Transformation of the Fundamental Relationships in Sequential Analysis

Abstract

For the class of distribution functions given by $$dP(x, \theta) = \exp \lbrack r(\theta)A(x) + s(\theta)B(x)\rbrack dw(x),$$ it is shown that a set of three transformations can be introduced which completely define the Sequential Probability Ratio Test for testing a hypothesis $H_0$ against $H_1$. When the observer specifies the threshold parameters $\theta_0$ and $\theta_1$ corresponding to the hypotheses $H_0$ and $H_1$ and the strength $\alpha, \beta$ of the test, he specifies the three transformations and hence the Sequential Test. However, there is an infinity of sets of parameter points $(\theta_0, \theta_1, \alpha, \beta)$ which satisfy the same transformations and hence define the same Sequential Test. The Operating Characteristic Function and the Average Sample Number Function are derived in terms of these transformations.