Unified Solutions of Integral Equations of SPRT for Exponential Random Variables

Abstract

The main characteristics of the sequential probability ratio test (SPRT) are described by acceptance probability (AP) and average sample number (ASN). The properties of the cumulative sum control procedure, CUSUM in short, which is a variant of the SPRT, are mainly determined by the average run length (ARL). The characteristics are obtained by solving Fredholm type integral equations. For the sake of mathematical simplicity, the models for the exponential random variables and the sum of the exponential random variables have been considered to test the effects of the design parameters of the SPRT and CUSUM. The solution of the integral equation for each case obtained by various methods in previous studies. In this article, we show that all the integral equations related to the characteristics of AP, ASN and ARL for the models of the exponential random variable and the sum of exponential random variables are defined in a framework, and their solutions are obtained precisely in a unified method. The suggested solutions for exponential distribution do not require any other numerical steps to obtain the unknown constants used in the solutions.

Key Words:

• Sequential sampling
• Average sample number
• Average run length

Acknowledgments

The author thanks Professor Vardeman of Iowa State University for kind discussions about this topic, and the anonymous referee for his/her insightful comments.