Nonparametric iterated-logarithm extensions of the sequential generalized likelihood ratio test

TL;DR

This paper introduces a nonparametric extension of the sequential GLR test with time-uniform confidence sequences, providing a unified analysis and practical construction methods for various distribution classes.

Contribution

It develops a simple analytic upper bound for the nonparametric GLR statistic, enabling time-uniform boundary crossing analysis and confidence sequence construction.

Findings

Derived a geometric interpretation of the GLR statistic.

Provided nonasymptotic bounds on expected sample sizes.

Constructed tunable, time-uniform confidence sequences.

Abstract

We develop a nonparametric extension of the sequential generalized likelihood ratio (GLR) test and corresponding time-uniform confidence sequences for the mean of a univariate distribution. By utilizing a geometric interpretation of the GLR statistic, we derive a simple analytic upper bound on the probability that it exceeds any prespecified boundary; these are intractable to approximate via simulations due to infinite horizon of the tests and the composite nonparametric nulls under consideration. Using time-uniform boundary-crossing inequalities, we carry out a unified nonasymptotic analysis of expected sample sizes of one-sided and open-ended tests over nonparametric classes of distributions (including sub-Gaussian, sub-exponential, sub-gamma, and exponential families). Finally, we present a flexible and practical method to construct time-uniform confidence sequences that are easily tunable to be uniformly close to the pointwise Chernoff bound over any target time interval.