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

TL;DR

This paper investigates how to optimally choose the data splitting ratio in the split likelihood ratio test within the universal inference framework, aiming to improve test power while maintaining finite-sample validity.

Contribution

It introduces the noncentral split chi-square distribution to analyze the split likelihood ratio test under local alternatives and proposes an optimal data splitting ratio.

Findings

Proposes a new class of noncentral split chi-square distributions.

Provides numerical methods to select the optimal data splitting ratio.

Enhances the power of split likelihood ratio tests without sacrificing finite-sample validity.

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.