Bounds on Negative Binomial Approximation to Call Function

TL;DR

This paper develops Stein's method for negative binomial approximation using call functions, providing error bounds and demonstrating applications to CDOs, advancing probabilistic approximation techniques.

Contribution

It introduces a novel Stein's method approach for negative binomial distribution with call functions and derives explicit error bounds for sums of dependent variables.

Findings

Derived explicit error bounds for negative binomial approximation

Applied bounds to collateralized debt obligations (CDO)

Compared new bounds with existing approximation bounds

Abstract

In this paper, we develop Stein's method for negative binomial distribution using call function defined by $f_z(k)=(k-z)^+=\max\{k-z,0\}$, for $k\ge 0$ and $z \ge 0$. We obtain error bounds between $\mathbb{E}[f_z(\text{N}_{r,p})]$ and $\mathbb{E}[f_z(V)]$, where $\text{N}_{r,p}$ follows negative binomial distribution and $V$ is the sum of locally dependent random variables, using certain conditions on moments. We demonstrate our results through an interesting application, namely, collateralized debt obligation (CDO), and compare the bounds with the existing bounds.