Performance analysis of Wald's SPRT with independent but non-stationary log-likelihood ratios

TLDR: Two inductive equations governing the OC and ASN are developed and they have non-unique solutions in the general case, but do have unique solutions in two special cases: (a) the log-likelihood ratios converge in distributions and (b) the logs have periodic distributions.

Abstract: The characteristics and behavior of Wald's sequential probability ratio test are revealed by two important functions — operating characteristic (OC) and average sample number (ASN). These two functions have been studied extensively under the assumption of independent and identically distributed (i.i.d.) log-likelihood ratios, which is too stringent for many applications. This paper relaxes the requirement of identical distribution. Two inductive equations governing the OC and ASN are developed. Unfortunately, they have non-unique solutions in the general case. They do have unique solutions in two special cases: (a) the log-likelihood ratios converge in distributions and (b) the log-likelihood ratios have periodic distributions. Numerical solutions for these two special cases are obtained. They are compared with the results of Monte Carlo simulations because existing methods for this problem setting are lacking.