Truncated sequential hypothesis tests

Any views or conclusions contained in this Memorandum should not be interpreted as representing the official opinion or policy of ARPA. DDC AVAILABILITY NOTICE Qualified requesters may obtain copies of this report from the Defense Documentation Center (DDC). 1700 MAIN Si ~SANTA MONICA • CAlifORNIA • • o. t t h-iii PREFACE In the operation of phased-array radars it is possible, in principle, to save power without sacrificing performance by not deciding in advance how long an observation should be made, e.g., through sequential testing. In practice, some limit must be put on the freedom of the designer to choose observation times-hence the test is truncated. This Memorandum describes and evaluates a class of such tests. The work was undertaken as basic research in technology applicable to the design of electronically scanned radars of potential use in ballistic missile defenses. It is part of a continuing study for ARPA on low-altitude defense against ballistic missiles. • \/ v SUMMARY Through a careful examination of the equations by which Wald determines the values of the boundaries for tests of sequential hypotheses, we are able to obtain interesting relationships for the conditional probability distributions of the stage at which the test terminates. These enable us to study sequential tests in which the boundaries are functions of the sample number. We are particularly concerned with tests with convergent boundaries, and investigate a set of boundaries which approach those of a truncated Wald test. Approximate expressions for the expected sample number and probabilities of error are obtained for the tests considered. The obtained approximations apply best for gently tapering slopes. Extensions of our method can be applied to evaluate various ad hoc schemes for truncating tests and to the theory of tests with a varying parameter. vii ACKNOWLEDGMENT We wish to thank Ivan Selin for the technical review of this Memorandum and for several helpful suggestions.