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.

翻译：群序贯分析中，数据分批收集分析直至满足预设停止准则。参数推断通常依赖于重复计算的最大似然估计量（MLEs）的渐近多元正态性，这一结果最早由Jennison和Trumbull（1997）在一般正则条件下严格证明。本文利用Stein方法，在相同条件下给出联合群序贯MLEs与相应正态分布之间光滑检验函数距离的最优阶非渐近界。我们的结果假设观测独立但允许异质性（即非同分布）数据。我们分析了当数据来自指数族时这些界的简化形式。最后，我们给出了多元Kolmogorov距离与光滑函数距离之间的一般关系结果，这不仅使我们的结果扩展到前一种度量，也可能具有独立研究价值。