Ben Berckmoes

On the sample mean after a group sequential trial

A popular setting in medical statistics is a group sequential trial with independent and identically distributed normal outcomes, in which interim analyses of the sum of the outcomes are performed. Based on a prescribed stopping rule, one decides after each interim analysis whether the trial is stopped or continued. Consequently, the actual length of the study is a random variable. It is reported in the literature that the interim analyses may cause bias if one uses the ordinary sample mean to estimate the location parameter. For a generic stopping rule, which contains many classical stopping rules as a special case, explicit formulas for the expected length of the trial, the bias, and the mean squared error (MSE) are provided. It is deduced that, for a fixed number of interim analyses, the bias and the MSE converge to zero if the first interim analysis is performed not too early. In addition, optimal rates for this convergence are provided. Furthermore, under a regularity condition, asymptotic normality in total variation distance for the sample mean is established. A conclusion for naive confidence intervals based on the sample mean is derived. It is also shown how the developed theory naturally fits in the broader framework of likelihood theory in a group sequential trial setting. A simulation study underpins the theoretical findings.

An Approach Theoretic Version of Anscombe’s Theorem with an Application in Biostatistics

We establish an approach theoretic version of Anscombe’s theorem, which we apply to justify the use of confidence intervals based on the sample mean after a group sequential trial.

A Note on Probability Metrics in a Categorical Setting

Probability metrics constitute an important tool in probability theory and statistics (Dall’Aglio et al. 1991; Rachev 1991; Zolotarev, Teor. Veroyatnost. i Primenen. 28(2), 264–6, 1983) as they are specific metrics on spaces of random variables which, by satisfying an extra condition, concord well with the randomness structure. But probability metrics suffer from the same instability under constructions as metrics. In Lowen (2015), as well as in former and related work which can be found in the references of Lowen (2015), a comprehensive setting was developed to deal with this. It is the purpose of this note to point out that these ideas can also be applied to probability metrics thus embedding them in a natural categorical framework, showing that certain constructions performed in the setting of probability theory are in fact categorical in nature. This allows us to deduce various separate results in the literature from a unified approach.