A method for designing three-hypothesis test problems and sequential schemes

In applications, a two-sided hypothesis test problem sometimes needs to be changed to a three-hypothesis one with the two alternative hypotheses properly selected. In this article, we obtain the hypothesis design and the three-hypothesis sequential test scheme under the Koopman-Darmois distribution by solving a system of equations that meet requirements on the error rates and average sample number. This method provides a useful guide for practitioners to design hypotheses in multihypothesis test problems with controlled error rates and sampling cost. Formulas of the scheme's error rates and average sample number are obtained using numerical quadrature for the discrete-time situation.