Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Power one sequential tests exist for weakly compact $\mathscr P$ against $\mathscr P^c$

Ashwin Ram, Aaditya Ramdas

Abstract

Suppose we observe data from a distribution $P$ and we wish to test the composite null hypothesis that $P\in\mathscr P$ against a composite alternative $P\in \mathscr Q\subseteq \mathscr P^c$. Herbert Robbins and coauthors pointed out around 1970 that, while no batch test can have a level $α\in(0,1)$ and power equal to one, sequential tests can be constructed with this fantastic property. Since then, and especially in the last decade, a plethora of sequential tests have been developed for a wide variety of settings. However, the literature has not yet provided a clean and general answer as to when such power-one sequential tests exist. This paper provides a remarkably general sufficient condition (that we also prove is not necessary). Focusing on i.i.d. laws in Polish spaces without any further restriction, we show that there exists a level-$α$ sequential test for any weakly compact $\mathscr P$, that is power-one against $\mathscr P^c$ (or any subset thereof). We show how to aggregate such tests into an $e$-process for $\mathscr P$ that increases to infinity under $\mathscr P^c$. We conclude by building an $e$-process that is asymptotically relatively growth rate optimal against $\mathscr P^c$, an extremely powerful result.

Power one sequential tests exist for weakly compact $\mathscr P$ against $\mathscr P^c$

Abstract

Suppose we observe data from a distribution and we wish to test the composite null hypothesis that against a composite alternative . Herbert Robbins and coauthors pointed out around 1970 that, while no batch test can have a level and power equal to one, sequential tests can be constructed with this fantastic property. Since then, and especially in the last decade, a plethora of sequential tests have been developed for a wide variety of settings. However, the literature has not yet provided a clean and general answer as to when such power-one sequential tests exist. This paper provides a remarkably general sufficient condition (that we also prove is not necessary). Focusing on i.i.d. laws in Polish spaces without any further restriction, we show that there exists a level- sequential test for any weakly compact , that is power-one against (or any subset thereof). We show how to aggregate such tests into an -process for that increases to infinity under . We conclude by building an -process that is asymptotically relatively growth rate optimal against , an extremely powerful result.

Paper Structure

This paper contains 8 sections, 12 theorems, 55 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Power-One Sequential Tests Against Simple Alternatives
  4. Power-One Sequential Tests against Composite $\mathcal{Q}$
  5. Weak lower semicontinuity not necessary for power-one testing
  6. Aggregating tests to a consistent e-process
  7. REGROW $e$-process against $\mathcal{P}^c$
  8. Conclusion

Key Result

Lemma 1

The map $(M, N)\mapsto\mathrm{KL}(M\|N)$ from $\mathcal{M}_1(\mathsf{X})\times\mathcal{M}_1(\mathsf{X})$ to $[0,\infty]$ is lower semicontinuous for the weak topology on both coordinates. $\blacktriangleleft$$\blacktriangleleft$

Theorems & Definitions (26)

  • Definition 1
  • Definition 2
  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Lemma 3
  • proof
  • Definition 3
  • Proposition 1
  • ...and 16 more