Finite-sample bounds to the normal limit under group sequential sampling

TL;DR

This paper provides finite-sample, non-asymptotic bounds on how close the joint MLEs in group sequential analysis are to a normal distribution, accommodating heterogeneity and extending to various metrics.

Contribution

It introduces optimal order bounds using Stein's method for group sequential MLEs, including non-i.i.d. data and exponential family cases, extending existing asymptotic results.

Findings

Derived non-asymptotic bounds for MLEs in group sequential analysis.

Extended bounds to heterogeneous data and exponential family models.

Connected multivariate Kolmogorov distance to smooth function distance.

Abstract

In group sequential analysis, data is collected and analyzed in batches until pre-defined stopping criteria are met. Inference in the parametric setup typically relies on the limiting asymptotic multivariate normality of the repeatedly computed maximum likelihood estimators (MLEs), a result first rigorously proved by Jennison and Turbull (1997) under general regularity conditions. In this work, using Stein's method we provide optimal order, non-asymptotic bounds on the distance for smooth test functions between the joint group sequential MLEs and the appropriate normal distribution under the same conditions. Our results assume independent observations but allow heterogeneous (i.e., non-identically distributed) data. We examine how the resulting bounds simplify when the data comes from an exponential family. Finally, we present a general result relating multivariate Kolmogorov distance to smooth function distance which, in addition to extending our results to the former metric, may be of independent interest.