Approximate bias calculations for sequentially designed experiments

A linear model is considered in which the design variables may be functions of previous responses and/or auxiliary randomisation. The model is observed successive times, where t is a stopping time, and interest lies in estimating the parameters of the model. Approximations are derived for the bias and variance of the maximum likelihood estimators of the parameters at time t. The derivations involve differentiating the fundamental identity of sequential analysis. The accuracy of the approximations is assessed by simulation for a multi-armed clinical trial model proposed by Coad (1995), two autoregressive models and the sequential design of Ford and Silvey (1980). Very weak expansions are used to justify the approximations.