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

We develop a nonparametric extension of the sequential generalized likelihood ratio (GLR) test and corresponding confidence sequences. By utilizing a geometrical interpretation of the GLR statistic, we present a simple upper bound on the probability that it exceeds any prespecified boundary function. Using time-uniform boundary-crossing inequalities, we carry out a nonasymptotic analysis of the sample complexity of one-sided and open-ended tests over nonparametric classes of distributions, including sub-Gaussian, sub-exponential, and exponential family distributions in a unified way. We propose a flexible and practical method to construct confidence sequences for the corresponding problem that are easily tunable to be uniformly close to the pointwise Chernoff bound over any target time interval.