Likelihood Ratio Computations of Operating Characteristics

The point of this paper is that likelihood ratios, especially in conjunction with suitable symmetry, yield at least partial operating characteristic (OC) information for procedures other than the sequential probability ratio test (SPRT). Unpublished work of W. J. Hall ([12], [13]) apparently contains related observations to this effect. We find it useful to discuss OC computations in terms of mutually 'conjugate' parametric values, which are considered at some length in Section 2; this term appears first in [17], and the concept also elsewhere ([11], [4], [6]), all in the context of the SPRT. In Section 3 the ideas are applied to the Wiener process, for the sorts of two-decision 'wedge' procedures discussed in [15], [8], [1], and [12], and, in Section 4, to modifications of the sorts of three-decision procedures discussed in [2], [20], and [3]. Section 5 contains certain special absorption probabilities for Brownian motion; Section 6 is devoted to the binomial case