Brownian Approximations and Truncated Tests

Abstract

The central role of the normal distribution in statistics arises because of its simplicity and usefulness as an approximation to other probability distributions. In sequential analysis considerable additional simplification results from approximating sums of independent random variables x _l + · · · + x _n in discrete time by a Brownian motion process W(t), 0 ≤ t < ∞ in continuous time. Although this approximation is rarely quantitatively adequate (cf. Section 5), its comparative simplicity leads to appreciable qualitative insight; and quantitatively it does provide a first, crude approximation which can often be used as a basis for subsequent refinement. This chapter is concerned primarily with sequential tests for the mean of a Brownian motion process. Various truncated modifications of the sequential probability ratio test will be introduced, and problems of estimates and attained significance levels relative to sequential tests will be discussed.