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

Abstract

The recently introduced framework of universal inference provides a new approach to constructing hypothesis tests and confidence regions that are valid in finite samples and do not rely on any specific regularity 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 noncentral 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.

Finite sample inference for empirical Bayesian methods

“…The lack of power due to the looseness of Markov's inequality was first mentioned and discussed in Wasserman et al (2020), where it is also pointed out that, in the universal inference setting, the logarithm of the analogous ratio statistics to (6) have tail probabilities that scale, in 𝛼, like those of 𝜒 2 statistics. The conservativeness of universal inference constructions is further discussed in the works of Dunn et al (2021), Tse andDavison (2022), andStrieder andDrton (2022), where the topic is thoroughly explored via simulations and theoretical results regarding some classes of sufficiently regular problems. We observe this phenomenon in the comparisons in Sections 3.1 (and further expanded in the Data S1).…”