Operating characteristic and average sample number functions of truncated sequential probability ratio test

The operating characteristic (OC) and average sample number (ASN) characterize the performance of the sequential probability ratio test (SPRT). In this paper, the ASN and OC of the truncated SPRT (TSPRT) are examined in a general setting, in which the log-likelihood ratios are not necessarily identically distributed and the bounds for the test can be time varying. Two inductive integral equations are derived for the OC and ASN respectively, which can be solved analytically by backward induction provided the convolution involved can be evaluated analytically. The initial value for backward induction can be readily determined by TSPRT's truncation strategy. Due to the difficulty of analytically evaluating the convolution in the induction process, numerical methods may be necessary. Numerical algorithms based on the system of linear algebraic equation method and the finite element analysis are developed. Examples are provided to illustrate our algorithms.