On the choice of the splitting ratio for the split likelihood ratio test

The recently introduced framework of universal inference provides a new approach to constructing hypothesis tests and confidence regions that are valid in finite samples do not rely on any specific regularity assumptions the underlying statistical model. At core methodology is split likelihood ratio statistic, which formed under data splitting compared cleverly selected critical value. As this value can be very conservative, it interesting mitigate potential loss power by careful choice according split. Motivated problem, we study test local alternatives introduce resulting class noncentral chi-square distributions. We investigate properties distributions use numerically examine propose an optimal for composite hypotheses different dimensions.