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.
View on arXiv