Stein's identity to the class of discrete distributions. This extended allowed us to develop recursive formulas for generating conditional moments. As examples and illustrations, we consider several classes of distributions including the exponential family, and we apply this result to demonstrate how to generate conditional moments. This has a large promise of applications in risk management.