Asymptotic Expansions for a Truncated Version of Lerche's Variation of a Sequential Problem

SYNOPTIC ABSTRACT

Lerche (1986) introduced a variation of the sequential problem of deciding the sign of the drift of a Browninan Motion. He demonstrated that the parabolic boundary, which corresponds to repeated significance tests at a fixed level, is optimal for that variation. However, he raised the question of how the optimal procedure changes when the test is truncated, i.e., stopping is required by some specified time t*.

Bather and Petkau have responded to that challenge by calculating the optimal stopping boundary numerically and by finding inner bounds on the boundary analytically. In this paper asymptotic expansions are derived for the boundary near the time of termination.

These expansions are somewhat unusual in that they represent a double series. In one direction the orders of magnitude of the terms change very slowly. In the other direction they change rapidly.

Key Words and Phrases:

• Sequential Test
• Brownian Motion
• Asymptotic Expansion
• Truncated Test
• Heat Equation