# Asymptotic properties of maximum likelihood estimators with sample size recalculation

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## Abstract

Consider an experiment in which the primary objective is to determine the significance of a treatment effect at a predetermined type I error and statistical power. Assume that the sample size required to maintain these type I error and power will be re-estimated at an interim analysis. A secondary objective is to estimate the treatment effect. Our main finding is that the asymptotic distributions of standardized statistics are random mixtures of distributions, which are non-normal except under certain model choices for sample size re-estimation (SSR). Monte-Carlo simulation studies and an illustrative example highlight the fact that asymptotic distributions of estimators with SSR may differ from the asymptotic distribution of the same estimators without SSR.

## Keywords

Adaptive designs Asymptotic distribution theory Interim analysis Local alternatives Maximum likelihood estimation Mixture distributions## Mathematics Subject Classification

62K99 62L05 62F05 62E20## Notes

### Acknowledgements

We thank Dr. Assaf P. Oron, an unknown referee and the editor for their very valuable feedback on our manuscript.

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