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

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