Berry–Esseen bound for the spacings-based test

Derive a Berry–Esseen bound for the goodness-of-fit test statistic constructed with the spacings-based design-density-free estimator, thereby extending the Gaussian approximation theory from the known-density test to the spacings version.

Background

The paper proves a non-asymptotic Gaussian approximation for the test statistic when the visit-time density $g$ is known. It also proposes a spacings-based version that avoids using $g$, and numerical experiments suggest that this feasible test performs at least as well as the known-density procedure.

The theoretical distributional approximation for the spacings-based statistic is not established. The authors explicitly conjecture that an analogous Berry–Esseen result should hold, leaving open the derivation of such a bound.