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：在小组顺序分析中,在达到预先界定的停止标准之前,将分批收集和分析数据。参数设置中的推论通常依赖于反复计算的最大概率估计器(MLEs)的有限无症状性多变常态,这是Jennison和Turbull(1997年)在一般正常条件下首次严格证明的结果。在这项工作中,利用Stein的方法,我们提供了最佳的顺序,对联合组相继 MLE 和同一条件下适当正常分布之间顺利测试功能的距离的非无症状界限。我们的结果假定独立观察,但允许差异性(即非身份分布的)数据。我们研究了当数据来自指数式组时所产生的界限是如何简化的。最后,我们提出了一个将多变量 Kolmogorov 距离与平稳功能距离相联系的一般结果,除了将我们的结果扩大到以前的指标外,还可能具有独立的兴趣。