AN IMPROVEMENT TO WALD'S APPROXIMATION FOR SOME PROPERTIES OF SEQUENTIAL TESTS

SUMMARY A SIMPLE modification of the Wald approximations to the operating characteristic and average sample number of a sequential test is given which provides estimates of their values for starting points of the test near the boundaries and an improved approximation for general starting points. In his book Sequential Analysis Wald (1947)gives an approximate formula for the operating characteristic of a sequential test or, equivalently, for the probability that a particle performing a linear random walk between two absorbing barriers is absorbed by a specified barrier. This formula is valid for starting points of the test not near either boundary and for mean paths inclined at not more than a small angle to the boundaries so that the overlap of the boundary at the end of the test is negligible. In this note we give a simple modification of Wald's formula which provides a better approximation and gives some information about values at the boundaries. Suppose that a Wald sequential test is to be carried out on a population for which the scores, xi, assigned to the observations are independent variates from a population with a continuous frequency functionf (x). Let the boundaries of the test be a, b (a 1). With this scoring system the boundaries in the Wald inspection diagram (Wald, 1947, p. 120) are the horizontal lines with ordinates a, b; in terms of the linear random walk, a, b are the positions of the two point boundaries. By considering expectations conditional upon the first observation, we find that the probability that the test ends at the lower boundary, P (Z), satisfies the equation (e.g., Kemperman, 1950) b