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The relative efficiency of sequential tests

Henri Doerks, Erik Ekström, Yuqiong Wang

Abstract

While many statistical procedures rely on a fixed sample size, sequential methods allow a decision-maker to adapt the sample size to achieve a given precision. In this way, sequential tests reduce the average number of observations required to achieve a given power of the test -- but by how much? To address this question, we focus on the scenario of testing the unknown drift of a Brownian motion, comparing the Wald sequential probability ratio test with tests that use a pre-determined fixed sample size. We provide precise bounds on the average reduction in sample size needed to achieve a desired precision. Specifically, we demonstrate that for symmetric error bounds, the sequential test reduces the average sample size by at least 36\% and by at most 75\%. Moreover, the reduction in sample size increases monotonically with the power of the test, meaning that the relative advantage of using a sequential test over a fixed sample size test grows as higher power is required. We also study the relative efficiency in the case with asymmetric error bounds, and we provide a lower bound in terms of the symmetric case.

The relative efficiency of sequential tests

Abstract

While many statistical procedures rely on a fixed sample size, sequential methods allow a decision-maker to adapt the sample size to achieve a given precision. In this way, sequential tests reduce the average number of observations required to achieve a given power of the test -- but by how much? To address this question, we focus on the scenario of testing the unknown drift of a Brownian motion, comparing the Wald sequential probability ratio test with tests that use a pre-determined fixed sample size. We provide precise bounds on the average reduction in sample size needed to achieve a desired precision. Specifically, we demonstrate that for symmetric error bounds, the sequential test reduces the average sample size by at least 36\% and by at most 75\%. Moreover, the reduction in sample size increases monotonically with the power of the test, meaning that the relative advantage of using a sequential test over a fixed sample size test grows as higher power is required. We also study the relative efficiency in the case with asymmetric error bounds, and we provide a lower bound in terms of the symmetric case.
Paper Structure (7 sections, 6 theorems, 74 equations, 2 figures, 1 table)

This paper contains 7 sections, 6 theorems, 74 equations, 2 figures, 1 table.

Table of Contents

  1. Introduction
  2. Testing an unknown drift of a Brownian motion
  3. Tests with a fixed sample size
  4. Sequential tests
  5. Dependence on the signal-to-noise ratio
  6. Relative efficiency
  7. The asymmetric case

Key Result

Theorem 3.1

The function $f$ is increasing on $(0,\frac{1}{2})$, with $f(0+)=\frac{1}{4}$ and $f(\frac{1}{2}-)=\frac{2}{\pi}$.

Figures (2)

  • Figure 1: The function $f(\alpha)$ for $\alpha\in(0,\frac{1}{2})$ (left), and $f(\alpha)$ for $\alpha\in(0,10^{-8})$ (right).
  • Figure 2: The function $F(\alpha,\beta)$ on $(0,\frac{1}{2})^2$ (left), and $F(\alpha,\beta)$ for $(\alpha,\beta)\in(0,\frac{1}{10})^2$ with logarithmic axes (right).

Theorems & Definitions (15)

  • Theorem 3.1
  • Remark 3.2
  • Remark 3.3
  • Lemma 3.4
  • Remark 3.5
  • proof
  • Proposition 3.6
  • proof
  • Remark 3.7
  • Theorem 4.1
  • ...and 5 more