Properties of Estimators in Exponential Family Settings with Observationbased Stopping Rules
- *Corresponding Author:
- Geert Molenberghs
I-BioStat, Universiteit Hasselt
Campus Diepenbeek, Agoralaan Building D
BE 3590 Diepenbeek Diepenbeek, Belgium
E-mail: [email protected]
Received Date: January 13, 2016 Accepted Date: January 18, 2016; Published Date: January 25, 2016
Citation: Milanzi E, Molenberghs G, Alonso A, Michael GK, Verbeke G, et al. (2016) Properties of Estimators in Exponential Family Settings with Observationbased Stopping Rules. J Biom Biostat 7:272. doi:10.4172/2155-6180.1000272
Copyright: © 2016 Milanzi E, et al. This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.
Often, sample size is not fixed by design. A key example is a sequential trial with a stopping rule, where stopping is based on what has been observed at an interim look. While such designs are used for time and cost efficiency, and hypothesis testing theory has been well developed, estimation following a sequential trial is a challenging, still controversial problem. Progress has been made in the literature, predominantly for normal outcomes and/or for a deterministic stopping rule. Here, we place these settings in a broader context of outcomes following an exponential family distribution and, with a stochastic stopping rule that includes a deterministic rule and completely random sample size as special cases. It is shown that the estimation problem is usually simpler than often thought. In particular, it is established that the ordinary sample average is a very sensible choice, contrary to commonly encountered statements. We study (1) The so-called incompleteness property of the sufficient statistics, (2) A general class of linear estimators, and (3) Joint and conditional likelihood estimation. Apart from the general exponential family setting, normal and binary outcomes are considered as key examples. While our results hold for a general number of looks, for ease of exposition, we focus on the simple yet generic setting of two possible sample sizes, N=n or N=2n.