## Bernoulli

• Bernoulli
• Volume 21, Number 2 (2015), 881-908.

### On the almost sure convergence of adaptive allocation procedures

#### Abstract

In this paper, we provide some general convergence results for adaptive designs for treatment comparison, both in the absence and presence of covariates. In particular, we demonstrate the almost sure convergence of the treatment allocation proportion for a vast class of adaptive procedures, also including designs that have not been formally investigated but mainly explored through simulations, such as Atkinson’s optimum biased coin design, Pocock and Simon’s minimization method and some of its generalizations. Even if the large majority of the proposals in the literature rely on continuous allocation rules, our results allow to prove via a unique mathematical framework the convergence of adaptive allocation methods based on both continuous and discontinuous randomization functions. Although several examples of earlier works are included in order to enhance the applicability, our approach provides substantial insight for future suggestions, especially in the absence of a prefixed target and for designs characterized by sequences of allocation rules.

#### Article information

Source
Bernoulli, Volume 21, Number 2 (2015), 881-908.

Dates
First available in Project Euclid: 21 April 2015

Permanent link to this document
https://projecteuclid.org/euclid.bj/1429624964

Digital Object Identifier
doi:10.3150/13-BEJ591

Mathematical Reviews number (MathSciNet)
MR3338650

Zentralblatt MATH identifier
1320.62189

#### Citation

Baldi Antognini, Alessandro; Zagoraiou, Maroussa. On the almost sure convergence of adaptive allocation procedures. Bernoulli 21 (2015), no. 2, 881--908. doi:10.3150/13-BEJ591. https://projecteuclid.org/euclid.bj/1429624964

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