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