Closed sequential tests of an exponential parameter

SUMMARY This paper is concerned with the application of several closed sequential procedures to the testing of an exponential parameter. In particular, the procedures of Armitage (1957), Schneiderman & Armitage (1962a,b) and Anderson (1960) are considered. In certain of these cases either the properties or bounds on the properties of the sequential test can be obtained without approximations. There are certain situations in which the risk of a long sequential experiment is unacceptable. Some common examples are to be found in the area of medical trials and in various types of industrial experimentation. Such situations have brought about the study of alternatives to the sequential probability ratio test (SPRT) in which a procedure's boundaries are closed. One class of such tests is the 'restricted' procedures of Armitage (1957) and the 'wedge' plans of Schneiderman & Armitage (1962a, b). Another test which also belongs to this category is a modification of the SPRT as described by Anderson (1960) and by Donnelly in his unpublished Ph.D. thesis. These tests, which are concerned with normal means, require an approximation of the random walk by a diffusion process for an evaluation of their properties. In this paper we shall apply the above type regions to the testing of the parameter from an exponential distribution and show that in certain cases the properties of the test can be obtained without approximations.