Inference via Randomized Test Statistics

TL;DR

This paper introduces a method using external randomization to improve the convergence of test statistics to their limiting distributions, leading to sharper statistical inference.

Contribution

It proposes a novel approach based on a central limit theorem for weighted sums, demonstrating improved convergence rates for rank-based and phi-divergence test statistics.

Findings

Randomization enforces convergence to chi-square distribution with high probability.

Convergence rate is improved to approximately O(1/n) with logarithmic factors.

Applicable to a family of rank-based and phi-divergence test statistics.

Abstract

We show that external randomization may enforce the convergence of test statistics to their limiting distributions in particular cases. This results in a sharper inference. Our approach is based on a central limit theorem for weighted sums. We apply our method to a family of rank-based test statistics and a family of phi-divergence test statistics and prove that, with overwhelming probability with respect to the external randomization, the randomized statistics converge at the rate $O(1/n)$ (up to some logarithmic factors) to the limiting chi-square distribution in Kolmogorov metric.