Modified Wald Formulation for Sequential Binary Hypothesis Testing in Statistically Periodic Processes

For the classical problem of sequential binary hypothesis testing (SHT) studied by Wald for independent and identically distributed (i.i.d.) data, the optimal test is the sequential probability ratio test (SPRT). When the data is no longer i.i.d., the SPRT's exact or strong optimality property is lost. However, the SPRT is asymptotically optimal for general non-i.i.d. data under mild conditions, as the error probabilities go to zero. In this paper, the problem of SHT is studied for a special class of non-i.i.d. processes, and an exact optimal solution is obtained. This special class is the class of statistically periodic processes encountered in many applications in science and engineering. The objective for SHT chosen in the paper is an average risk criterion containing time-varying penalties for collecting samples and making decision errors. It is shown that the optimal solution is the SPRT with a time-varying sequence of thresholds. It is further shown that a constant threshold variant is asymptotically optimal. The constant threshold test is then applied to electrocardiogram (ECG) data to perform energy-efficient detection of heart arrhythmia.